diff --git "a/7tE4T4oBgHgl3EQfcwyw/content/tmp_files/load_file.txt" "b/7tE4T4oBgHgl3EQfcwyw/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/7tE4T4oBgHgl3EQfcwyw/content/tmp_files/load_file.txt" @@ -0,0 +1,3182 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf,len=3181 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='05086v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='GR] 12 Jan 2023 Sublinear Rigidity of Lattices in Semisimple Lie Groups Ido Grayevsky Abstract Let G be a real centre-free semisimple Lie group without compact factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I prove that irreducible lattices in G are rigid under two types of sublinear distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I show that if Λ ≤ G is a discrete subgroup that sublinearly covers a lattice, then Λ is itself a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I use this result to prove that the class of lattices in groups that do not admit R-rank 1 factors is SBE complete: if Λ is an abstract finitely generated group that is Sublinearly BiLipschitz Equivalent (SBE) to a lattice in G, then Λ can be homomorphically mapped into G with finite kernel and image a lattice in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This generalizes the well known quasi-isometric completeness of lattices in semisimple Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1 Introduction The quasi-isometric rigidity and classification of irreducible lattices in semisimple Lie groups was established in the 1990’s by the accumulated work of many authors - Pansu [48], Schwartz [52], Kleiner-Leeb [32], Eskin [22], Drut¸u [16], to name a few which are closely related to this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' See [23] for a concise survey of the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 (Quasi-Isometric Completeness, Theorem I in [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real finite-centre semisimple Lie group without compact factors, Γ ≤ G an irreducible lattice and Λ an abstract finitely generated group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ and Λ are quasi-isometric, then there is a group homomorphism Φ : Λ → G with finite kernel whose image Λ′ := Φ(Λ) is a lattice in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Put differently, there is a lattice Λ′ ≤ G and a finite subgroup F ≤ Λ such that the sequence 1 → F → Λ → Λ′ → 1 is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, Λ′ is uniform if and only if Γ is uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the standard terminology of metric rigidity, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 states that the classes of uniform and non- uniform lattices in a group G as in the statement are quasi-isometrically complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The main goal of the current work is to generalize this result to a sublinear setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Sublinear BiLipschitz Equivalences (SBE) are a sublinear generalization of quasi-isometries, brought forward by Cornulier [13] in the past decade or so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A function u : R≥0 → R≥1 is sublinear if limr→∞ u(r) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a, b ∈ R≥0 denote a ∨ b : max{a, b}, and for a pointed metric space (X, x0, dX) and x, x1, x2 ∈ X, let |x|X := dX(x, x0), |x1 − x2|X := dX(x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let (X, dX, x0), (Y, dY , y0) be pointed metric spaces, L ∈ R>0 a constant, u : R≥0 → R≥1 a sublinear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A map f : X → Y is an (L, u)-SBE if the following conditions are satisfied: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' f(x0) = y0, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ∀x1, x2 ∈ X, 1 L|x1 − x2|X − u � |x1|X ∨ |x2|X � ≤ |f(x1) − f(x2)|Y ≤ L|x1 − x2|X + u � |x1|X ∨ |x2|X � , 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ∀y ∈ Y ∃x ∈ X such that |y − f(x)|Y ≤ u(|y|Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1 A map is an SBE if it is an (L, u)-SBE for some L and u as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I prove SBE-completeness for irreducible lattices in groups without R-rank 1 factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 (SBE-Completeness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real centre-free semisimple Lie group without compact or R-rank 1 factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let Γ ≤ G be an irreducible lattice and Λ an abstract finitely generated group, both considered as metric spaces with some word metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume there is an (L, u)-SBE f : Λ → Γ with u a subadditive sublinear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then there is a group homomorphism Φ : Λ → G with finite kernel whose image Λ′ := Φ(Λ) is a lattice in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, Λ′ is uniform if and only if Γ is uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The main ingredients in the proof are of independent interest: the first is geometric rigidity for the corresponding symmetric space, stating that every self SBE of such a space is sublinearly close to an isom- etry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This generalizes Kleiner and Leeb’s result [32] on self quasi-isometries of symmetric spaces, as well as Eskin’s [22] and Drutu’s [16] results on self quasi-isometries of non-uniform lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For the definition of the ‘compact core’ of a lattice see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 (Sublinear Geometric Rigidity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a symmetric space of noncompact type without R-rank 1 factors, Γ ≤ Isom(X) an irreducible lattice and X0 ⊂ X the compact core of Γ in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For any (L, u)-SBE map f : X0 → X0 there is a sublinear function v = v(L, u) and an isometry g : X → X such that d � q(x), g(x) � ≤ v(|x|) for all x ∈ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 actually requires a stronger version of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4, formulated in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Notice that Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 both hold for uniform as well as non-uniform lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I remark that ‘generalized quasi-isometries’ already appeared in the context of geometric rigidity, as a technical tool in Eskin and Farb’s work [21] [22] on quasi-isometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Indeed much of their work is carried for maps which are even more general than SBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It seems however that their approach cannot yield a sublinear bound as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4, which is necessary for the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The second ingredient is a property I call sublinear rigidity, stating that a discrete subgroup Λ ≤ G which sublinearly covers a lattice is itself a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Sublinear rigidity holds for groups of any R-rank, and is the cornerstone of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Its proof contains the bulk of the original ideas that appear in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a function u : R≥0 → R>0 and a subset Y ⊂ X, define the u-neighbourhood of Y to be Nu(Y ) := {x ∈ X | d(x, Y ) ≤ u(|x|)} A subset Y ⊂ X is said to sublinearly cover Z ⊂ X if Z ⊂ Nu(Y ) for some sublinear function u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For the definition of a Q-rank 1 lattice, see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 (Sublinear Rigidity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real centre-free semisimple Lie group without compact factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let Γ ≤ G be an irreducible lattice, Λ ≤ G a discrete subgroup that sublinearly covers Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ is of Q-rank 1, assume further that Λ is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then Λ is a lattice in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The notion of irreducibility in the non-standard context of a general (non-lattice) subgroup is explained Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Sublinear neighbourhoods arise naturally in the presence of SBE maps: the essential difference between a quasi-isometry and an SBE is that ‘far away in the space’, the ‘additive’ error term of an SBE gets larger and larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One is led to consider metric neighbourhoods that grow - sublinearly, yet unboundedly - with the distance to some (arbitrary) fixed base point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The hypothesis that u is sublinear is optimal in the sense that u could not be taken to be an arbitrary linear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Indeed, the geometric meaning of Γ ⊂ Nu(Λ) is that for every element γ ∈ Γ, the ball BG � γ, u(|γ|) � ‘of sublinear radius about γ’ must intersect Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Observe that if f is the identity function f(r) = r, then by definition f(|g|) = f � d(g, eG) � = d(g, eG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular G lies in the f-neighbourhood of the trivial subgroup which is, after all, not a lattice in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For uniform lattices and for lattices with Kazhdan’s property (T) one can however relax the assumption of sublinearity: 2 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 (Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real centre-free semisimple Lie group without compact factors, Γ ≤ G an irreducible lattice, Λ ≤ G a discrete subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix ε > 0 and assume Γ ⊂ Nu(Λ) for the function u(r) = ε · r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ is uniform and ε < 1, then Λ is a uniform lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ has Kazhdan’s property (T) then there is a constant ε(G) such that if ε < ε(G) then Λ is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 generalizes the case where Γ lies in a bounded neighbourhood ND(Λ) for some D > 0, proved by Eskin and Schwartz in a slightly modified version (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the bounded case, the result is much stronger and states that Λ must be commensurable to Γ (except in groups locally isomorphic to SL2(R), see [52]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the sublinear setting I can only prove a limited commensurability result, which stems from a reduction to the bounded setting: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G, Γ and Λ be as in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ is uniform then so is Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ is of Q-rank 1 then: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ ̸⊂ ND(Λ) for any D > 0, then also Λ is of Q-rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ ⊂ ND(Λ) for some D > 0 and in addition Γ sublinearly covers Λ, then Λ is commensurable to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I stress that the case where Γ and Λ each sublinearly covers the other arises naturally in the context of SBE-completeness, see Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The most interesting case in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 is when Γ is of Q-rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In that case, the proof is entirely geometric, relying on the following key proposition which might be of independent interest: Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the setting of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6, assume that Γ is a Q-rank 1 lattice which does not lie in any bounded neighbourhood of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then there exists a horosphere H based at the rational Tits building associated to Γ such that � Λ ∩ StabG(H) � x0 intersects H in a cocompact metric lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, the bounded horoball HB does not intersect the orbit Λ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 Outline of Proof My proofs rely and draw on the works on the quasi-isometric rigidity for non-uniform lattices, due to Schwartz [52] in R-rank 1, and to Drut¸u [16] and Eskin [22] independently in groups of R-rank greater than 1, often called higher rank groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The geometric proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 for Q-rank 1 lattices is quite delicate and involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For this reason I give here a detailed sketch of the arguments and of the ideas one should have in mind when reading the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' What is written here is a good enough account if one wishes to understand the main ideas while avoiding the technical details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I end the section with a brief sketch of the proofs for SBE-completeness and for sublinear rigidity in the case of property (T) groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Strategy for Q-rank 1 Lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote dγ := d(γ, Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The novel case is when {dγ}γ∈Γ is unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The rationale for the proof comes from a conjecture of Margulis, recently proved in full generality by Benoist and Miquel ([5], see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Their result states that a discrete subgroup in a higher rank Lie group is a lattice as soon as it intersects a horospherical subgroup in a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This result could be seen as an algebraic converse to the geometric structure of a Q-rank 1 lattice, whose orbit in X intersects some parabolic horospheres in a cocompact (metric) lattice (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 is the geometric analogue of the Benoist-Miquel criterion, and basically completes the proof in the higher R-rank case (some non-trivial translation work is needed, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Also, it easily follows from Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 that every Γ-conical limit point is also Λ-conical (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof for R-rank 1 groups is then a simple use of a criterion of Kapovich-Liu for geometrically finite groups ([30], see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I now give a detailed description of the proof of Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3 The ABC of Sublinear Constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix a point x0 ∈ X = G/K, identify Γ and Λ with Γ · x0 and Λ · x0 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Observe that by definition of dγ the interior of balls of the form B(γx0, dγ) does not intersect Λ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I call such balls (or general metric sets) Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, these balls intersect Λ · x0 (only) in the bounding sphere: call such balls (sets) tangent to Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The Λ-free and, respectively, Γ-free regions in X are the main objects of interest in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 is known in the case of bounded {dγ}γ∈Γ, it makes sense to think about large Λ-free regions as ‘problematic’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The state of mind of the proof relies on two easy observations that complete each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The sublinear constraint implies that dγn → ∞ forces |γn| → ∞, suggesting that ‘problematic’ Λ-free regions should appear only ‘far away’ in the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand Λ is a group, and being Λ-free is a Λ-invariant property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular any metric situation that can be described in terms of the Λ-orbit (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' B(γ, dγ) is a Λ-free ball tangent to Λ) can be translated back to x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This means that ‘problematic’ regions could actually be found near x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The moral of these observations can be formulated into a general principle that lies in the heart of the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The sublinear constraint dγ ≤ u(|γ|) gives rise to many other constraints of ‘sublinear’ nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Each such constraint actually yields a uniform constraint inside any fixed bounded neighbourhood of x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since Λ and Γ are groups, many of these uniform bounds which are produced ‘near’ x0 turn out to be global bounds that depend only on the group and not on a specific orbit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Put differently: the trick is to describe metric situations in terms of the Γ and Λ orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One then uses the group invariance in order to move these metric situations around the space to a place where the sublinear constraint can be exploited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The Argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The above paragraph should more or less suffice the reader to produce a complete proof for uniform lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For non-uniform lattices, denote by λγ the closest Λ-orbit point to the point γ ∈ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For H a cusp horosphere of Γ, let ΓH := {γ ∈ Γ | γx0 ∈ H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The ultimate goal is to show that the metric lattice ΓH · x0 yields a metric lattice that is more or less {λγx0}γ∈ΓH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One proceeds by the following steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finding Λ-free horoballs (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1): The arbitrarily large Λ-free balls B(γn, u(|γn|)) are translated to x0, and the compactness of the unit tangent space at x0 yields a converging direction which is the base point at infinity of a Λ-free horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Translating by Λ, this yields Λ-free horoballs tangent to every Λ-orbit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Controlling angles (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2): For every γ ∈ Γ with dγ uniformly large enough, one associates a point ξ at X(∞) such that ξ is the base point of a Λ-free horoball tangent to λγx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The angle between the geodesics [λγx0, γx0] and [λγX0, ξ) is shown to be small as dγ grows large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is used to show that arbitrarily large dγ give rise to arbitrarily deep Γ-orbit points inside Λ-free horoballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A key step is Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8, producing a Γ-free Euclidean cylinder between λγx0 and γx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Λ-cocompact horospheres (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3): One uses uniform bounds near x0 to prove that every Λ-free horoball that is (almost) tangent to Λ must lie in a uniformly bounded neighbourhood of Λ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If dγ is large enough for some γ that lies on a horosphere HΓ of a cusp of Γ, then any γ′ ∈ ΓHΓ also admits large dγ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since the bounds from the previous steps only depend on dγ′, all λγ′ are forced to lie on the same horosphere HΛ parallel to HΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One concludes that Λ · x0 intersects HΛ on the nose in a cocompact metric lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Property (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Sublinear rigidity for groups with property (T) is established by the criterion that a discrete subgroup there is a lattice if and only if it has the same exponential growth rate as a lattice (Leuzinger [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is quite straightforward that sublinear distortion cannot affect this growth rate (Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' SBE Rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The general scheme for SBE-completeness is parallel to the quasi-isometry case: each λ ∈ Λ naturally gives rise to an SBE X0 → X0 of the compact core of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Each self SBE is close to an isometry by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4, allowing to embed Λ as a discrete subgroup of isometries in G that sublinearly 4 covers Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof for Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 heavily relies on Drut¸u’s argument for quasi-isometries [16], which uses the properties of the induced biLipschitz map on the asymptotic cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As SBE also induce such a biLipschitz map - indeed that was a main motivation for Cornulier to study SBE [13] - it is possible to generally follow Drut¸u’s argument also in the SBE setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 Possible Improvements, Related and Future Work The proof suggests three natural improvements to the statement of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One could probably relax the assumption of trivial centre and allow finite centre, if the same relaxation is applicable in Leuzinger’s work on property (T) groups [36] and in Prasad’s work [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, the proofs for uniform lattices and for Q-rank 1 lattices hold also for groups G with finite centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The irreducibility of Λ may be derived directly from the irreducibility of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lastly, in view of the geometric characterization of Q-rank (see Corollary D in [35]), it is reasonable that the Q-rank of Λ should equal that of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There are problems that arise naturally from this work which seem to require new ideas: Question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real finite-centre semisimple Lie group without compact factors that admits a R- rank 1 factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Are the classes of uniform and non-uniform lattices of G SBE-complete?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, is this true when G is of R-rank 1?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' See Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 below for a discussion on the case of R-rank 1 factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' SBE of R-rank 1 symmetric spaces is the main focus in Pallier’s work [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' He investigated the sublinearly large scale geometry of hyperbolic spaces, and proved that two R-rank 1 symmetric spaces that are SBE are homothetic, answering a question of Drut¸u (see Remarks 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='16 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='17 in [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Also in this context Pallier and Qing [47] recently showed that the sublinear Morse boundary is an SBE invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Another problem is to find a non-trivial example of the setting of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6: Question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real finite-centre semisimple Lie group without compact factors, Γ ≤ G a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Does there exist a finitely generated group that is SBE to Γ but not quasi-isometric to it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Or, at least, not known to be quasi-isometric to one?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Does there exist Λ ≤ G discrete that ε-linearly covers Γ but which does not sublinearly cover it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the side of the proof, it would be very interesting if the geometric ideas that prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 for Q-rank 1 lattices could be applied to any Q-rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' While there are apparent places where the proof uses the unique geometry of Q-rank 1 lattices, most of the geometric arguments leading to Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 seem to be susceptible to the higher Q-rank setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Such a generalization of the proof would definitely shed more light on the mysterious lattice arising from growth considerations in property (T) groups, and in particular on the question of commensurability of Λ and Γ in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It would also be interesting to see whether one can push the geometric argument forward in order to establish a complete geometric analogue of the Benoist-Miquel criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Namely, can one find a direct geometric proof that Λ admits finite co-volume (perhaps similarly to Schwartz’s argument in the bounded case, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lastly, one could possibly relate this work to the work of Fraczyk and Gelander [24], who proved that a discrete subgroup (of a higher rank simple Lie group) is a lattice if and only if it has bounded injectivity radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' While their result seem very much related to the condition Γ ⊂ Nu(Λ), the nature of their work does not give explicit bounds on the injectivity radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Specifically, given r > 0 one cannot tell directly from their results how ‘far’ one must wander in X in order to find a point with injectivity radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Perhaps one could use the sublinear results of this work to say something about the relation of |x|X and InjRad(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 Acknowledgments This paper is based on my DPhil thesis, supervised by Cornelia Drut¸u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I thank her for suggesting the question of SBE-completeness and for guiding me in my first steps in the theory of Lie groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I thank Uri Bader, Tsachik Gelander and the Midrasha on groups at the Weizmann institute, where I learned the basics 5 of symmetric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I thank my thesis examiners Emmanuel Breuillard and Yves Cornulier for their careful inspection and numerous remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I thank Elon Lindenstrauss for telling me about Leuzinger’s result [36] on property (T), and Or Landesberg, Omri Solan, Elyasheev Leibtag, Itamar Vigdorovich and Tal Cohen for many discussions on different aspects of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally I thank Gabriel Pallier for explaining his examples of some unusual SBE in R-rank 1, and for his interest in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2 Preliminaries For standard definitions and facts about fundamental domains, see Chapter 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The facts about fundamental domains for Q-rank 1 lattices appear in Raghunathan’s book [50] and in Prasad’s work on rigidity of Q-rank 1 lattices [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In notations and generalities I follow: Borel’s book on algebraic groups [6], Helgason’s books on Lie groups and symmetric spaces [27, 28], and Eberline’s book on the geometry of symmetric spaces of noncompact type [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 Generalities on Semisimple Lie Groups and their Lattices Let G be a real centre-free semisimple Lie group without compact factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A discrete subgroup Γ ≤ G is a lattice if Γ\\G carries a finite volume G-invariant measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Equivalently, Γ is a lattice if Γ\\X is a finite volume Riemannian manifold, where X = G/K is the symmetric space of noncompact type corresponding to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A lattice is irreducible if its projection to every simple factor of G is dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The group G can be viewed as an algebraic group via the adjoint representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If G is of R-rank greater than 1, then by the Margulis arithmeticity theorem every irreducible lattice of G is arithmetic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The Q-rank Γ is the Q-rank of the Q-structure associated to (G, Γ) be the arithmeticity theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A result of Prasad [49] states that if G admits a R-rank 1 factor, then a non-uniform irreducible lattice of G is of Q-rank 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The group G has Kazhdan’s property (T) if and only if it does not admit an SO(n, 1) or an SU(n, 1) factor, and an irreducible lattice Γ ≤ G has property (T) if and only if G has property (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Together with Prasad’s result, I may conclude: Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real centre-free semisimple Lie group without compact factors, and Γ ≤ G an irreducible lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then at least one of the following occurs: (a) G has property (T) (b) Γ is a non-uniform Q-rank 1 lattice (c) Γ is uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 is therefore an immediate result of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 Cusps and the Rational Tits Building The facts about symmetric spaces of noncompact type can be found in Eberline’s book [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since the geometry of Q-rank 1 lattices resembles that of lattices in R-rank 1, the reader could for the most part simply have the image of the hyperbolic plane in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If one wishes to see flats that are not geodesics, then a product of two hyperbolic planes is enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Even the product of the hyperbolic plane and R is helpful, albeit this space has a Euclidean factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 Basic Geometry of Symmetric Spaces of Noncompact Type Visual Boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The visual boundary X(∞) of X is the set of equivalence classes of geodesic rays, where two geodesic rays are equivalent if their Hausdorff distance is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a ray η : [0, ∞) → X, η(∞) denotes the equivalence class of η in X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There are two natural topologies on X(∞) that will be of use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The cone topology is the one given by viewing X(∞) as the set of all geodesic rays emanating from some fixed base point x0, with topology induced by the unit tangent space at x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is a natural topology on X := X ∪ X(∞) such that X is the compactification of X and the induced topology on X(∞) is the cone topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A well known fact about geodesic rays in nonpositively curved spaces, stating that two ‘close’ geodesic rays fellow travel: 6 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Given time T and ε > 0, there is an angle α = α(T, ε) so that if η1, η2 are two geodesic rays with η1(0) = η2(0) = x for some x ∈ X and ∡x(η1, η2) ≤ α then dX � η1(t), η2(t) � < ε for all t ≤ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The Tits metric on X(∞) is defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Given a totally geodesic submanifold Y ⊂ X, let Y (∞) ⊂ X(∞) be the subset of all points that admit a geodesic ray η lying inside Y (or, equivalently, those points that admit a ray lying at bounded Hausdorff distance to Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For any ξ1, ξ2 ∈ X(∞) there exists a flat F ⊂ X such that ξ1, ξ2 ∈ F(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Define dT (ξ1, ξ2) ∈ [0, π] to be the angle between two geodesic rays η1, η2 ⊂ F emanating from some point x ∈ F and with η1(∞) = ξ1, η2(∞) = ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is a well defined metric on X(∞), called the Tits metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The pair � X(∞), dT � is a geodesic metric space, and isometries of X act on it by isometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I will use the following relation between the cone and the Tits topologies: Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 in [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a symmetric space of noncompact type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The Tits metric on X(∞) is semicontinuous with respect to the cone topology: for any ξ, ζ ∈ X(∞) and every ε > 0, there exists neighbourhoods of the cone topology U, V ⊂ X(∞) of ξ and ζ respectively such that for all ξ′ ∈ U, ζ′ ∈ V one has ∡(ξ′, ζ′) ≥ ∡(ξ, ζ) − ε Moreover, for any flat F ⊂ X, the cone topology and the Tits topology coincide on F(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Busemann Functions, Horoballs and Horospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Horoballs and horospheres play a crucial role in the proof, a role which stems from their role in the geometric description of the compact core of non-uniform lattices (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='20 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A Busemann function on X is any function of the form fη(x) = lim t→∞ d � x, η(t) � − t for some geodesic ray η of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A horoball HB ⊂ X is an open sublevel set of a Busemann function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A horosphere H ⊂ X is a level set of a Busemann function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Two equivalent geodesic rays η1, η2 give rise to Busemann function which differ by a constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' fη1 − fη2 = C for some C ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If HB is the sublevel set of fη, then η(∞) is called the base point of the horoball HB (and respectively of the horosphere H = ∂HB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The base point of a horoball is well defined, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' it depends only on η(∞) and not on η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For every choice of x ∈ X, ξ ∈ X(∞) there is a unique horosphere H based at ξ with x ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I denote this horosphere by H(x, ξ) and the bounded horoball HB(x, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The following proposition collects some basic properties that will be of use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 (Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5 in [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let x ∈ X, ξ ∈ X(∞), and let H = H(x, ξ), HB the horoball bounded by H and f the Busemann function based at ξ with f(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For any point y ∈ X, let η be the bi-infinite geodesic determined by the geodesic [y, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then PH(y) = η ∩ H, where PH(y) is the unique point closest to y on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For any point y ∈ X, f(y) = ±d � y, PH(y) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, f(y) is negative if and only if y ∈ HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If x′ ∈ X, then the horospheres H = H(x, ξ) and H′ = H(x′, ξ) are equidistant: if y ∈ H, y′ ∈ H′, then d(y, H′) = d(y′, H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Such horospheres are called parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A Busemann function fη thus naturally determines a filtration of X by the co-dimension 1 manifolds {Ht}t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By convention I usually assume that Ht := {x ∈ X | fη(x) = −t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The stabilizer of a point ξ ∈ X(∞) acts transitively on the set of horospheres based at ξ, so every two such horospheres are isometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, there is a close relation between the induced metrics on horospheres with the same base point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Briefly, if dH denotes the induced distance on a horosphere H ⊂ X, then dH � PH(x), PH(y) � for any two points x, y ∈ H′ can be bounded uniformly below and above as a function of the distance dH′(x, y) and the curvature bounds on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' See Heintze-Im hof [26] for precise statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 7 Using the above properties one can show that two horospheres are parallel if and only if they are based at the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular for every x ∈ X, ξ ∈ X(∞) it holds that StabG � H(x, ξ) � ⊂ StabG(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In addition, If H, H′ are two parallel horospheres based at the same ξ ∈ X(∞) and A ⊂ H is a cocompact metric lattice in H, then πH′(A) is a cocompact metric lattice in H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a point ξ ∈ X(∞) and a flat F with ξ ∈ F(∞), one readily observes that every horoball HB based at ξ intersects F in a Euclidean half space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular for every ζ ∈ X(∞) with dT (ξ, ζ) < π 2 , for every geodesic ray η with η(∞) = ζ and every horoball HB based at ξ there is some T for which η↾t>T ⊂ HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Parabolic and Horospherical Subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The isometries of X are classified into elliptic, hyperbolic, and parabolic isometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Most significant for this paper are the parabolic isometries, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' those g ∈ G whose displacement function x �→ gx does not attain a minimum in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Every such isometry fixes (at least) one point in X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A group P ≤ Isom(X) is called geometrically parabolic if it is of the form Gξ := StabG(ξ) for some ξ ∈ X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Such groups act transitively on X, and in particular act transitively on the set of geodesic rays in the equivalence class of ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The same holds also for the identity component G◦ ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' An element g ∈ Gξ acts by permutation on the set of horoballs based at ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This permutation is a translation with respect to the filtration of the space X by horospheres based at ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Put differently, if {Ht}t∈R is a filtration of X by horospheres based at ξ, then for every g ∈ Gξ there is l(g) ∈ R such that gHt = Ht+l(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is quite clear from all of the above that for every horosphere based on ξ, the group GH := StabG(H) acts transitively on H, and the same holds for G◦ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I now present a fundamental structure theorem for geometrically parabolic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote g := Lie(G), and let g = k ⊕ p be a Cartan decomposition defined using the maximal compact subgroup K ≤ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall that the Lie exponential map exp : g → G gives rise to a family of 1-parameter subgroups of the form exp(tX) for each X ∈ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 in [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let x ∈ X(∞), and let X ∈ p be the tangent vector of the unit speed geodesic [x0, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ht ξ be the 1-parameter subgroup defined by t �→ exp(tX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then an element g ∈ G fixes ξ if and only if limt→∞ h−t ξ ght ξ exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 (Langlands Decomposition, Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='25 in [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ξ ∈ X(∞) and ht ξ as in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let F be a flat containing [x0, ξ) and A ≤ G the maximal abelian subgroup such that Ax0 = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote Gξ := StabG(ξ), and define Tξ : Gξ → G by g �→ limn→∞ h−n ξ ghn ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then Tξ is a homomorphism, and there are subgroups Nξ, Aξ, Kξ ≤ Gξ such that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Aξ = exp � Z(X) ∩ p � , where Z(X) is the centralizer of X in g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, every element a ∈ Aξ lies in some conjugate Ag = gAg−1 with the property that [x0, ξ) ⊂ F g := Agx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Kξ ≤ K = StabG(x0) is the compact subgroup fixing the bi-infinite geodesic determined by [x0, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' KξAξ = AξKξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Nξ = Ker(Tξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is a connected normal subgroup of Gξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Gξ = NξAξKξ, and the indicated decomposition of an element is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' G = NξAξK, and the indicated decomposition of an element is unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In case ξ is a regular point at X(∞), this decomposition is the Iwasawa decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Gξ has finitely many connected components, and G◦ ξ = (KξAξ)◦Nξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Gξ is self normalizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Viewing G as an algebraic group, the geometrically parabolic subgroups are exactly the (algebraically) non-trivial parabolic subgroups, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' proper subgroups of G that contain a normalizer of a maximal unipotent subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 is a geometric formulation of the algebraic Langlands decomposition of parabolic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall that a horospherical subgroup is the unipotent radical of a non-trivial parabolic group, or equivalently groups of the form Ug := {u ∈ G | limn→∞ g−nugn = idG}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The latter implies that Nξ a horospherical subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 8 Limit Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' An important set associated to a discrete group ∆ ≤ G acting by isometries on X is the limit set L∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By definition L∆ := ∆ · x ∩ X(∞), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' it is the intersection with X(∞) of the closure, in the compactification X = X ∪ X(∞), of an orbit ∆ · x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is clear that L∆ does not depend on the choice of x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The limit set of any lattice is always the entire X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In fact, much more is true: Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ∆ ≤ G = Isom(X), and SX the unit tangent bundle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A vector v ∈ SX is ∆-periodic if there is δ ∈ ∆ and s > 0 such that δη(t) = η(t + s) for all t ∈ R, where η is the bi-infinite geodesic determined by the vector v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a flat F ⊂ X (including geodesics), denote ∆F := {δ | δF = F}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The flat F is called a ∆-periodic flat if there exists a compact set C ⊂ F such that ∆F C = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 (Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 in [20], Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3′ in [41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ ≤ G = Isom(X) is a lattice, then: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The subset in SX of Γ-periodic vectors is dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let F ⊂ X be any flat, η any bi-infinite geodesic in F, and denote v = ˙η(0) ∈ SX the initial velocity vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is a sequence vn ∈ SX of regular vectors such that (a) limn→∞ vn = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' (b) The bi-infinite geodesics ηn determined by vn are all Γ-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' (c) Denote by Fn the (unique) flat containing ηn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Each Fn is Γ-periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Put differently, the set of Γ-periodic flats is dense in the set of flats of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' See Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='25 for the notion of regular tangent vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Points in the limit set of a group are classified according to how the orbit approaches them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ∆ ≤ G be a discrete subgroup, ξ ∈ L∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The point ξ is called conical if for some (hence any) x and some (hence every) geodesic ray η with η(∞) = ξ there is a number D = D(x, η) such that for every T ∈ R>0 there is t > T for which B � η(t), D � ∩ ∆ · x ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since ∆ is discrete, this is equivalent to ∆ · x ∩ ND(η) being infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The point ξ is called horospherical if for every horoball HB based at ξ and every x ∈ X, ∆ · x ∩ HB is non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, a conical limit point is horospherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The point ξ is non-horospherical if it is not horospherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As a corollary of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9, one has: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ ≤ Isom(X) is a lattice, then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The set of Γ-conical limit points is dense in X(∞) with the cone topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' LΓ = X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I finish this section with some results on geometrically finite subgroups of isometries in R-rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a R-rank 1 symmetric space and ∆ ≤ Isom(X) a discrete subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote by Hull(∆) the closed convex hull in X = X ∪X(∞) of the limit set L∆, and Hull(∆) = X ∩Hull(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By virtue of negative curvature, Hull(∆) is the union of all geodesics η such that η(∞), η(−∞) ∈ L∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The convex core of ∆ is defined to be ∆\\Hull(∆) ⊂ ∆\\X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', the quotient of Hull(∆) by the ∆-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='14 (Bowditch [9], see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 in [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a R-rank 1 symmetric space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' a symmetric space of pinched negative curvature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A discrete group ∆ ≤ G = Isom(X) is geometrically finite if for some δ > 0, the uniform δ-neighbourhood in ∆\\X of the convex core Nδ � ∆\\Hull(∆) � , has finite volume and there is a bound on the orders of finite subgroups of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A group is geometrically infinite if it is not geometrically finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 9 Immediately from the definition of geometrical finiteness, one gets a simple criterion for a subgroup to be a lattice: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a R-rank 1 symmetric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If ∆ ≤ Isom(X) is geometrically finite and admits L∆ = X(∞), then ∆ is a lattice in Isom(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Sublinear distortion does not effect the limit set, as the following lemma shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let Γ, Λ ≤ G be discrete subgroups and u : R≥0 → R>0 a sublinear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ ⊂ Nu(Λ), then LΓ ⊂ LΛ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' every Γ-limit point is a Λ-limit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, if Γ is a lattice then LΛ = X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By definition, one has to show that given a point ξ ∈ X(∞) and a sequence γn ∈ Γ such that γnx0 → ξ (in the cone topology on X), there is a corresponding sequence λn ∈ Λ with λnx0 → ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Define λn := λγn to be the closest point to γn in Λ, and to ease notation denote xn := γnx0, x′ n = λnx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let also ηn := [x0, xn] and η′ n := [x0, x′ n] be unit speed geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally, let Tn denote the time in which ηn terminates, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ηn(Tn) = xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Convergence in the cone topology xn → ξ is equivalent to the fact that the geodesics ηn converge to η := [x0, ξ) uniformly on compact sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This means in particular that Tn → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Non-positive curvature guarantees that the functions Fn(t) := d � η(t), ηn(t) � , F ′ n(t) := d � η(t), η′ n(t) � and Gn := d � ηn(t), η′ n(t) � are convex (F ′ n is just a notation, completely unrelated to the derivative of Fn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since Gn(0) = Fn(0) = F ′ n(0) = 0 any of these functions is either constant 0 or monotonically increasing, so proving uniform convergence of η′ n to η amounts to proving limn F ′ n(T ) = 0 for every T ∈ R≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Triangle inequality gives F ′ n(T ) ≤ Fn(T ) + Gn(T ), and by assumption limn Fn(T ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Notice that Gn(Tn) = d(xn, x′ n) ≤ u(|γn|) = u(Tn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Writing T = T Tn · Tn, convexity of Gn implies Gn(T ) ≤ (1 − T Tn )Gn(0) + T Tn Gn(Tn) ≤ 0 + T Tn u(Tn) = T · u(Tn) Tn As limn Tn = ∞ it follows from sublinearity that limn Gn(T ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that η′ n converge to η uniformly on compact sets, therefore ξ lies in the limit set of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 I prove that in the setting of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12, the set of Λ-conical limit points contains the set of Γ-conical limit points (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It holds that the conical limit points of Γ are dense in X(∞) (in the cone topology, see Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12) and therefore LΛ = LΓ = X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, every Γ-limit point is a Λ-limit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The strength of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='16 is that it does not assume anything on Γ other than that it is sublinearly covered by Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='16 does not require Γ to be a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 Cusps, Compact Core, and the Rational Tits Building In this section I present some of the structure theory of non-compact quotients of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The focus is on the structure of ‘cusps’ in noncompact finite volume quotients of symmetric spaces, and the ‘location’ of cusps on the visual boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Cusps and Compact Core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Consider V = Γ\\X, for Γ ≤ G a non-uniform lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is a locally symmetric space of finite volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The term ‘cusps’ is an informal name given to those areas in a locally symmetric space through which one can ‘escape to infinity’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Another description is that cusps are the ends of the complement of a large enough compact set in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In strictly negative curvature, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' in R-rank 1 locally symmetric spaces, these cusps have a precise description as submanifolds of the form C × R≥0 for a compact manifold C, and metrically (C, t) gets narrower as t → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There are finitely many cusps, each corresponding to a point at X(∞) called a ‘parabolic point’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Introduction in [3] or [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A fundamental feature of the cusps is that one can ‘chop’ them out of the quotient manifold V and get a compact manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This could be done in such a way so that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The lifts of the chopped parts to the universal cover X are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Each cusp is covered in X by the Γ-orbit of a horoball, that is, the lift of a cusp is the Γ-orbit of a horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The respective base points are called parabolic points of Γ in X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Γ acts on X \\ � � i∈I HBi � cocompactly, where {HBi}i∈I is the set of horoballs coming from the lifts of cusps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since there are only finitely many cusps and Γ discrete, there are exactly countably many such horoballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' See for example Section 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 in [17] where this is illustrated in the case of the real hyperbolic spaces Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Formally, one has;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='18 (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 in [19], see also Introduction therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume X is of R-rank 1, and Γ ≤ G a non-uniform lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The space V = Γ\\X has only finitely many (topological) ends and each end is parabolic and Riemannian collared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, each cusp is a quotient of a horoball HB based at a parabolic limit point ξ such that Γ ∩ Gξ acts cocompactly on H = ∂HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For symmetric spaces of higher rank, a similar construction is available (see [37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By removing a countable family of horoballs from X, one obtains a subspace on which Γ acts cocompactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There are two main differences from the situation in R-rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One is that an orbit map γ �→ γx is a quasi-isometric embedding of Γ (with the word metric) into X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='19 (Lubozki-Mozes-Raghunathan, Theorem A in [38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a semisimple Lie group of higher R-rank, dG a left invariant metric induced from some Riemannian metric on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let Γ an irreducible lattice, dΓ the corresponding word metric on Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then dG↾Γ×Γ and dΓ are Lipschitz equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This result plays a significant preliminary role in the proofs of quasi-isometric rigidity for non-uniform lattices in higher rank symmetric spaces in both [16] and [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The second difference in higher rank spaces is that the horoballs could not in general be taken to be disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' However, in the special case of Q-rank 1 lattices the horoballs can be taken to be disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall that the Q-structure of (G, Γ) is a Q-structure on G = G(R) in which Γ is an arithmetic lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The following theorem sums up the relevant properties for Q-rank 1 lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='20 (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 in [34], see also Remarks 3 and 4 in [37], Section 13 in [50], and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 in [49]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume X is of higher rank, and Γ ≤ G an irreducible torsion-free non-uniform lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the locally symmetric space V = Γ\\X there exists a continuous and piece-wise real analytic exhaustion function h : V → [0, ∞) such that, for any s > 0, the sublevel set V(s) := {h < s} is a compact submanifold with corners of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover the boundary of V(s), which is a level set of h, consists of projections of subsets of horospheres in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The Γ-action on the above set of horospheres has finitely many orbits, and the following conditions are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The corresponding horoballs bounded by these horospheres can be taken to be disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For each such horosphere H the action of Γ ∩ StabG(H) on H is cocompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The Q-structure of (G, Γ) is of Q-rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The lattice Γ is of Q-rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the setting of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='18 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='20, the horoballs and horospheres that appear in the statement are called (global) horoballs (horospheres) of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Base points of these are called parabolic limit points of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The above geometric characterization of Q-rank 1 lattices is all that I use in order to prove the key Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The compact core of Γ is the complement in X of the horoballs of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The group Γ acts on it cocompactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The following corollary describes the orbit of Q-rank 1 lattices in X, and especially some finiteness properties which I will use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 11 Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let Γ ≤ G be a Q-rank 1 lattice, x ∈ X, and ξ a parabolic limit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is a unique horosphere H based at ξ such that both following conditions hold: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Γ · x ∩ H is a cocompact metric lattice in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Γ · x ∩ HB = ∅, where HB is the horoball bounded by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Call H an x-horosphere of Γ at ξ, and the corresponding bounded horoball an x-horoball of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, one has: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For every C there exists a bound K = K(x, C) so that B(x, C) intersects at most K x-horospheres of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is D = D(x) > 0 such that H ⊂ ND(Γ · x ∩ H) for any x-horosphere H of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The constant D is called the compactness number of (Γ, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is a number N = N(Γ) such that every point x ∈ X admits exactly N x-horospheres of Γ that intersect x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' These are called the horospheres of (Γ, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since Γ is a Q-rank 1 lattice, the stabilizer in Γ of a parabolic limit point ξ ∈ X(∞) acts cocompactly on each horosphere based at ξ, and in particular on Hx := H(x, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let D(x, H) be such that Hx ⊂ ND(Γ · x ∩ Hx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A priori D(x) depends on H, but the fact that Γ acts by isometries implies that for every horosphere of the form H′ := γH = H(γx, γξ) for some γ ∈ Γ, one has H′ ⊂ ND(Γ · x ∩ H′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' So D(x) depends only on the Γ-orbit of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since Γ acts on the set of parabolic limit points with finitely many orbits (that is to say Γ\\X has finitely many cusps) one may take D = D(x) to be the maximum of the respective bounds on each orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This gives the required compactness number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Thinking of horoballs of γ as lifts of ends of the complement of some compact subset of V = Γ\\X, one sees that there is some horoball HB based at ξ so that Γ · x /∈ HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote H = ∂HB, and y = PHB(x) ∈ H be the projection on the closed convex set that is the closure of the horoball HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally, Let η = [x, y] and denote l = d(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The existence of the required horosphere is equivalent to the fact that the following non-empty set admits a maximum: Hx,ξ := {t ∈ [0, l] | Γ · x ∩ H � η(t), ξ � ̸= ∅} Indeed 0 ∈ Hx,ξ and it is a bounded set, so it admits a supremum T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, this set is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If t were an accumulation point, then for any small ε > 0 the geodesic segment η↾(t−ε,t+ε) would intersect D-cocompactly infinitely many horospheres of (Γ, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Orbit points on different horospheres are in particular different points, therefore the set B � η(t), D + ε � ∩ Γ · x would be infinite, contradicting discreteness of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that T is a maximum, and that H � η(T ), ξ � is the unique desired horosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The argument above generally shows that there cannot be an accumulation point in X of x-horospheres of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, for every C, the ball B(x, C) intersects only finitely many x-horospheres of Γ, say K(C), proving item 1 in the ‘moreover’ statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For the last statement, simply note that the horoballs of Γ are the Γ-translates of finitely many horoballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the terminology of the statement, infinitely many horospheres of (Γ, x) imply that infinitely many of them are in the same Γ-orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Suppose these are {Hn}n∈N, with base points ξn that are evidently pairwise different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally let γn ∈ Γ for which γnH1 = Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since Γ ∩ StabG(Hn) acts cocompactly on Hn, there is γ′ n ∈ Γ ∩ StabG(Hn) that maps γnx to B(x, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Discreteness of Γ implies that the set γ′ nγnx is finite, hence infinitely many of these points are the same point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' However γ′ nγnξ1 = ξn and therefore γ′ nγn ̸= IdX, contradicting the fact that Γ is torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For the most part, I am interested in a fixed base point x0 and the x0-horospheres and horoballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By a slight abuse of terminology I omit x0 and call these objects ‘horospheres of Γ’ and ‘horoballs of Γ’, respectively, denoting the associated cocompactness number DΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='22 allows to upgrade a metric lattice of H to a metric lattice coming from StabG(H) only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 12 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume that a torsion-free discrete group ∆ ≤ G = Isom(X) admits the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is a bound N such that at each point x ∈ ∆ · x0 there are at most N horoballs that are tangent to x and do not intersect ∆ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is a horosphere H ⊂ X such that (a) The set ∆ · x0 ∩ H is a cocompact metric lattice in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' (b) The horoball HB bounded by H is ∆-free, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ∆ · x0 ∩ HB ⊂ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then � ∆ ∩ StabG(H) � x0 is also a cocompact metric lattice in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is the Pigeonhole Principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix x ∈ ∆ · x0 ∩ H, and let {HBi}i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=',N} be the finite set of horoballs tangent to x that do not intersect ∆ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g that H is the bounding horosphere of HB1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix a ∆-orbit point δ0x0 ∈ ∆ · x0 ∩ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Up to translating by some element of ∆, I may assume x = x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For any other ∆-orbit point δx0 ∈ ∆ · x0 ∩ H let i(δ) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' , N} be the index of the horoball δ−1HB1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Notice that from hypothesis 1 it indeed follows that δ−1HB1 ∈ {HBi}i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=',N}, because the action of ∆ is by isometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Define δi to be the element in ∆ for which: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' i(δi) = i, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' δ−1 i (HB1) = HBi 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' d(δix0, x0) is minimal among all such δ ∈ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' , N} there is a δ ∈ ∆ with i = i(δ), while for others there might not be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I will only care about those i for which there is such δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g that these are i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' M} for M ≤ N, and let L := max1≤i≤M{d(x0, δix0)} < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For such an index i0, the ∆-orbit points that share the same i(δ) = i0 are in the same ∆ ∩ StabG(H) orbit, namely {δx0 | i(δ) = i0} ⊂ � ∆ ∩ StabG(H) � δi0x0 Indeed, if δ−1HB1 = HBi0 then by definition δδ−1 i0 HB1 = δHBi0 = HB1, hence δδ−1 i0 ∈ StabG(H) is an element mapping δi0x0 to δx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let now δx0 ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Its distance from the orbit � ∆ ∩ StabG(H) � x0 is � ∆ ∩ StabG(H) � invariant, therefore d � δx0, � ∆ ∩ StabG(H) � x0 � = d � δi(δ)x0, � ∆ ∩ StabG(H) � x0 � ≤ d(δi(δ), x0) The right-hand side is uniformly bounded by L, proving that (∆ · x0 ∩ H) ⊂ NL �� ∆ ∩ StabG(H) � x0 � The fact that ∆ · x0 ∩ H is a cocompact metric lattice in H renders (∆ ∩ StabG(H) � x0 a cocompact metric lattice in h as well, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Real and Rational Tits Buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The location of the parabolic points in X(∞) also plays an important role in the geometry of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In case Γ is an arithmetic lattice, the natural framework to consider these points is the so called rational Tits building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is a building structure on the subset of parabolic points at X(∞), sometimes referred to as ‘rational points’ in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' They are exactly those points in X(∞) whose stabilizers are Q-defined (algebraic) parabolic groups of G (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 for more details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I present this object, denoted WQ(Γ), together with the more familiar real Tits building structure on X(∞) with the Tits metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The main goal is to present the results of Hattori [25], that give a good description of the rational Tits building in terms of conical and horospherical limit points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In case G is of R-rank 1, by WQ(Γ) I mean the (countable) set of parabolic limit points of Γ (so that X(∞) \\ WQ(Γ) is comprised of conical limit points only, see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 13 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A geodesic η is said to be regular if it is contained in a unique maximal flat F ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The point η(∞) ∈ X(∞) is called a regular point of X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A point ξ ∈ X(∞) is singular if it is not regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Regularity does not depend on the choice of representative geodesic ray η for ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A Weyl chamber of X(∞), or an open spherical chamber, is any connected component in the Tits topology of X(∞) \\ S, where S ⊂ X(∞) is the subset of singular points at X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='26 (Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 in [29] and Section 8 in [3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The Weyl chambers induce a simplicial complex structure on X(∞) that is a spherical Tits building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The apartments of the building are exactly the sets of the form F(∞) ⊂ X(∞) for all flats F ⊂ X, and the chambers are exactly the Weyl chambers at X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, the Tits metric completely determines the building structure, and vice versa, and � X(∞), dT � is a metric realization of the Tits building at X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' None of the rich theory of buildings is used directly in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Given a non-uniform lattice of Γ ≤ G the rational Tits building WQ(Γ) is a building structure on the subset of parabolic points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is not in general a sub-building of the real spherical building.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Flats of X correspond to real maximal split tori in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since G is an algebraic group defined over Q, one can consider the maximal Q-split tori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The rational flats of X are then the G(Q)-orbits of maximal Q-split tori of G, and the rational boundary are all points ξ ∈ X(∞) such that ξ ∈ FQ(∞) for some rational flat FQ(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One defines regular rational directions and rational Weyl chambers in an analogous way to the real case, this time taking only rational flats into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For further details details see [29], and Section 2 in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='27 (Theorem A in [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X = G/K be a symmetric space of non-compact type and of higher rank, and let Γ ≤ Isom(X) be an irreducible non-uniform lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then WQ(Γ) does not include horospherical limit points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The π 2 -neighbourhood N π 2 � WQ(Γ) � := {ξ ∈ X(∞) | dT � ξ, WQ(Γ) � < π 2 } does not include conical limit points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In Q-rank 1 , the converse statement also holds: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='28 (Theorem B in [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X = G/K be a symmetric space of non-compact type of higher rank and Γ ≤ Isom(X) be an irreducible non-uniform lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let V = {ξ ∈ X(∞) | dT � ξ, WQ(Γ) � ≥ π 2 } Suppose that Γ is of Q-rank 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then V consists of conical limit points only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In groups of R-rank 1 one has the following well known fact: Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='29 (Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 in [10], see also Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='29 in [17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a symmetric space of noncompact type and of R-rank 1, Γ ≤ Isom(X) a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then every ξ ∈ X(∞) is either conical or non-horospherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' When Γ is of Q-rank 1 , the following holds: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' WQ(Γ) = {ξ ∈ X(∞) | N π 2 (ξ) does not contain conical limit points} 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Any two points ξ, ξ′ ∈ WQ(Γ) are at Tits distance = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In view of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='29, for R-rank 1 both statements hold trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In higher rank, both follow from the following observation: for any point ξ′ ∈ WQ(Γ) and any point ζ ∈ X(∞) with d(ζ, ξ′) = π 2 , ζ is conical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To see this notice that ζ lies on the boundary of a horosphere based at ξ′: take a flat F ⊂ X with ξ′, ζ ∈ F(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Any geodesic with limit ζ is contained in (a Euclidean) horosphere based at ξ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that Γ is cocompact on the horospheres based at WQ(Γ) implies that ζ is conical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 14 The second item of the corollary follows: let ξ, ξ′ ∈ WQ(Γ) and c : [0, α] a Tits geodesic joining them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is a flat F ⊂ X containing both ξ, ξ′ as well as c ⊂ F(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Every point ζ that is at Tits distance π 2 from either ξ or ξ′ is conical, and no point inside the π 2 neighbourhood of either ξ or ξ′ is conical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In F(∞) the Tits metric is the same as the Tits metric on the Euclidean space of an equal rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore one may prolong the geodesic c so that c(0) = ξ, c(α) = ξ′ and c(π) = ξ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If dT (ξ, ξ′) < π, then there is a point along this prolonged geodesic that is at Tits distance exactly π 2 from ξ (so it is conical by the first paragraph), but at Tits distance strictly less than π 2 from ξ′ (so it cannot be conical by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore dT (ξ, ξ′) = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For the first item, one containment is just Hattori’s Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For the other containment, pretty much the same argument from above works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume for some ξ ∈ X(∞) that N π 2 (ξ) consists of non-conical limit points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular ξ itself is not conical, and by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='28 it holds that d(ξ, WQ(Γ)) < π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ξ′ ∈ WQ(Γ) be a point realizing this distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As above, this gives rise to a flat F containing ξ, ξ′ and another point ζ that is at Tits distance π 2 from ξ′ but at Tits distance strictly less than π 2 from ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The first forces ζ to be conical, and the latter forces it to be non-conical, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Hattori’s characterization relies on a simple lemma which will also be of use in the sequel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It relates the (linear) penetration rate of a geodesic into a horoball to the Tits distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='31 (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 in [25]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a symmetric space of higher rank and of noncompact type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let η1, η2 : [0, ∞) → X be two geodesic rays, α := dT � η1(∞), η2(∞) � and b2 the Busemann function corresponding to η2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then there exists a positive constant C1, depending only on η1 and η2, such that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If α > π 2 , then for all t ≥ 0 b2 � η1(t) � ≥ −t · cos α − C1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If α = π 2 , then b2 � η1(t) � is monotone non-increasing in t and −C1 ≤ b2 � η1(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If α < π 2 , then for all t ≥ 0 b2 � η1(t) � ≤ −t · cos α − C1 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If X is a symmetric spaces of R-rank 1, maximal flats are geodesics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore every two points ξ, ζ ∈ X(∞) admit dT (ξ, ζ) = π, and it is clear from the strict negative curvature that Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='31 is true also in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3 Uniform Lattices In this section I prove: Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a semisimple Lie group without compact factors and with finite centre.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let Γ ≤ G be a lattice, Λ ≤ G a discrete subgroup such that Γ ⊂ Nu(Λ) for some sublinear function u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ is uniform, then Λ is a uniform lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The focus of this paper is on sublinear distortion, however for uniform lattices (and also for lattices that have property (T), see Section 5), a slightly stronger result holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I call this ε-linear rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let f, g : R≥0 → R>0 be two monotonically increasing functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Call f asymptotically smaller than g if lim sup f g ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote this relation by f ⪯∞ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the setting of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1, the conclusion holds also under the relaxed assumption that u(r) ⪯∞ εr for any 0 < ε < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Clearly Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 implies Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From now and until the end of this section, the standing assumptions are those of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 15 Lattice Criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A discrete group is a uniform lattice if and only if it admits a relatively compact fundamental domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The criterion I use is the immediate consequence that if Γ is uniform and u is bounded (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Γ ⊂ ND(Λ) for some D > 0), then Λ is a uniform lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Outline of Proof and Use of ε-Linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The goal is to show that the ε-linearity of u forces Γ ⊂ ND(Λ) for some D > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' that Γ actually lies inside a bounded neighbourhood of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof is by way of contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If there is no such D > 0 then there are arbitrarily large balls that do not intersect Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof goes by finding such large Λ-free balls that are all tangent to some fixed arbitrary point x ∈ X (see Figure 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The ε-linearity then gives rise to concentric Γ-free balls that are arbitrarily large, contradicting the fact that Γ is a uniform lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The main difference from the non-uniform case is that for a non-uniform lattice Γ, the space X does admit arbitrarily large Γ-free balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This situation requires different lattice criteria and much extra work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Still the proof for the uniform case, though essentially no more than a few lines, lies the foundations for and presents the logic of the much more involved case of Q-rank 1 lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 Notations and Terminology Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a metric space, Y, Z ⊂ X two closed subsets of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The closest point projection of Y to Z is the set theoretic map pZ : Y → Z defined by pZ(y) := zy, where zy ∈ Z is any point realizing the distance d(y, Z) = d(y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If X is a proper metric space and Z discrete, then there are at most finitely many such points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In any case of multiple points, pZ chooses one arbitrarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The particular case of interest from now on is where the metric space is the pointed symmetric space (X, x0), the two subsets are the orbits Γ · x0 and Λ · x0, and the projection is pΛ·x0 : Γ · x0 → Λ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To ease notation I often denote this projection by pΛ (there is no risk of ambiguity since the subgroups Λ and Γ are always considered in the context of their respective orbits in X and not in G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The following definitions will be used repeatedly in both this section and in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It mainly fixes terminology and notation of the geometric situation illustrated in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let H ≤ G = Isom(X)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A set U ⊂ X is called H-free if H · x0 ∩ Int(U) = ∅, where Int(U) is the topological interior of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' That is, U is called H-free if its interior does not intersect the H-orbit H ·x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote PΛ(γx0) = PΛ(γ) = λγx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' dγ := d(γx0, λγx0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Bγ := B(γx0, dγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is a Λ-free ball centred at γx0 and tangent to λγx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' x′ γ := λ−1 γ γx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Notice |x′ γ| = dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' B′ γ := λ−1 γ Bγ = B(x′ γ, dγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is Λ-free as a Λ-translate of the Λ-free ball Bγ, and is tangent to x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For s ∈ R>0 and a ball B = B(x, r), denote sB := B(x, sr), the rescaled ball with same centre and radius sr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a sequence γn, denote by λn, dn, Bn, B′ n, x′ n the respective λγn, dγn, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There exists S = S(x, u) ∈ (0, 1) such that for every s ∈ (0, S) there is R = R(s, S) such that if r > R and B = B(y, r) is a Λ-free ball tangent to x, then sB is Γ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, the existence of arbitrarily large Λ-free balls that are all tangent to a fixed point x ∈ X implies the existence of arbitrarily large Γ-free balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 16 x0 x′ γ = dγ = s · dγ Γ − free Λ − free γx0 = dγ Λ − free λγx0 Lλ−1 γ Figure 1: Basic Setting and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A Λ-free ball about γx0 of radius dγ, translated by λ−1 γ to a ball tangent to x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The linear ratio between |x′ γ| = dγ and the Λ-free radius forces the red ball to be Γ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is a slightly stronger version of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8 if u is sublinear: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G, Γ, Λ and u be as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 (in particular, u is a sublinear function and Γ ⊂ Nu(Γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For every x ∈ X and every s ∈ (0, 1) there exists R = R(x, s) > 0 such that for every r > R, if B = B(y, r) is a Λ-free ball tangent to x then sB is Γ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I omit the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9, which is a slightly simpler version of the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof is more easily read if one assumes x = x0 and u(r) = εr so I begin with this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume B = B(y, R) is Λ-free for some y ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The assumption x = x0 means that |y| = d(y, x0) = r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume that for a given s ∈ (0, 1), the ball sB intersects Γ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This gives rise to an element γ ∈ Γ such that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' |γ| = d(γx0, x0) ≤ d(y, x0) + d(γx0, y) = (1 + s)r (triangle inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, d(γx0, Λ · x0) ≤ ε(1 + s)r 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' B � γx0, (1 − s)r � ⊂ B(y, r), so it is Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that for s for which sB ∩ Γ · x0 ̸= ∅, one has the inequality (1 − s)r ≤ ε(1 + s)r, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1−s 1+s ≤ ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The number ε is fixed and smaller than 1, while 1−s 1+s limit to 1 monotonically from below as s > 0 tend to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that there is a segment (0, S) ⊂ (0, 1) such that for all s ∈ (0, S), sB is Γ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume now that x ̸= x0 and u(r) ⪯∞ εr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As above, if γx0 ∈ B(y, sr) then it is the centre of a Λ-free ball of radius (1 − s)r, and so (1 − s)r ≤ u(|γ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I wish to use the ε-linear bound on u as I did before, only this time u is only asymptotically smaller than εr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To circumvent this I just need to show that |γ| is large enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Indeed since B � γx0, (1 − s)r � is Λ-free it does not contain x0 ∈ Λ · x0 and in particular (1 − s)r ≤ d(x0, γx0) = |γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For some R1(s) = R1(s, u) one therefore has for all r > R1(s) (1 − s)r ≤ u(|γ|) ≤ ε|γ| On the other hand |y| ≤ d(x, y) + d(x, x0) = r + |x|, and consequently |γ| ≤ (1 + s)r + |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For r > R1(s) one has 17 (1 − s)r ≤ u(|γ|) ≤ ε|γ| ≤ ε � (1 + s)r + |x| � This means that s for which Γ · x0 ∩ B(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' sr) ̸= ∅ must admit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' for all r > R1(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1 − s 1 + s + |x| r = (1 − s)r (1 + s)r + |x| ≤ ε < 1 (1) The rest of the proof is just Calculus 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' and concerns with finding S = S(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ε,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' u) ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1) so that for any s ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' S) there is R(s) such that all r > R(s) satisfy ε < 1 − S 1 + S + |x| r ≤ 1 − s 1 + s + |x| r (2) The lemma readily follows from inequalities 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Explicitly, fix ε′ > ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As before, monotonic approach of 1−s 1+s to 1 allows to fix S ∈ (0, 1) for which ε < ε′ < 1−s 1+s for all s ∈ (0, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Next note that for any fixed s ∈ (0, S), limr→∞ 1−s 1+s+ |x| r = 1−s 1+s, and that the approach in monotonically increasing with r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since ε < ε′, this limit implies that for some R2 > R1(S), all r > R2 admit ε < 1−S 1+S+ |x| r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally notice that for any fixed r the function 1−s 1+s+ |x| r is again monotonically increasing as s tends to 0 from above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore inequality 2 holds for every s ∈ (0, S) and all r > R2(S) (capital S is intentional and important).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To conclude the proof, notice that if moreover r > R1(s) (again lowercase s is intentional and important) then inequalities 1,2 both hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This means that for any s ∈ (0, S) there is R(s) := max{R1(s), R2(S)} such that r > R(s) ⇒ B(y, sr) is Γ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The constants R1(s), R2(S) have the desired dependencies, hence so does R(s), proving the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 (Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is a uniform bound on {dγ}γ∈Γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', Γ ⊂ ND(Λ) for some D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, Λ is a uniform lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4 Q-rank 1 Lattices In this section I prove: Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a semisimple Lie group without compact factors and with finite centre, Γ ≤ G an irreducible non-uniform Q-rank 1 lattice, Λ ≤ G a discrete irreducible subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ ⊂ Nu(Λ) for some sublinear function u, then Λ is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, if Γ ̸⊂ ND(Λ) for any D > 0, then Λ is also of Q-rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ ⊂ ND(Λ) for some D > 0, then Λ could be a uniform lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' An obvious obstacle for that is if Λ ⊂ Nu′(Γ) for some sublinear function u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This condition turns out to be sufficient for commensurability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a semisimple Lie group without compact factors and with finite centre, Γ ≤ G an irreducible non-uniform Q-rank 1 lattice, Λ ≤ G a discrete subgroup such that Γ ⊂ ND(Λ) for some D > 0, and Λ ⊂ Nu(Γ) for some sublinear function u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then Λ ⊂ ND′(Γ) for some D′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As a result of Eskin’s and Schwartz’s arguments (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3), I conclude: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the setting of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 and unless G is locally isomorphic to SL2(R), Λ is com- mensurable to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8 is a result of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 Strategy Lattice Criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I use three different lattice criteria, depending on the R-rank of G and on whether or not Γ ⊂ ND(Λ) for some D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' My proof for Q-rank 1 lattices is motivated by a criterion of Benoist and Miquel, resolving a conjecture of Margulis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It can be viewed as an algebraic converse to the geometric structure of the compact core described in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='16 in [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a semisimple real algebraic Lie group of real rank at least 2 and U be a non-trivial horospherical subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ∆ be a discrete Zariski dense subgroup of G that contains an indecomposable lattice ∆U of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then ∆ is a non-uniform irreducible arithmetic lattice of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' See Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='47 for the precise meaning of an indecomposable horospherical lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For R-rank 1 groups, one has the following theorem by Kapovich and Liu, stating that a group is geometrically finite so long as ‘most’ of its limit points are conical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall L(∆) is the limit set of ∆ ≤ Isom(X), and Lcon(∆) is the set of its conical limit points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5 in [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a R-rank 1 symmetric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A discrete subgroup ∆ ≤ Isom(X) is geometrically infinite if and only if the set L(∆) \\ Lcon(∆) of non-conical limit points has the cardinality of the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As a direct corollary I obtain the following criterion: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a R-rank 1 symmetric space, Γ ≤ G = Isom(X) a non-uniform lattice and Λ ≤ G a discrete subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If L(Λ) = X(∞) and Lcon(Γ) ⊂ Lcon(Λ), then Λ is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since Γ is a lattice, L(Γ) = X(∞) and it is geometrically finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 implies the cardinality of X(∞) \\ Lcon(Γ) is strictly smaller than the continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The assumption Lcon(Γ) ⊂ Lcon(Λ) implies the same holds for Λ, and in particular that Λ is geometrically finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The assumption that L(Λ) = X(∞) implies that Λ is geometrically finite if and only if it is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a R-rank 1 symmetric space, Γ ≤ G = Isom(X) a non-uniform lattice and Λ ≤ G a discrete subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ ⊂ ND(Λ) for some D > 0, then Λ is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By definition of the limit set and of conical limit points, it is clear that every Γ-limit point is a Λ-limit point, and every Γ-conical limit point is also Λ-conical limit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 that Λ is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Also in higher rank the inclusion Γ ⊂ ND(Λ) implies that Λ is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This result is due to Eskin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 (Eskin, see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a semisimple Lie group without compact factors and of higher rank, Γ ≤ G an irreducible non-uniform lattice, Λ ≤ G a discrete subgroup such that Γ ⊂ ND(Λ) for some D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then Λ is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 was used in the proof of quasi-isometric rigidity for higher rank non-uniform lattices in [16], [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the (earlier) R-rank 1 case, Schwartz [52] used an analogous statement, which requires one extra assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 (Schwartz, see Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 in [52] and Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real simple Lie group of R-rank 1 and with finite centre, Γ ≤ G an irreducible non-uniform lattice, Λ ≤ G a discrete subgroup such that both Γ ⊂ ND(Λ) and Λ ⊂ ND(Γ) for some D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then Λ is a lattice and commensurable to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 should be viewed as a generalization of the bounded case depicted in Theo- rems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10, which were known to experts in the field in the late 1990’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Complete proofs for these statements were never given in print, and I take the opportunity to include them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3, where I also prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I thank Rich Schwartz and Alex Eskin for supplying me with their arguments and allowing me to include them in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I also thank my thesis examiner Emmanuel Breuillard for encouraging me to find and make these proofs public.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 19 Outline of Proof and Use of Sublinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lattices of Q-rank 1 admit a concrete geometric structure (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This structure is manifested in the geometry of an orbit of such a lattice in the corresponding symmetric space X = G/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One important geometric property is the existence of a set of horoballs which the orbit of the lattice intersects only in the bounding horospheres, and in each such horosphere the orbit forms a (metric) cocompact lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 reduce the proof to the case where Γ ̸⊂ ND(Λ) for any D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In that case, the essence lies in proving the existence of horospheres in X which a Λ-orbit intersects in a cocompact lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is proved purely geometrically, using the geometric structure of Q-rank 1 lattices and the sublinear constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Together with some control on the location of these horospheres, I prove two major statements: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Λ · x0 intersects a horosphere H ⊂ X in a cocompact lattice (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Every Γ-conical limit point is also a Λ-conical limit point (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The R-rank 1 case of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 follows directly from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 using the second item above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The higher rank case requires a bit more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' namely it requires to deduce from the above items that Λ meets the hypotheses of the Benoist-Miquel Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To that end I use a well known geometric criterion (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='50) in order show that Λ is Zariski dense, and a lemma of Mostow (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='36) to show that Λ intersects a horospherical subgroup in a cocompact lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Outline for Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 is the core of the original mathematics of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is devoted to proving that Λ · x0 intersects some horospheres in a cocompact lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof is quite delicate and somewhat involved, and I include a few figures and a detailed informal overview of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The figures are detailed and may take a few moments to comprehend, but I believe they are worth the effort.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 deals with the case where Γ ⊂ ND(Λ), and elaborates on Schwartz’s and Eskin’s proofs of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 is devoted to the translation of the geometric results of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 to the algebraic language used in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Though the work is indeed mainly one of translation, some of it is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally, in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5 I put everything together for a complete proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I highly recommend the reader to have a look at the uniform case in Section 3 before reading this one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 A Λ-Cocompact Horosphere Recall that dγ := d(γx0, λγx0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In this section I prove: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If {dγ}γ∈Γ is unbounded, then there exists a horosphere H based at WQ(Γ) such that � Λ∩StabG(H) � x0 intersects H in a cocompact metric lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, the bounded horoball HB is Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Throughout Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 the standing assumptions are that {dγ}γ ∈ Γ is unbounded, and Γ is an irreducible Q-rank 1 lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The Argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof is by chasing down the geometric implications of unbounded dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' These implications are delicate, but similar in spirit to the straight-forward proof for uniform lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof consists of the following steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Unbounded dγ results in Λ-free horoballs HBΛ tangent to Λ-orbit points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Each such horoball is based at WQ(Γ), giving rise to corresponding horoballs of Γ, denoted HBΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If dγ is large, then γx0 must lie deep inside a unique Λ-free horoball tangent to λγx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I use: (a) A bound on the distance d(HΛ, HΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' (b) A bound on the angle ∠λγx0([λγx0, γx0], [λγx0, ξ)), where ξ ∈ X(∞) is the base point of a suitable Λ-free horoball tangent to λγx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There exist horospheres of Γ, say HΓ, such that if γx0 ∈ HΓ then large dγ implies large Λ-free areas along the bounding horosphere of some HBΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If HBΛ is boundedly close to some Λ-orbit point, then HΛ is almost Λ-cocompact, that is HΛ ⊂ ND(Λ·x0) for some universal D = D(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Together with the previous step, this yields a uniform bound on dγ along certain horospheres of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally I elevate the almost cocompactness to actual cocompactness and show HΛ ⊂ ND(Λ · x0 ∩ HΛ) for some D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This immediately elevates to HΛ ⊂ (Λ ∩ StabG(HΛ)) · x0, proving the proposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The Properties of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The geometric properties of Γ that are used in the proof are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In higher rank, the characterization of WQ(Γ) using conical / non-horospherical limit points (Corol- lary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In R-rank 1, the dichotomy of limit points being either non-horospherical or conical (The- orem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Γ-cocompactness along the horospheres of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For every point x ∈ X and C > 0 there is a bound K(C) on the number of horospheres of Γ that intersect B(x, C) (Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 Λ-Free Horoballs I retain the notations and objects defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There exists a Λ-free horoball tangent to x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since {dγ}γ∈Γ is unbounded, there are γn ∈ Γ with dn = dγn = d(γn, λn) → ∞ monotonically, where λn ∈ Λ is a Λ-orbit point closest to γn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote x′ n = λ−1 n γnx0, ηn := [x0, x′ n], and vn ∈ Sx0X the initial velocity vectors vn := ˙ηn(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The tangent space Sx0X is compact, so up to a subsequence, vn converge monotonically in angle to a direction v ∈ Sx0X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let η be the unit speed geodesic ray emanating from x0 with initial velocity ˙η(0) = v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote ξ := η(∞) the limit point of η in X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I claim that the horoball HB := ∪t>0B � η(t), t � , based at ξ and tangent to x0, is Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let t > 0 and consider η(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For every ε > 0, there is some angle α = α(t, ε) such that any geodesic η′ with ∠x0(η, η′) < α admits d � η(t), η′(t) � < ε/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The convergence vn → v implies d � η(t), ηn(t) � < ε/2 for all but finitely many n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, B �� η(t), t � ⊂ Nε � B � ηn(t), t �� for all such n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a fixed t ≤ dn, it is clear from the definitions that B � ηn(t), t � ⊂ B′ n = B(x′ n, dn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One has dn → ∞, and so for a fixed t > 0 it holds that t < dn for all but finitely many n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that for any fixed t > 0 there is n large enough such that B � η(t), t � ⊂ Nε � B � ηn(t), t �� ⊂ NεB′ n I conclude that for every ε > 0, HB ⊂ � n Nε(B′ n) = Nε � � n B′ n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This implies that any point in the interior of HB is contained in the interior of one of the Λ-free balls B′ n, proving HB is Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Suppose HB is a Λ-free horoball, based at some point ξ ∈ X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then ξ ∈ WQ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For any geodesic η with limit ξ, the size d(x0, γx0) of the Γ-orbit points γx0 that lie boundedly close to η grows linearly in the distance to any fixed horosphere based at ξ, and in particular to H = ∂HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The sublinear constraint d(γx0, λγx0) ≤ u(|γ|) together with the fact that HB is Λ-free imply that the size of such γ is bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In R-rank 1 every limit point is either conical or in WQ(Γ), proving the lemma in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For higher rank, the above argument actually shows more: it shows that a point ξ′ ∈ N π 2 (ξ) is not conical, because every geodesic with limit ξ′ ∈ N π 2 (ξ) entres HB at a linear rate (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Hattori’s characterization of WQ(Γ) (Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='30) implies ξ ∈ WQ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 21 Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Given a Λ-free horoball HBΛ, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='14 gives rise to a horoball of Γ based at the same point at X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Call this the horoball corresponding to HBΛ, and denote it by HBΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The corresponding horosphere is denoted HΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the course of my work I had had a few conversations with Omri Solan regarding the penetration of geodesics into Λ-free horoballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assuming Λ ⊂ Nu(Γ) implies that Λ preserves WQ(Γ) (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is the case in the motivating setting where Λ is an abstract finitely generated group that is SBE to Γ, see Claim 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 in Chapter 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the case of SL2(R) Omri suggested to use the action of Λ on the Bruhat-Tits tree of SL2(Qp) (for all primes p) and the classification of these elements into elliptic and hyperbolic elements (separately for each p) in order to deduce that Λ actually lies in SL2(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' We did not pursue that path nor its possible generalization to the SLn case and general Bruhat-Tits buildings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 A Γ-orbit point Lying Deep Inside a Λ-Free Horoball I established the existence of Λ-free horoballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It may seem odd that the first step in proving Λ · x0 is ‘almost everywhere’ is proving the existence of Λ-free regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' But this fits perfectly well with the algebraic statement that non-uniform lattices must admit unipotent elements (see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 in [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The goal of this section is to obtain some control on the location of the Λ-free horoballs, in order to conclude that some γx0 lies deep inside HBΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This results in yet more Λ-free regions, found on the bounding horosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I need one property of sublinear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I thank Panos Papazoglou for noticing a mistake in the original formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let u be a sublinear function, f, g : R≥0 −→ R>0 two positive monotone functions with limx→∞ f(x) + g(x) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If for all large enough x it holds that f(x) ≤ u � f(x) + g(x) � , then for every 1 < s and all large enough x it holds that f(x) ≤ u � s · g(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular f(x) ≤ u′� g(x) � for some sublinear function u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume as one may that u is non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By definition of sublinearity limx→∞ u � f(x)+g(x) � f(x)+g(x) = 0, so by hypothesis limx→∞ f(x) f(x)+g(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This means that for every ε > 0 one has f(x) ≤ ε · g(x) for all large enough x, resulting in f(x) ≤ u � f(x) + g(x) � ≤ u � (1 + ε) · g(x) � Notice that for a fixed s > 0, the function u′(x) = u(sx) is sublinear, as lim x→∞ u(sx) x = lim x→∞ s · u(sx) sx = 0 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let C > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is L = L(C) such that if HBΛ is any Λ-free horoball tangent to a point x ∈ B(x0, C) then d(HΛ, HΓ) ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, there is a sublinear function u′ such that: L(C) ≤ � u′(C) if HBΓ ⊂ HBΛ C if HBΛ ⊂ HBΓ Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If HBΛ ⊂ HBΓ, then clearly d(HΛ, HΓ) ≤ C, simply because HBΓ is Γ-free and in particular cannot contain x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore HΓ must separate HΛ from x0 and in particular d(HΛ, HΓ) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume that HBΓ ⊂ HBΛ, and denote l = d(HΛ, HΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The horoball HBΓ is a horoball of Γ, hence Γ · x0 is DΓ-cocompact along HΓ and there is an element γ ∈ Γ with |γ| ≤ C + l + DΓ and γx0 ∈ HΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since HBΛ is Λ-free one has l ≤ d(γx0, λγx0) ≤ u(|γ|) ≤ u(C + l + DΓ) C, DΓ are fixed, so this inequality can only occur for boundedly small l, say l < L′(C) (DΓ is a universal constant and may be ignored).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Consult figure 2 for a geometric visualization of this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 22 Rad = C x0 Λ − free HBΛ ξ Γ − free HBΓ ≤ L(C) γx0 PHΓ(x0) ≤ L(C) + DΓ Figure 2: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A Λ-free horoball HBΛ intersects a ball of radius C about x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The associated Γ-free horoball HBΓ is boundedly close, essentially due to the uniform cocompactness of Γ · x0 along the Γ horospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It remains to show that L′(C) is indeed sublinear in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' First define L(C) to be the minimal L that bounds the distance d(HΛ, HΓ) for all possible HBΛ tangent to a point x ∈ B(x0, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is indeed a minimum, since by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='22 there are only finitely many horoballs of Γ intersecting B � x0, C + L′(C) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For every C there is thus a horoball HBΓ C and an element γ ∈ Γ such that γx0 ∈ HΓ, d(HΛ C, HΓ C) = L(C) and |γ| ≤ C + L(C) + DΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that HBΛ C is Λ-free implies L(C) = d(HΛ C, HΓ C) ≤ u(|γ|) ≤ u � C + L(C) + DΓ � Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='17 implies that L(C) ≤ u′(C) for some sublinear function u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The following is an immediate corollary, apparent already in the above proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For every C > 0 there is a bound K = K(C) and a fixed set ξ1, ξ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ξK ∈ WQ(Γ) ⊂ X(∞) so that every Λ-free horoball HBΛ which is tangent to some point x ∈ B(x0, C) is based at ξi for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', K}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, for any specific point x ∈ NC(Λ · x0) there are at most K Λ-free horoballs tangent to x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let HBΛ be a horoball tangent to a point x ∈ B(x0, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='18 bounds d(HBΛ, HBΓ) by L(C), hence HBΓ is tangent to a point x′ ∈ B � x0, C + L(C) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='22 there are only finitely many possibilities for such HBΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular there are finitely many base-points for these horoballs, say ξ1, ξ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' , ξK(C) ∈ WQ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally, recall that a horoball is determined by a base point and a point x ∈ X tangent to it, so the last statement of the corollary holds for any x ∈ B(x0, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' But the property in question is Λ-invariant so the same holds for any point x ∈ Λ · B(x0, C) = NC(Λ · x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The bound on d(HBΛ, HBΓ) given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='18 further strengthen the relation between HBΛ and HBΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The ultimate goal is to show that the HBΛ-s play the role of the Γ-horoballs in the geometric structure of Q-rank 1 lattices, namely to show that Λ · x0 is cocompact on the HΛ-s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This requires to actually find Λ-orbit points somewhere in X, and not just Λ-free regions as was done up to now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As one might suspect, these points arise as λγx0 corresponding to points γx0 ∈ HΓ, which exist in abundance since Γ · x0 ∩ HΓ is a cocompact lattice in HΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The hope is that a Λ-free horoball HBΛ tangent to λγx0 would correspond to a horoball of Γ tangent to γx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This would have forced all the λγ to actually lie on the same bounding horosphere, and {λγx0 | γx0 ∈ HΓ} would then be a cocompact lattice in HΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This hope turns out to be more or less true, but it requires 23 some work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The goal of the rest of this section is to establish a relation between a Λ-free horoball HBΛ tangent to λγx0 and γx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I start with some notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In light of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='19, there is a finite number N of Λ-free horoballs tangent to x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' {HBΛ i }N 1=1 are the Λ-free horoballs tangent to x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ξi ∈ WQ(Γ) is the base point of HBΛ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' vi ∈ Sx0X is the unit tangent vector in the direction ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ηi := [x0, ξi) is the unit speed geodesic ray emanating from x0 with limit ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular vi = d dtηi(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' HBΓ i is the horoball of Γ that corresponds to HBΛ i , based at ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' HBΛ λ,i, ξi λ, ηi λ are the respective λ-translates of the objects above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For example, HBΛ λ,i := λ · HBΛ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' H decorated by the proper indices denotes the horosphere bounding HB, the horoball with respective indices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' HΛ i := ∂HBΛ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For an angle α > 0 and a tangent vector v0 ∈ SxX, define (a) The α-sector of v in SxX is the set {v ∈ SxX | v ∈ Nα(v0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall that the metric on SxX is the angular metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' (b) The α-sector of v in X are all points y ∈ X for which the tangent vector at 0 of the unit speed geodesic [x, y] lies in the α-sector of v in SxX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For every angle α ∈ (0, π 2 ) there exists D = D(α) such that if dγ > D then for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' , N}, γx0 lies inside the α-sector of vi λγ at λγx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Furthermore whenever α is uniformly small enough, there is a unique such i = i(γ), independent of α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Translation by the isometry λ−1 γ preserves angles and distances, so it is enough to prove that there is an i for which x′ γ := λ−1γx0 lies inside the α-sector of vi, and that this i is unique if α is uniformly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume towards contradiction that there is α ∈ (0, π 2 ) and a sequence γn ∈ Γ, λn := λγn ∈ Λ with dγn unbounded, and x′ n := λ−1 n γnx0 not lying in the union of the α-sectors of vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By perhaps taking smaller α I may assume all the α-sectors of the vi in Sx0X are pairwise disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This can be done because there are only finitely many vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Compactness of Sx0X allows me to take a converging subsequence v′ n := ˙ [x0, x′n], with limit direction v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote by η′ the geodesic ray emanating from x0 with initial velocity v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The exact same argument of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='13 proves that η′(∞) is the base point of a Λ-free horoball tangent to x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' But this means v′ = vi for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' , N}, contradicting the fact that all v′ n lie outside the α-sectors of the vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This proves that there is a bound D = D(α) such that if dγ > D then x′ γ lies within the α-sector of some vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof clearly shows that whenever α is small enough so that the α-sectors of the vi are disjoint, x′ γ lies in the α-sector of a unique vi as soon as dγ > D(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='13 I used compactness of Sx0X to induce a converging subsequence of directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='21 actually shows that the fact there are finitely many Λ-free horoballs tangent to x0 implies a posteriori that there was not much choice in the process - all directions [x0, x′ γ] must fall into one of the finitely many directions vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Next, I want to control the actual location of certain points with respect to the horoballs of interest, and not just the angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This turns out to be a more difficult of a task than one might suspect, since control on angles does not immediately give control on distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall that large Λ-free balls near x0 imply large concentric Γ-free balls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The precise quantities and bounds are given by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8 (one can use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 to obtain a slightly cleaner statement).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 24 Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let S ∈ (0, 1) be the constant given by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8, and let s ∈ (0, S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is a bound D = D(s) such that dγ > D implies that γx0 lies sdγ deep in HBΛ λγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof is a bit delicate but very similar to that of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In essence, I use the Γ-free balls near x0 to produce a Γ-free cylinder, which would force a certain geodesic not to cross a horosphere of Γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' force it to stay inside a Γ-free horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='21 it is only required to show that x′ = λ−1 γ γx0 is sdγ deep inside HBΛ i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I start with proving that x′ ∈ HBΓ i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I learned the hard way that even this is not a triviality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall the notation Bγ = B(γx0, dγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The ball B′ γ = λ−1 γ Bγ is a Λ-free ball of radius dγ about x′ = λ−1 γ γx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote by x′ t the point at time t along the unit speed geodesic η′ := [x0, x′].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It holds that |x′ t| = t and, for t ≤ dγ, x′ t is the centre of a Λ-free ball of radius t tangent to x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The constant s is fixed and by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8 there is T ′ = T ′(s) such that if t > T ′, the ball sB � x′ t, t � is Γ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The next goal is to show that x′ T ∈ HBΓ for some adequate T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For any time T > 0, let α = α(ε, T ) be the angle for which d � η(T ), ηi(γ)(T ) � < ε for every η in the α-sector of vi(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By perhaps taking smaller α I may assume that α is uniformly small as stated in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let D(α) be the bound given by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='21 guaranteeing D(α) < dγ ⇒ d � x′ T , ηi(γ)(T ) � < ε For my needs in this lemma ε may as well be chosen to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I now choose a specific time T for which I want x′ T and ηi(γ)(T ) to be close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There are only finitely many Λ-free horoballs {HBΛ i }i∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=',N} tangent to x0, giving rise to a uniform bound L = maxi∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=',N}{d(HΛ i , HΓ i )} on the distance d(HΛ i(γ), HΓ i(γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix T to be any time in the open interval (T ′ + L + ε, dγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that L + ε < T implies that ηi(γ)(T ) lies at least ε-deep inside HBΓ i(γ), and therefore η′(T ) ∈ HBΓ i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall that any point on HΓ is DΓ-close to a point γx0 ∈ HΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By perhaps enlarging T and shrinking α if necessary, I may assume that DΓ < sT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Thus for all T < t ≤ dγ, x′ t is the centre of a Γ-free ball of radius st > sT > DΓ, hence {x′ t}T ≤t≤dγ does not cross a horosphere of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since x′ T ∈ HBΓ i(γ), this implies that x′ t stays in HBΓ i(γ) for all T < t ≤ dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular x′ dγ = x′ ∈ HBΓ i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To get the result of the proposition, recall that sB′ γ = B(x′, sdγ) is Γ-free, so x′ must be at distance at least sdγ − DΓ from any horosphere of Γ, and in particular from HΓ i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In terms of Busemann functions, this means that bηi(γ)(x′) ≤ −sdγ + DΓ whenever one can find such T ′ + L + ε < T < dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since HBΛ i(γ) is tangent to x0, the corresponding horoball HBΓ i(γ) lies inside it, and so x′ lies (sdγ − DΓ)-deep inside HBΛ i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A close look at the argument yields the desired bound D = D(s) such that the above holds whenever dγ > D(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To help the reader take this closer look, I reiterate the choice of constants and their dependencies as they appear in the proof: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let T ′ = T ′(s) the constant from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8 and L = maxi∈{1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=',N}{d(HBΛ i , HBΓ i )}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix T > T ′ + L + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix α = α(1, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix D(s) = max{D(α), T + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I remark, for the reader worried about the DΓ which appears in the final bound but not in the statement, that (a) DΓ is a fixed universal constant and may as well be ignored, and (b) the discrepancy can be formally corrected by taking a slightly larger s < s′ to begin with and as a result perhaps enlarging the bound D for dγ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Also note that L = L(Λ) is a universal constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 Intersection of Λ-Free Regions and the Existence of a Λ-Cocompact Horosphere In this section I find Λ-orbit points that lie close to the bounding horosphere of a Λ-free horoball HBΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In order to find such points I need to make sure HBΛ is not contained inside a much larger Λ-free horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I introduce the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A Λ-free horoball HBΛ is called maximal if it is tangent to a point x = λx0 ∈ Λ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is called ε-almost maximal if d(Λ · x0, HΛ) < ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It may happen that a discrete group admits free but not no maximally free horoballs - see discussion in section 4 of [55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In any case it is clear that any Λ-free horoball can be ‘blown-up’ to an ε-almost maximal Λ-free horoball, for every ε > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, every two ε-almost maximal horoballs based at the same point ξ ∈ WQ(Γ) lie at distance at most ε of one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For my needs any fixed ε would suffice, and I fix ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is DΛ > 0 such that if HBΛ is 1-almost maximal Λ-free horoball then HΛ ⊂ NDΛ(Λ·x0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' d(x, Λ · x0) ≤ DΛ for all x ∈ HΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Notice that Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='26 does not state Λ · x0 even intersects HΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I start with a short sketch of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Consider a 1-maximal horoball and a point x on its bounding horosphere with d(x, Λ · x0) = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One may translate this situation to x0, which results in a Λ-free horoball HBΛ intersecting the (closed) D-ball about x0 at a point w with B(w, D) Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof differs depending on whether HBΓ ⊂ HBΛ or the other way round, since I use the bounds from 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='18: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If HBΓ ⊂ HBΛ, there is a sublinear bound on d(HBΛ, HBΓ), which readily yields a bound on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' if HBΛ ⊂ HBΓ there is a bound on d(x0, HBΓ) that is independent of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' So there are only finitely many possibilities for HBΓ, independent of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Hence there are only finitely many possible base points for HBΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' These in turn correspond to possible base points for such HBΛ, and this finiteness yields a bound on the distance d(HBΓ, HBΛ) < L that is independent of D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The rest of the proof is quite routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let HBΛ be a 1-almost maximal Λ-free horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By definition there is λ ∈ Λ and z ∈ HΛ such that d(λx0, z) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I show that if there is some z′ ∈ HΛ for which d(z′, Λ · x0) ≥ D, then D must be uniformly small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Exactly how small will be set in the course of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix D > 1 and assume that there is z′ ∈ HΛ with d(z, Λ · x0) ≥ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Up to sliding z′ along HΛ, the continuity of the function x �→ d(x, Λ · x0) together with Intermediate Value Theorem allows to assume that d(z′, Λ · x0) = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let λ′ ∈ Λ be the element for which d(z′, λ′x0) = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Translating by λ′−1 yields 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A Λ-free horoball HBΛ 0 := λ′−1HBΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A point w := λ′−1z′ ∈ HΛ 0 for which |w| = d(w, x0) = d(w, Λ · x0) = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume first that HBΓ 0 ⊂ HBΛ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='18 there is a sublinear function u′ such that d(HΓ 0 , HΛ 0 ) ≤ u′(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This yields a point γx0 ∈ HΓ 0 for which d(w, γx0) ≤ u′(D) + DΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Thus |γx0| ≤ D + u′(D) + DΓ and the reverse triangle inequality gives D − � u′(D) + DΓ � ≤ d(w, λγx0) − d(w, γx0) < d(γx0, λγx0) Together with the bound d(γx0, λγx0) ≤ u(|γx0|) and rearranging, one obtains D ≤ u � D + u′(D) + DΓ � + u′(D) + DΓ The right hand side is clearly a sublinear function in D, hence this inequality may hold only for boundedly small D, say D < D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that HBΓ 0 ⊂ HBΛ 0 may occur only when D < D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Notice that D1 depends only on u and u′, and not on HBΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 26 w x0 Λ − free B(w, D) τ(t0) τ Λ − free HBΛ ξ Γ − free HBΓ γx0 Figure 3: Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='26, case HBΛ ⊂ HBΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The red horosphere of Γ is trapped between x0 and HΛ, and is at distance t0 from x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A Γ-orbit point on the red horosphere close to x0 allows to use sublinearity to get a bound on t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume next that HBΛ 0 ⊂ HBΓ 0 , and that the containment is strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since x0 ∈ Γ · x0, the geodesic τ := [x0, w] is of length D and intersects HΓ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote by t0 ∈ [0, D) the time in which τ intersects HΓ 0 , and let w′ := τ(t0) ∈ HΓ 0 be the intersection point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular |w′| = t0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is clear that B(w′, t0) is Λ-free, as a subset of the ball B(w, D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Again there is γx0 ∈ B(w′, DΓ) ∩ HΓ 0 and so |γx0| ≤ t0 + DΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By reverse triangle inequality t0 − DΓ ≤ d(w′, λγx0) − d(w′, γx0) ≤ d(γx0, λγx0) and the sublinear constraint gives t0 − DΓ ≤ u(t0 + DΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This can only happen for boundedly small t0, say t0 < T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that if HBΛ 0 ⊂ HBΓ 0 , then HBΓ 0 is a horoball of Γ tangent to some point y ∈ B(x0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='22 there are finitely many horoballs of Γ tangent to points in B(x0, T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular there is a finite set {ξ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' , ξ′ K} ∈ WQ(Γ) of possible base points for HBΓ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This set depends only on T , and since the choice of T was completely independent of D, the set of possible base points is independent of D as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let � HBΓ i be the horoball of Γ based at ξ′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I can now bound the distance d(HBΓ 0 , HBΛ 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let 1 ≤ i ≤ K be an index for which there is a Λ-free horoball based at ξ′ i that is contained in � HBΓ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is thus some 1-almost-maximal Λ-free horoball based at ξ′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix an arbitrary such 1-almost-maximal Λ-free horoball � HBΛ i for each such i, and let Li := d(� HBΛ i , � HBΓ i ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally, define L := max{Li} + 1 among such i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As stated in Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='25, d(HBΛ 0 , � HBΛ i ) ≤ 1 for some i, therefore d(HBΓ 0, HBΛ 0 ) ≤ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall |w| = D and B(w, D) is Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It holds that d(w, HΓ 0 ) ≤ L, and so there is γx0 ∈ HΓ 0 for which d(w, γx0) ≤ L + DΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular |γx0| ≤ D + L + DΓ (in fact it is clear that |γx0| ≤ T + DΓ, but this won’t be necessary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Reverse triangle inequality gives D − (L + DΓ) ≤ d(w, λγx0) − d(w, γx0) ≤ d(γx0, λγx0) and from the sublinear constraint I conclude D − (L + DΓ) ≤ u(D + L + DΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since L, DΓ are fixed constants independent of D, this can only hold for boundedly small D, say D < D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, one gets a uniform bound DΛ := max{D1, D2} such that x ∈ HΛ ⇒ d(x, Λ · x0) < DΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Every Γ-conical limit point is a Λ-conical limit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 27 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ξ ∈ X(∞) be a Γ-conical limit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let η : R≥0 → X be a geodesic with η(∞) = ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By definition there is a bound D > 0 and sequences tn → ∞, γn ∈ Γ such that d � γnx0, η(tn) � < D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Consider the corresponding λn := λγn and λnx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If dn is uniformly bounded, then ξ is Λ-conical by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Otherwise, assume dn is monotonically increasing to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For some fixed s ∈ (0, 1) it holds that for all but finitely many n ∈ N, γnx0 is sdn deep inside HBΛ n := HBΛ λn,i(γn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I assume dn is large enough so that sdn > D, and in particular η(tn) ∈ HBΛ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ξn ∈ WQ(Γ) be the respective base points of HBΛ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The point ξ is Γ-conical, and by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='28 π 2 ≤ d(ξ, WQ(Γ)) ≤ d(ξ, ξn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof differs depending on whether the above inequality is strict or not for any n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume first that for some m ∈ N, d(ξ, ξm) = π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By item 2 of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='31, d � HΛ m, η(t) � is uniformly bounded, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', there is C > 0 such that for every t > 0 there is xt ∈ HΛ m for which d � xt, η(t) � < C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='26, d(xt, Λ · x0) < DΛ, hence d � η(tn), xtn � ≤ C + DΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This means that ξ is Λ-conical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Otherwise, for all n ∈ N it holds that π 2 < d(ξ, ξn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that η(tn) ∈ HBΛ n together with Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='31 implies that at some later time the geodesic ray η leaves HBΛ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Thus there is sn > tn for which η(sn) ∈ HΛ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since HBΛ n are maximal Λ-free horoballs, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='26 gives rise to points λnx0 such that d � λnx0, η(sn) � ≤ DΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This renders ξ as a Λ-conical limit point, as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I now prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The strategy is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For HBΛ = HBΛ λγ,i(γ), one uses Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='23 to get that HBΓ ⊂ HBΛ and that the distance d(HΛ, HΓ) is large with dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The horosphere HΓ admits a Γ-cocompact metric lattice, and so the projections of these metric lattice points onto HΛ form a cocompact metric lattice in HΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It remains to show that for each γ′x0 ∈ Γ · x0 ∩ HΓ, the corresponding λ′ = λγ′ indeed lies on the same HΛ and boundedly close to the projection PHΛ(γ′x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is done by putting together all the geometric facts obtained up to this point, specifically Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One delicate fact that will be of use is that two maximal Λ-free horoballs that are based at the same point must be equal, because none of them can contain a Λ-orbit point while on the other hand both bounding horospheres intersect Λ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix s > 0 for which Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='23 yields a corresponding bound D(s), and let γ ∈ Γ such that sdγ > 2 · � DΛ + D(s) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Consider the (maximal) Λ-free horoball HBΛ λγ,i(γ) based at ξi(γ) λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I show that Λ · x0 ∩ HΛ λγ,i(γ) is a cocompact metric lattice in HΛ λγ,i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I keep the subscript notation because the proof is a game between HBΛ λγ,i(γ) and another Λ-free horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let HBΓ λγ,i(γ) be the Γ-horoball corresponding to HBΛ λγ,i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I can conclude that HBΓ λγ,i(γ) ⊂ HBΛ λγ,i(γ), because the choice of dγ > D(s) guarantees γx0 is sdγ deep inside HBΛ λγ,i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular HBΛ λγ,i(γ) is not Γ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, it holds that L := d(HΛ λγ,i(γ), HΓ λγ,i(γ)) ≥ sdγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let γ′ ∈ Γ be any element in the cocompact metric lattice Γ · x0 ∩ HΓ λγ,i(γ), and consider two associated points: (a) λ′x0 = λγ′x0 and (b) the projection of γ′x0 on HΛ λγ,i(γ), denoted p′ γ := PHΛ λγ ,i(γ)(γ′x0) ∈ HΛ λγ,i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The horoball HBΛ λγ,i(γ) is a maximal Λ-free horoball so it is also 1-almost maximal, hence d(p′ γ, Λ · x0) ≤ DΛ and the following holds: sdγ ≤ L ≤ dγ′ ≤ d(γ′x0, p′ γ) + d(p′ γ, Λ · x0) ≤ L + DΛ (3) Consider ξi(γ′) λ′ , and assume towards contradiction that ξi(γ′) λγ′ ̸= ξi(γ) λγ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Both points lie in WQ(Γ) and therefore must be at Tits distance π of each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore the fact that γ′x0 lies in HBΛ λγ,i(γ) implies that the geodesic [γ′x0, ξi(γ′)] leaves HBΛ λγ,i(γ) at some point z ∈ HΛ λγ,i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that D(s) ≤ sdγ ≤ dγ′ implies that γ′x0 lies s2dγ deep inside HBΛ λγ′ ,i(γ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore the point z also lies at least s2dγ deep inside HBΛ λγ′ ,i(γ′), and therefore z is the centre of a Λ-free horoball of radius at least s2dγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By choice of dγ the point z therefore admits a 2DΛ neighbourhood that is Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' But z lies on HΛ λγ,i(γ), a maximal horosphere of Λ, contradicting Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that ξi(γ′) λγ′ = ξi(γ) λγ , so HBΛ λγ,i(γ) and 28 γx0 dγ γ′x0 λγ′x0 HBΛ λγ′ ,i(γ′) z′ := πHBΛ λγ′ ,i(γ′)(γ′x0) D1 ���� 1 ξi(γ′) λγ′ ξi(γ) λγ ≥ 1 2dγ z ≥ 1 2 dγ′ Λ − free ball B � z, 1 2dγ � HBΛ λγ,i(γ) HBΓ λγ,i(γ) λγx0 Figure 4: Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assuming towards contradiction that ξi(γ) λγ ̸= ξi(γ′) λγ′ results in a point z ∈ HBΛ λγ,i(γ) (blue coloured and bold faced in the bottom part of the figure) admitting a large Λ-free neighbourhood, contradicting almost cocompactness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' HBΛ λγ′ ,i(γ′) are two Λ-free horoballs that are tangent to a Λ · x0 point and based at the same point at ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This implies HBΛ λγ,i(γ) = HBΛ λγ′ ,i(γ′), and in particular λγ′x0 ∈ HΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally, it is clearly seen from Inequality 3 that d(λ′x0, p′ γ) ≤ d(λ′x0, γ′x0) + d(γ′x0, p′ γ) ≤ dγ′ + L ≤ L + DΛ + L The element γ′x0 ∈ Γ · x0 ∩ HΓ λγ,i(γ) was as arbitrary element, and the above argument shows that the corresponding Λ-orbit points satisfy: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' λ′x0 all lie on HΛ λγ,i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Each p′ γ is 2L + DΛ close to the point λ′x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This shows that the cocompact metric lattice {p′ γ | γ′x0 ∈ HΓ λγ,i(γ)} lies in a bounded neighbourhood of the set of points Λ · x0 ∩ HΛ λγ,i(γ), proving that Λ · x0 ∩ HΛ λγ,i(γ) is a cocompact metric lattice in HΛ λγ,i(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='24 elevates this to � Λ ∩ StabG(HΛ λγ,i(γ)) � x0 ∩ HΛ λγ,i(γ) being a cocompact metric lattice in HΛ λγ,i(γ), completing the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 The Bounded Case Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12 is enough in order to prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 in the case Γ ̸⊂ ND(Λ) for any D > 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' in case Γ does not lie in a bounded neighbourhood of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The case where Γ and Λ lie at bounded Hausdorff distance, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' where Γ ⊂ ND(Λ) and Λ ⊂ ND(Γ), arose naturally in the context of the quasi-isometric classification of non-uniform lattices in the works of Schwartz [52] (R-rank 1), Drut¸u [16] and Eskin [22] (higher rank).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I restate the theorems in the bounded setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 29 Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='28 (Eskin [22], Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real centre-free semisimple Lie group without compact factors and of higher rank, Γ ≤ G an irreducible non-uniform lattice, Λ ≤ G a discrete subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ ⊂ ND(Λ) for some D > 0, then Λ is a lattice, and if moreover Λ ⊂ ND(Γ) then Λ and Γ are commensurable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='29 (Schwartz [52], Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real simple Lie group of R-rank 1, Γ ≤ G an irreducible non-uniform lattice, Λ ≤ G a discrete subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If both Γ ⊂ ND(Λ) and Λ ⊂ ND(Γ) for some D > 0, then Λ is a lattice, and if moreover G is not locally isomorphic to SL2(R), then Λ is commensurable to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The notable difference between the two statements is that for higher rank groups, the inclusion Λ ⊂ ND(Γ) is only required to prove commensurability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In view of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8, this allows me to omit that assumption from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Notice also that for groups with property (T) the result easily follows from the (much more recent) result by Leuzinger in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the context of commensurability in the sublinear setting, I can only prove a limited result, Namely that Λ is commensurable to Γ if Γ is an irreducible Q-rank 1 lattice and both Γ ⊂ ND(Λ) and Λ ⊂ Nu(Γ) for some constant D > 0 and a sublinear function u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is done via a reduction to the bounded case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real semisimple Lie group without compact factors and with finite centre, Γ ≤ G an irreducible lattice of Q-rank 1, Λ ≤ G a discrete subgroup, and u a sublinear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ ⊂ ND(Λ) for some D > 0 and Λ ⊂ Nu(Γ), then actually Λ ⊂ ND′(Γ) for some D′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, if G is of R-rank 1, the conclusion holds under the relaxed assumption that u(r) ⪯∞ εr for some ε < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' While the setting of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 is indeed rather limited, the situation that both Γ ⊂ Nu(Λ) and Λ ⊂ Nu(Γ) arises naturally from the motivating example of SBE-rigidity in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Notice however that Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 is not known for groups G that admit R-rank 1 factors, which is the only setting for which I can prove Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 A Reduction I start with the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The first step is to establish the fact that Λ must preserve WQ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real semisimple Lie group without compact factors and with finite centre, Γ ≤ G an irreducible non-uniform lattice of Q-rank 1, Λ ≤ G a discrete subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume that Γ ⊂ Nu(Λ) and that Λ ⊂ Nu′(Γ) for sublinear functions u, u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then Λ · WQ(Γ) ⊂ WQ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, if G is of R-rank 1, the conclusion holds under the relaxed assumption that u′(r) ⪯∞ εr for some ε < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof is similar to the argument of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='14, and uses the linear penetration rate of a geodesic into a horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ξ ∈ WQ(Γ), and let HΓ be a horosphere bounding a Γ-free horoball HBΓ with HΓ ∩Γ·x0 a metric lattice in HΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume first that u′ is sublinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since HBΓ is Γ-free and Λ ⊂ Nu′(Γ), I can conclude that Λ · x0 ∩ HBΓ ⊂ Nu′(HΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall (Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='31) that every geodesic ray η with limit point ξ′ ∈ N π 2 (ξ) penetrates HBΓ at linear rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore for every such geodesic ray η and every sublinear function v there is R = R(η, v) > 0 for which Nv(η↾r>R) is Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand, let λ ∈ Λ, and assume towards contradiction that λξ /∈ WQ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then by Proposi- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='30 there is a Γ-conical limit point ξ′ ∈ N π 2 (λξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The hypothesis that Γ ⊂ Nu(Λ) then implies that for every R > 0, Nu(η↾r>R) ∩ Λ · x0 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Translating by λ−1 yields a contradiction to the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that λξ ∈ WQ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I now modify the argument to include u′(r) ⪯∞ εr when G is of R-rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In this case, the only point ξ′ ∈ N π 2 (λξ) is λξ itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore by the same argument as above, the assumption that λξ /∈ WQ(Γ) implies that the u-sublinear neighbourhood of every geodesic ray with limit point ξ intersects Λ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', for every η with limit point ξ and every R > 0 it holds that Nu(η↾r>R) ∩ Λ · x0 ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand, every such geodesic penetrates HBΓ at 1-linear rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This amounts to the following fact: if v′(r) = εr for some ε ∈ (0, 1), then for some R > 0, the set Nu(η↾r>R)∩Nv′(HΓ) = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is a contradiction to Λ ⊂ Nu′(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 30 Proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume towards contradiction that there is a sequence λn such that d(λnx0, Γ · x0) > n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall that Γ·x0 is a cocompact metric lattice in the compact core of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This implies that there is a number D′ > 0 such that any λ ∈ Λ for which λx0 /∈ ND′(Γ · x0) must lie at least 1 2D′-deep inside a horoball of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I can assume that for all n ∈ N there are corresponding horoballs of Γ, which I denote HBΓ n, for which λn · x0 ∈ HBΓ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that Γ ⊂ ND(Λ) then implies that ND(Λ · x0) covers a cocompact metric lattice in HΓ n, namely the metric lattice Γ · x0 ∩ HΓ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the terminology of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2, HΓ n is almost Λ-cocompact, or D-almost Λ-cocompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I first prove that every horoball of Γ contains a Λ-free horoball (this is of course immediate if Λ ⊂ NC(Γ) for some C > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume towards contradiction that there is a horoball HBΓ of Γ that does not contain a Λ-free horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote HΓ := ∂HBΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the notations of the previous paragraph, I can assume without loss of generality that HBΓ = HBΓ n for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote by ξ the base point of HBΓ, fix some arbitrary x ∈ HΓ and consider the geodesic ray η := [x, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The constraint that Λ ⊂ Nu(Γ) implies that for every R > 0 there is some L > 0 for which the ball B � η(L + t), R � is Λ-free, for all t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, for all large enough n ∈ N (depending on R), the horosphere H(ξ, λnx0) that is parallel to HΓ and that passes through λnx0 contains a point that is the centre of Λ-free ball of radius R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This property is Λ-invariant, as well as the fact that HBΓ is based at WQ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, these two properties hold for the horoballs HBn := λ−1 n HBΓ, whose respective base points I denote ξn := λ−1 n ξ ∈ WQ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix R = D +2DΓ (recall that DΓ is such that every horosphere H of Γ admits H ⊂ NDΓ(Γ·x0 ∩H)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let L = L(D+2DΓ) be the corresponding bound from the previous paragraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For every n > L the horoball HBn has bounding horosphere Hn that admits a point zn ∈ Hn for which B(zn, D+2DΓ) is Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, the same is true for every horosphere that is parallel to Hn which lies inside HBn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since Γ ⊂ ND(Λ), this means that every horosphere that lies inside HBn admits a point that is the centre of a Γ-free ball of radius 2DΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that none of those horospheres could be the horosphere of Γ corresponding to the parabolic limit point ξn ∈ WQ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since x0 ∈ Hn it must therefore be that Hn is a horosphere of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' But this contradicts the fact that zn ∈ Hn and B(zn, 2DΓ) is Γ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This shows that no horoball of Γ contains a sequence of Λ-orbit points that lie deeper and deeper in that horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Put differently, it shows that every horoball of Γ contains a Λ-free horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I remark that the above argument shows something a bit stronger, which I will not use but which I find illuminating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It proves that as soon as d(λx0, Γ · x0) is uniformly large enough, say more than M, then λx0 must lie on a (D + 2DΓ)-almost Λ-cocompact horosphere parallel to HΓ, where HΓ is the bounding horosphere of any horoball of Γ in which λx0 lies (recall that it must lie in at least one such horoball).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand if d(λx0, Γ · x0) < M, then since every point in the Γ-orbit lies on a horosphere of Γ one concludes that λx0 lies on a horosphere H based at WQ(Γ) that is (M + D)-almost Λ-cocompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I can now assume that every HBΓ n contains a Λ-free horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular it contains a 1-almost maximal Λ-free horoball HBΛ n (see Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By definition there is a point λ′ nx0 that is at distance at most 1 from HΛ n = ∂HBΛ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Up to enlarging d(λnx0, Γ · x0) or decreasing it by at most 1, I can assume λn = λ′ n to begin with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Consider HBn := λ−1 n HBΛ n with Hn = ∂HBn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is a sequence of horoballs, each of which contains a Λ-free horoball at depth at most 1, based at corresponding parabolic limit points ξn ∈ WQ(Γ), and tangent to points that are at distance at most 1 from x0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', Hn ∩ B(x0, 1) ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since Γ ⊂ ND(Λ) I conclude that each of the HBn contain a horoball of depth at most D + 1 that is Γ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore the horoball of Γ that is based at ξn must have its bounding horosphere intersecting B(x0, D + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='22 there are only finitely such horoballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that there are finitely many points ξ′ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' , ξ′ K ∈ WQ(Γ) such that for every n ∈ N there is i(n) ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' , K} with ξn = ξ′ i(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From the Pigeonhole Principle there is some ξ′ ∈ {ξ′ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' , ξ′ K} for which ξn = ξ′ for infinitely many n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Passing to a subsequence I assume that this is the case for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To begin with the HBΓ n are horoballs of Γ, and therefore as in the first case the bounding horospheres HΓ n are D + 2DΓ-almost Λ-cocompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is a Λ-invariant property and therefore the same holds for the λ−1 n translate of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' These are the horospheres which are based at ξ′ and lie outside HBn at distance d(λnx0, Γ · x0) > n − 1 from Hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' They form a sequence of outer and outer horospheres based at the same point at WQ(Γ), all of which are D + 2DΓ-almost Λ-cocompact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is a contradiction, since the union of such horospheres intersect every horoball of X, contradicting the existence of Λ-free horoballs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Formally, 31 take some ζ ∈ WQ(Γ) different from ξ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since both ξ′ and ζ lie in WQ(Γ), they admit dT (ζ, ξ′) = π and there is a geodesic η with η(−∞) = ξ′ and η(∞) = ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let HBΓ ζ be the horoball of Γ that is based at ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By the first step of this proof, every such horoball must contain a Λ-free horoball HBΛ ζ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore there is some T > 0 such that for all t > T the point η(t) lies 2(D + 2DΓ) deep in HBΛ ζ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that for all t > T , B � η(t), 2D � is Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand, for arbitrarily large t it holds that the horosphere based at ξ′ and tangent to η(t) is D+2DΓ-almost Λ-cocompact, and in particular d � η(t), Λ·x0 � < D+2DΓ, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that Λ ⊂ ND′(Γ) for some D′ > 0, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the setting of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='30, Λ is a lattice commensurable to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 The Arguments of Schwartz and Eskin The R-rank 1 case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The statement of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 is a slight modification of his original formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' His framework leads to a discrete subgroup ∆ ≤ G such that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Every element of ∆ quasi-preserves the compact core of the lattice Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Namely, each element of ∆ is an isometry of X that preserves WQ(Γ) and that maps every horosphere of Γ to within the D = D(∆) neighbourhood of some other horosphere of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It holds that Γ ⊂ ND(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From these two properties Schwartz is able to deduce that ∆ has finite covolume, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' that ∆ is a lattice in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Here is a sketch of his argument, which works whenever Γ is a Q-rank 1 lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the setting described above, ∆ is a lattice in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof sketch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Consider X′ 0 := � g∈∆ g · X0, where X0 is the compact core of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This space serves as a ‘compact core’ for ∆: the fact that ∆ quasi-preserves X0 implies that X′ 0 ⊂ ND(X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is a ∆-invariant space, and therefore one gets an isometric action of ∆ on X′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This action is cocompact: the reason is that Γ acts cocompactly on X0, and Γ ⊂ ND(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Formally, every point in X′ 0 is D-close to a point in X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Every point in X0 is DΓ-close to a point in Γ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Every point in Γ · x0 is D-close to a point in ∆ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore the ball of radius 2D + DΓ contains a fundamental domain for the action of ∆ on X′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It remains to see that the action of ∆ on X \\ X′ 0 is of finite covolume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As a result of the cocompact action of ∆ on X′ 0, there is B := B(x0, R) so that X′ 0 ⊂ ∆·B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' X′ 0 is the complement of a union of horoballs, which one may call horoballs of ∆, with bounding horospheres of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that Γ is of Q-rank 1 means that the horoballs of Γ are disjoint, and therefore those of Λ are almost disjoint: there is some C > 0 such that for every horosphere H of Λ and every point x ∈ H, d(x, X′ 0) < C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Up to enlarging the radius of B by C, I can assume that H ⊂ ∆ · B for every horosphere H of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Each horoball of Λ is based at WQ(Γ), and each lies uniformly boundedly close to the corresponding horoballs of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='22 one therefore sees that there are finitely many horoballs of ∆ that inter- sect B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote them by HB1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' , HBN, their bounding horospheres by Hi = ∂HBi, and their intersection with B by Bi := B ∩ HBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let also ξi ∈ WQ(Γ) denote the base point of each HBi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Each Bi is pre-compact and therefore the projection of each Bi on Hi is pre-compact as well (this is a consequence e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' of the results of Heintze-Im hof recalled in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let Di ⊂ Hi be a compact set that contains this projection, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' PHi(Bi) ⊂ Di ⊂ Hi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular B ∩ Hi ⊂ Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Observe now that for every horoball HB of ∆, with bounding horosphere H = ∂HB, the ∆-orbit of every point x ∈ H intersects some Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' First notice that for x ∈ H the choice of B implies that the ∆-orbit of x must intersect B, say gx ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular gH ∩ B ̸= ∅, and since gHB is a horoball of ∆ then by definition gH = Hi for some i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One concludes that indeed gx ⊂ Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, let y ∈ X is any point that lies inside a horoball HB of ∆, and x = PH(y) its projection on the bounding horosphere H = ∂HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By the previous paragraph there is some g ∈ ∆ and i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' , N} for which gx ∈ Di, and therefore it is clear that gy lies on a geodesic emanating from Di to ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 32 Finally, define Cone(Di) to be the set of all geodesic rays that emanate from Di and with limit point ξi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The previous paragraph proves that �N i=1 Cone(Di) contains a fundamental domain for the action of ∆ on X \\ X′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, the fact that Di ⊂ Hi is compact readily implies that each Cone(Di) has finite volume, and so this fundamental domain is of finite volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To conclude, B ∪ � �N i=1 Cone(Di) � is a set of finite volume and it contains a fundamental domain for the ∆-action on X, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof of commensurability of ∆ and Γ is given in full in [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is one essential difference between Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='34, namely the assumption that Λ ⊂ ND(Γ) rather than quasi-preserving the compact core of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In Schwartz’s work, the fact that ∆ ⊂ ND(Γ) is not relevant (even though it easily follows from the construction of his embedding of ∆ in G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' He only uses the two properties described above, namely the quasi-preservation of X0 and Γ ⊂ ND(∆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The assumption that Λ quasi-preserves the compact core of Γ does not feel appropriate in the context of my thesis, while the metric condition Λ ⊂ ND(Γ) seems much more natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is a stronger condition as I now show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='32, Λ · WQ(Γ) ⊂ WQ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let HΓ 1 be a horosphere of Γ, based at ξ ∈ WQ(Γ), and let γx0 ∈ HΓ 1 be some point on the metric lattice of Γ · x0 on HΓ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is an element λ ∈ Λ such that d(λx0, γx0) < D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, since Λ ⊂ ND(Γ) one knows that the parallel horoball that lies D-deep inside HBΓ 1 is Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let λ′ ∈ Λ be an arbitrary element of Λ, and consider λ′ · HΓ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The last statement in the previous paragraph is Λ-invariant, and so the horoball that lies D-deep inside λ′ · HBΓ 1 is Λ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that Γ ⊂ ND(Λ) then implies that the parallel horoball that lies 2D deep inside λ′HBΓ 1 is Γ-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let HΓ 2 be the horosphere of Γ that is based at λ′ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The last statement amounts to saying that HΓ 2 lies at most 2D-deep inside λ′HBΓ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand, one has d(λ′λx0, λ′HΓ 1 ) = d(λx0, HΓ 1 ) ≤ D, so there is a Λ-orbit point that lies within D of λ′HΓ 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The parallel horoball that lies D-deep inside HBΓ 2 must also be Λ-free, so I conclude that HΓ 2 must be contained in the parallel horoball to λ′HBΓ 1 which contains it and that is at distance D from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that d(λ′HΓ 1 , HΓ 2 ) ≤ 2D, and so that Λ quasi-preserves X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is interesting to note that Schwartz’s arguments are similar in spirit to my arguments in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In fact, one could also prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 using the same type of arguments that appear repeatedly in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2, namely by moving Λ-free horoballs around the space, specifically the proof of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I do not present it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Higher rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Eskin’s proof is ergodic, and based on results of Mozes [42] and Shah [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I produce it here without the necessary preliminaries, which are standard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To prove that Λ is a lattice amounts to finding a finite non-zero G-invariant measure on Λ\\G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Theorem 2 in [42], if P ≤ G is a parabolic subgroup then every P-invariant measure on Λ\\G is automatically G-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix a minimal parabolic subgroup P ≤ G and let µ0 be some fixed probability measure on Λ\\G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since P is amenable it admits a tempered Følner sequence Fn ⊂ P, and one can average µ0 along each Fn to get a sequence of probability measures µn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The weak* compactness of the unit ball in the space of measures on Λ\\G implies that there exists a weak* limit µ of the µn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The measure µ is automatically a finite P-invariant measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It remains to show that µ is not the zero measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To see this it is enough to show that for some compact set CΛ ⊂ Λ\\G and some Λg = x ∈ Λ\\G, one has 0 < lim inf n 1 |Fn| � Fn 1CΛ(xp−1)dp (4) Fix some compact neighbourhood CΓ ⊂ Γ\\G of the trivial coset Γe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The hypothesis Γ ⊂ ND(Λ) implies that there is a corresponding compact neighbourhood CΛ ⊂ Λ\\G of the trivial coset Λe such that for any p ∈ P, it holds that Γgp−1 ∈ CΓ ⇒ Λgp−1 ∈ CΛ (simply take CΛ to be the D + 1-blowup of CΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The action of P on Γ\\G is uniquely ergodic, therefore 0 < µΓ(CΓ) = lim n 1 |Fn| � Fn 1CΓ(Γp−1)dp 33 where µΓ denotes the natural G-invariant measure on Γ\\G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The defining property of CΛ ensures that Inequality (4) is satisfied, implying that µ is a non-zero P-invariant probability measure on Λ\\G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that µ is also G-invariant, and that Λ is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If moreover Λ ⊂ ND(Γ), one may use Shah’s Corollary [54] to conclude that Λ is commensurable to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 Translating Geometry into Algebra The goal of this section is to prove that the results of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 imply that Λ satisfies the hypotheses of the Benoist-Miquel criterion Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Namely, that Λ is Zariski dense, and that it intersects a horospherical subgroup in a cocompact indecomposable lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' These are algebraic properties, and the proof that Λ satisfies them is in essence just a translation of the geometric results of Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 to an algebraic language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The geometric data given by Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 is that for some horosphere H bounding a Λ-free horoball, Λ ∩ StabG(H) · x0 intersects H in a cocompact metric lattice (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12), and that the set of Λ-conical limit points contains the set of Γ-conical limit points (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Note that since K is compact the former implies that Λ ∩ StabG(H) is a uniform lattice in StabG(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 A Horospherical Lattice I assume that Λ ∩ StabG(H) is a lattice in StabGH, and I want to show that Λ intersects a horospherical subgroup U of G in a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This step requires quite a bit of algebraic background, which I give below in full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In short, the first goal is to show that StabG(H) admits a subgroup U ≤ StabG(H) that is a horospherical subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A lemma of Mostow (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='36 below) allows to conclude that Λ intersects U in a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='36 (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 in [40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let H be a Lie group having no compact connected normal semisimple non-trivial Lie subgroups, and let N be the maximal connected nilpotent normal Lie subgroup of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let Γ ≤ H be a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then N/N ∩ Γ is compact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the original statement Mostow uses the term ‘analytic group’, which I replaced here with ‘connected Lie subgroup’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This appears to be Mostow’s definition of an analytic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Section 10, Chapter 1 in [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In Chevalley’s Theory of Lie Groups, he defines a Lie group as a locally connected topological group whose identity component is an analytic group (Definition 1, Section 8, Chapter 4 in [12]), and proves (Theorem 1, Section 4, Chapter 4 therein) a 1-1 correspondence between analytic subgroups of an analytic group and Lie subalgebras of the corresponding Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='36 lays the rationale for the rest of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Explicitly, I prove that StabG(H) admits a subgroup that is a horospherical subgroup U of G (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='39), and that U is maximal connected nilpotent normal Lie subgroup of StabG(H) (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In order to use Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='36, I show that the horospherical subgroup Nξ is a maximal normal nilpotent connected Lie subgroup of StabG(H)◦, and that StabG(H)◦ admits no compact normal factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This requires to establish the structure of StabG(H)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the notation ht ξ = exp(tX) and Aξ = exp � Z(X) ∩ p � of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7, define A⊥ ξ to be the codimension-1 submanifold of Aξ that is orthogonal to {hξ(t)}t∈R (with respect to the Killing form in the Lie algebra).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Every element a ∈ A⊥ ξ stabilizes H = H(x0, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' An element in Aξ is an element that maps x0 to a point on a flat F ⊂ X that contains the geodesic ray [x0, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If a ∈ A⊥ ξ , then the geodesic [x0, ax0] is orthogonal to [x0, ξ), and lies in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From Euclidean geometry and structure of horospheres in Euclidean spaces, it is clear that ax0 ∈ H(x,ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since a ∈ Gξ, this means aH = H(ax0, ξ) = H(x, ξ) = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let H be a horosphere based at ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then StabG(H)◦ = (KξA⊥ ξ )◦Nξ, and in particular it contains a horospherical subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, StabG(H)◦ is normal in StabG(ξ)◦ and acts transitively on H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 34 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Clearly (KξA⊥ ξ )◦Nξ is a codimension-1 subgroup of StabG(ξ)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since StabG(H) ̸= StabG(ξ) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ht ξ /∈ StabG(H) for t ̸= 0), it is enough to show that (KξA⊥ ξ )◦Nξ ≤ StabG(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let kan ∈ (KξA⊥ ξ )◦Nξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It fixes ξ, so it is enough to show that kanx0 ∈ H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since k ∈ Kξ and kx0 = x0, it stabilizes H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From Claim 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 a ∈ StabG(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' So it remains to check that Nξ stabilizes H, but this is more or less the definition: fixing a base point x0, the horospheres based at ξ are parameterized by R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote them by {Ht}t∈R, where H = H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In this parameterization, any element g ∈ Gξ acts on {Ht}t∈R by translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I can thus define for g ∈ StabG(ξ) the real number l(g) to be that number for which gHt = Ht+l(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Clearly l � hξ(t) � = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The element n fixes ξ, so one has h−t ξ nht ξH0 = h−t ξ Ht+l(n) = Ht+l(n)−t = Hl(n) The fact that n ∈ Ker(Tξ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' that limt→∞ h−t ξ nht ξ = eG readily implies that necessarily l(n) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that (KξA⊥ ξ )◦Nξ = StabG(H)◦, as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Next recall that StabG(H)◦ acts transitively on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let x, y ∈ H, and consider g ∈ StabG(ξ)◦ with gx = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Writing an element g ∈ Gξ as kata⊥n ∈ Kξht ξA⊥ ξ Nξ, the argument above shows that kht ξa⊥nH0 = H0 if and only if t = 0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', if and only if g ∈ StabG(H)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally, let g ∈ StabG(ξ) and h ∈ StabG(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By the discussion above h · Ht = Ht for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Clearly −l(g) = l(g−1), and therefore ghg−1H0 = ghH−l(g) = g · H−l(g) = H0 Therefore StabG(H) is normal in StabGξ, and the same is true for the respective identity components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' StabG(H)◦ is a connected Lie group with no connected compact normal semisimple non- trivial Lie subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Every compact subgroup of G fixes a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let H ≤ G be some closed subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is standard to note that a normal N ≤ H that fixes a point x ∈ X must fix every point in the orbit H · x: hnh−1hx = hx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since H = StabG(H)◦ acts transitively on H, it shows that a normal compact subgroup of StabG(H)◦ fixes every point in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' An isometry fixing a horosphere pointwise while fixing its base point is clearly the identity, proving the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The following fact is well known but I could not find it in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A horosphere in X is not convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let H′ be some horosphere in X, with base point ζ ∈ X(∞), and assume towards contradiction that it is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix x ∈ H′ and a′ t the one parameter subgroup with η′(∞) = a′ tx, and denote H′ t = H(a′ tx, ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let eG ̸= n ∈ Nζ (Nζ defined with respect to a′ t in a corresponding Langlands decomposition), and consider the curve η′ n(t) := a′ tnx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I claim that this is a geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the one hand, the fact that H′ is convex implies that the geodesic segment [x, nx] is contained in H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore a′ t[x, nx] = [a′ tx, a′ tnx] ⊂ H′ t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' More generally it is clear that because a′ tH′ s = H′ s+t it holds that H′ t is convex for every t as soon as it is convex for some t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand, for every point y ∈ [x, nx], d(y, H′ t) = t, and more generally for any y ∈ [a′ snx, a′ sx] it holds that d(y, H′ t) = |s − t|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular this is true for η′ n(t) = yt := a′ tnx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I get that d � η′ n(t), η′ n(s) � = |s − t|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore η′ n is a geodesic (to be pedantic one has to show that η′ n is a continuous curve, which is a result of the fact that a′ t is a one parameter subgroup of isometries).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Clearly d � η′ n(t), η′(t) � = d(a′ tnx, a′ tx) = d(nx, x) and therefore η′ n is at uniformly bounded distance to η′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This bounds d(ηn, η′ n) as bi-infinite geodesics, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' for all t ∈ R, not just as infinite rays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The Flat Strip Theorem (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='13, Chapter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 in [11]), then implies that the geodesics ηn, η′ n bound a flat strip: an isometric copy of R × [0, l] (where l = d(x, nx)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Up to now I did not use the fact that n ∈ Nξ, only that the point nx lies on a geodesic that is contained in H′ = H′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore the entire bi-infinite geodesic that is determined by [x, nx] lies on a 2-dimensional flat F that contains η′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The two elements n, a′ t therefore admit nx, a′ tx ∈ F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is a fact that two such elements must commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I can conclude therefore that [n, a′ t] = eG, which contradicts the fact that that n ∈ Nζ = Ker(Tζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 35 Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='42 (Theorem 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='13 in [51]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let N be a connected real Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then Lie(N) is a nilpotent Lie algebra if and only if N is a nilpotent Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='43 (Proposition 13, Section 4, Chapter 1 in [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the notation of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7, nξ = Lie(Nξ) is a maximal nilpotent ideal in gξ = Lie(Gξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The presentation of nξ in [8] is given by means of the root space decomposition of StabG(ξ), that appears in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='13 in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There are two main objects in the literature that are referred to as the nilpotent radical or the nilradical of a Lie algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' These are: (a) the maximal nilpotent ideal of the Lie algebra, and (b) the intersection of the kernels of all irreducible finite-dimensional representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 13 in Section 4 of Chapter 9 in [7] shows that in the case of Lie algebras of parabolic Lie groups, these notions coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Nξ is a maximal connected nilpotent normal Lie subgroup of the identity connected com- ponent StabG(H)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='42 implies Nξ is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since StabGH ⊳ StabG(ξ), every normal subgroup of StabG(H) containing Nξ is in fact a normal subgroup of StabG(ξ), still containing Nξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It remains to prove maximality of Nξ among all connected nilpotent normal Lie subgroups of StabG(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Any such subgroup N ′ ⊳ StabG(ξ) gives rise to an ideal n′ of gξ = Lie � StabG(ξ) � , and by Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='42 it is a nilpotent ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='43 it is contained in nξ = Lie(Nξ), implying that N ′ ≤ Nξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A lattice in StabG(H) intersects the horospherical subgroup Nξ in a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='40 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='45 imply that the pair Nξ ⊳ StabG(H) satisfy the hypotheses of Mostow’s Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 Indecomposable Horospherical Lattices The Benoist and Miquel criterion requires the horospherical lattice to be indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is shown in [5] that if this lattice is contained in a Zariski dense discrete subgroup, then the indecomposability condition is equivalent to irreducibility of the ambient group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The precise definitions and statements are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='47 (Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='14 in [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a semisimple real algebraic Lie group G and U a horospherical subgroup of G, let ∆U be a lattice in U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ∆U is irreducible if for any proper normal subgroup N of G◦, one has ∆U ∩ N = {e}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ∆U is indecomposable if one cannot write G◦ as a product G◦ = N ′N ′′ of two proper normal subgroups N ′, N ′′ ⊳ G with finite intersection such that the group ∆′ U := (∆U ∩ N ′)(∆U ∩ N ′′) has finite index in ∆U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='48 (See Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 in [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a semisimple real algebraic Lie group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A discrete subgroup Λ ≤ G is said to be irreducible if, for all proper normal subgroups N ⊳ G, the intersection Λ ∩ N is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='49 (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 in [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a semisimple real algebraic Lie group, U ⊂ G a non-trivial horospherical subgroup, and ∆U ≤ U a lattice of U which is contained in a discrete Zariski dense subgroup ∆ of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then the following are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ∆ is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ∆U is irreducible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ∆U is indecomposable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 36 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 Zariski Density The last requirement is for Λ to be Zariski dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I use a geometric criterion which is well known to experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='50 (Proposition 2 in [31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a symmetric space of noncompact type, G = Isom(X)◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A subgroup ∆ ≤ G is Zariski dense if and only if: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ∆ does not globally fix a point in X(∞), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ∆ ̸≤ StabG(ζ) for any ζ ∈ X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The identity component of the Zariski closure of ∆ does not leave invariant any proper totally geodesic submanifold in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the proof I use several facts - mostly algebraic, and two geometric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I warmly thank Elyasheev Leibtag for his help and erudition in algebraic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The first property I need is very basic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ∆ ≤ G be a discrete subgroup, and let H ≤ G be the Zariski closure of ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then ∆ ∩ H◦ is of finite index in ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' H◦ is normal and of finite index in H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The following fact is probably known to experts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It appears in a recent work by Bader and Leibtag[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='52 (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 in [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let k be a field, G a connected k algebraic group, P ≤ G = G(R) a parabolic subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then the centre of G contains the centre of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Still on the algebraic side, I need a Theorem of Dani, generalizing the Borel Density Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='53 (See [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let S be a real solvable algebraic group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If S = S(R) is R-split, then every lattice ΓS ≤ S is Zariski dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is a fact (see Theorem 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 and Section 18 in [6]) that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Every unipotent group over R is R-split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a field k of characteristic 0, a solvable linear algebraic k-group is k-split if and only if its maximal torus is k-split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Finally I need two geometric facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The first is a characterization determining when does a unipotent element belongs to Nζ for some ζ ∈ X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='55 (Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8 in [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a symmetric space of noncompact type and of higher rank, n ∈ G = Isom(X)◦ a unipotent element, and ζ ∈ X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The following are equivalent: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For Nζ as in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7, n ∈ Nζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For some geodesic ray η with η(∞) = ζ it holds that limt→∞ d � nη(t), η(t) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For every geodesic ray η with η(∞) = ζ it holds that limt→∞ d � nη(t), η(t) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The last property I need is a characterization of the displacement function for unipotent elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='56 (See proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 in [4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a symmetric space of noncompact type, ζ ∈ X(∞) some point and n ∈ Nζ an element of the unipotent radical of StabG(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The displacement function x �→ d(nx, x) is constant on horospheres based at ζ, and for every ε > 0 there is a horoball HBε based at ζ such that d(nx, x) < ε for every x ∈ HBε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume that: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' � Λ ∩ StabG(H) � x0 is a cocompact metric lattice in a horosphere H ⊂ X bounding a Λ-free horoball.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 37 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Every Γ-conical limit point is a Λ-conical limit point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then Λ is Zariski dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I show the criteria of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='50 are met, starting with Λ ̸≤ StabG(ζ) for any ζ ∈ X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To this end, I first prove that Λ · x0 is not contained in any bounded neighbourhood of any horosphere H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ξ′ be the base point of H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Hattori’s Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='31 (and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='32), it is enough to find a Λ-conical limit point ζ′ with dT (ξ′, ζ′) ̸= π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Take some ζ′′ ∈ X(∞) at Tits distance π of ξ′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' take a flat F on which ξ′ lies and let ζ′′ be the antipodal point to ξ′ in F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix ε = π 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3, there are neighbourhoods of the cone topology U, V ⊂ X(∞) of ξ′, ζ′′ (respectively) so that every point ζ′ ∈ V admits dT (ξ′, ζ′) ≥ dT (ξ′, ζ′′)− π 4 = 3 4π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall that the set of Γ-conical limit points is dense (in the cone topology), so the second hypothesis implies there is indeed a Λ-conical limit point in V and therefore at Tits distance different (in this case larger) than π 2 from ξ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that Λ · x0 is not contained in any bounded metric neighbourhood of any horosphere of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume towards contradiction that Λ ≤ StabG(ζ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I show that this forces Λ ∩ Nζ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Propo- sition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='55 it is enough to find a unipotent element λ ∈ Λ and a geodesic η with η(∞) = ζ such that limt→∞ d � λη(t), η(t) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let F be a maximal flat with ξ, ζ ∈ F(∞), x ∈ F some point and X, Y ∈ a ≤ p two vectors such that exp(tY ) = η(t) for the unit speed geodesic η = [x, ζ), and exp(tX) = η′(t) for the unit speed geodesic η′ = [x, ξ) (where a ≤ p a maximal abelian subalgebra in a suitable Cartan decomposition g = p ⊕ k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let StabG(ξ) = KξAξNξ be the decomposition described in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 with respect to Y (notice that Nξ does not depend on choice of Y , see item 3 of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 in [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The assumption that Λ ≤ StabG(ζ) implies that for any λ ∈ Λ the distance d � λη(t), η(t) � either tends to 0 as t → ∞ or is uniformly bounded for t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the latter case there is some constant c > 0 for which d � λη(t), η(t) � = c for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As in the proof of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='41, the Flat Strip Theorem (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='13, Chapter 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 in [11]) implies that λ and at := exp(tY ) commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From the first hypothesis of the statement and Mostow’s result (Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='46) I know that Λ ∩ Nξ is a cocompact lattice in Nξ (attention to subscripts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore Λ ∩ Nξ is Zariski dense in Nξ (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, since commuting with an element is an algebraic property, an element g ∈ G that commutes with Λ∩Nξ must also commute with its Zariski closure, namely with Nξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This means that if at commutes with all Λ∩Nξ then it commutes with Nξ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' atn = nat for all t ∈ R and all n ∈ Nξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I know that at ∈ Aξ commutes with both Kξ and Aξ (Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7) therefore if at also commutes with Nξ then at lies in the centre of StabG(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This means that at is central in G (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a group G with compact centre this cannot happen, so there is indeed some unipotent element λ ∈ Λ ∩ Nξ for which limt→∞ d � λη(t), η(t) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='55 that Λ ∩ Nζ ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The first paragraph of the proof implies in particular that Λ·x0 does not lie in any bounded neighbourhood of a horosphere H′ based at ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The assumption that Λ ⊂ StabG(ζ) implies that every λ ∈ Λ acts by translation on the filtration {H′ t}t∈R by horospheres based at ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore as soon as Λ · x0 ̸⊂ Ht for some t ∈ R one concludes that ζ is a horospherical limit point of Λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' that every horoball based at ζ intersects the orbit Λ · x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='56 it holds that for a unipotent element g ∈ Nζ the displacement function x �→ d(gx, x) depends only on the horosphere H′ t in which x lies and that, for xt ∈ H′ t it holds that limt→∞ d(gxt, xt) = 0 (up to reorienting the filtration t ∈ R so that η(t) ∈ H′ t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a non-trivial element λζ ∈ Λ ∩ Nζ the previous paragraph therefore yields a sequence of elements λn ∈ Λ such that limn→∞ d(λζλnx0, λnx0) = 0, contradicting the discreteness of Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that Λ ̸≤ StabG(ζ) for every ζ ∈ X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume that H := � Λ Z�◦, the identity connected component of the Zariski closure of Λ, stabilizes a totally geodesic submanifold Y ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='51, Λ0 := Λ ∩ H is of finite index in Λ, therefore Λ0 ∩ StabG(H) is also a cocompact lattice in StabG(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that � Λ0 ∩ StabG(H) � x0 is a cocompact metric lattice in H readily implies that � Λ0 ∩ StabG(H) � y is a cocompact metric lattice in Hy = H(y, ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This goes to show that there is no loss of generality in assuming x0 ∈ H ∩ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It follows that Λ0 ∩ StabG(H) · x0 ⊂ Y ∩ H, and therefore H ⊂ ND(Y ) for some D > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A horosphere is a codimension-1 submanifold, implying that Y is either all of X or of codimension-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The latter forces Y = H, which is impossible since H is not totally geodesic (H is not convex, see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that H does not stabilize any totally geodesic 38 proper submanifold, and hence that Λ is Zariski dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 I now complete the proof of the main sublinear rigidity theorem for Q-rank 1 lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If {dγ}γ∈Γ is bounded, then Λ is a lattice by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 or Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9, depending on the R-rank of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If {dγ}γ∈Γ is unbounded, then Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='27 both hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In R-rank 1 the proof again follows immediately from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='16 and Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In higher rank, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 allows one to conclude that Λ is an irreducible, discrete, Zariski dense subgroup that contains a horospherical lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4, this renders Λ a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is a Q-rank 1 lattice as a result of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The sublinear nature of the hypothesis in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 induces coarse metric constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A horospherical lattice on the other hand is a very precise object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is not clear how to produce unipotent elements in Λ, or even general elements that preserve some horosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof sketched above produces a whole lattice of unipotent elements in Λ (this is Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='46);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' it is also the only proof that I know which produces even a single unipotent (or parabolic) element in Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 5 Lattices with Property (T) Recall that a lattice in a locally compact group G has property (T) if and only if G has property (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In this section I prove: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real centre-free semisimple Lie group without compact factors, Γ ≤ G a lattice, Λ ≤ G a discrete subgroup such that Γ ⊂ Nu(Λ) for some sublinear function u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ has property (T), then Λ is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As in the case of uniform lattices, lattices with property (T) admit the stronger version of ε-linear rigidity, for suitable ε = ε(G): Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real centre-free semisimple Lie group without compact factors, Γ ≤ G a lattice and Λ ≤ G a discrete subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ has property (T) then there exists ε = ε(G) > 0 depending only on G such that if Γ ⊂ Nu(Λ) for some function u(r) ⪯∞ εr, then Λ is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Clearly Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 implies Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I thank Emmanuel Breuillard for suggesting this generaliza- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From now and until the end of this section, the standing assumptions are those of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lattice Criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For groups with property (T) I use a criterion by Leuzinger, stating that being a lattice is determined by the exponential growth rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The formulation requires a definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Given a pointed metric space (X, dX, x0), denote: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' bX(r) = |B(x0, r)| 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' bu X(r) = supx∈X |B(x, r)| When a group ∆ acts on a pointed metric space X, the orbit ∆ · x0 together with the metric induced from X is a pointed metric space (∆ · x0, dX↾∆·x0, x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In this setting b∆·x0(r) = |BX(xo, r) ∩ ∆ · x0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' When the action is by isometries, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ∆ ≤ Isom(X), it is straightforward to observe that this quantity does not depend on the centre of the ball, and so bu ∆·x0(r) = b∆·x0(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The pointed metric spaces of interest in this section are the Γ and Λ orbits in the symmetric space X = G/K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 39 Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a symmetric space and ∆ ≤ G = Isom(X)◦ a subgroup of isometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The critical exponent of ∆ is defined to be δ(∆) := lim sup r→∞ log � b∆·x0(r) � r Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Throughout this section there is no risk of ambiguity, and I allow myself to ease notation and let b∆(r) = b∆·x0(r), bu ∆(r) = bu ∆·x0(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To a semisimple Lie group G one can associate a quantity ∥ρ∥, where ρ = ρ(G) is the half sum of positive roots in the root system of (g, a) (see Section 2 in [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 (Theorem 2 in [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real centre-free semisimple Lie group without compact factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ∆ be a discrete, torsion-free subgroup of G that is not a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If G has Kazhdan’s property (T ), then there is a constant c∗(G) (depending on G but not on ∆) such that δ(∆) ≤ 2∥ρ∥ − c∗(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is known that the critical exponent of a discrete subgroup ∆ ≤ G is bounded above by 2∥ρ∥ (see Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 in [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, every lattice Γ ≤ G admits δ(Γ) = 2∥ρ∥ (Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5 in [36], Theorem C in [1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Combining these facts with Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 yield: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 (Theorem B in [36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real centre-free semisimple Lie group without compact factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let ∆ be a discrete, torsion-free subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If G has Kazhdan’s property (T ), then ∆ is a lattice iff δ(∆) = 2∥ρ∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Line of Proof and the Use of ε-Linearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 goes by showing that ε-linear distortion cannot decrease the exponential growth rate by much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This fact is essentially manifested in a proposition by Cornulier [13], stated here in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is the only use I make of ε-linearity, and the computations involved are straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 is then an immediate consequence of Leuzinger’s criterion Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 In his study on SBE maps, Cornulier proves the following growth discrepancy result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8 (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 in [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X, Y be two pointed metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let u be a non-decreasing sublinear function and p : X → Y a map such that for some L, R0 > 0: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' |p(x)| ≤ max(|x|, R0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' p(|x|) ≤ |x| for all large enough x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' dY � p(x), p(x′) � ≥ 1 LdX(x, x′) − u(max{|x|, |x′|}) Then for all r > R0, bY (r) ≥ bX(r)/bu X � L · u(r) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I need a slightly modified version of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8: Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X, Y be two pointed metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let u be a non-decreasing function that admits u(r) ⪯∞ εr for some ε < 1, and p : X → Y a map such that for some L, R0 > 0: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' |p(x)| ≤ max(|x| + u(|x|), R0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' |p(x)| ≤ |x| + u(|x|) for all large enough x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' dY � p(x), p(x′) � ≥ 1 LdX(x, x′) − u(max{|x|, |x′|}) Then for all r > R0, bY (r) ≥ bX � r − u(r) � /bu X � L · u � r − u(r) �� Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Repeat verbatim the proof for Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 40 Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let u(r) ⪯∞ ε · r for some ε < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume that Γ ⊂ Nu(Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then δ(Λ) ≥ (1 − 4ε) · δ(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Restricting to ε < 1 2 stems from a 2 factor that appears in the proof and could possibly be dropped using a slightly more sophisticated approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For my needs this is more than enough, since in any case I eventually restrict attention to a small interval around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 can be formulated in a slightly more general fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Using the notation δ(W) := lim supr→∞ log � bW (r) � r for a general subset W in a general metric space X, the following general version holds: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let (X, x0) be a pointed metric space, Y, Z ⊂ X two subsets, and u(r) ⪯∞ εr for some ε < 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume that bu Z(r) = bZ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Z ⊂ Nu(Y ), then δ(Y ) ≥ (1 − 4ε) · δ(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, if u is sublinear, then δ(Z) = δ(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 holds even when the group G does not have property (T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 follows from Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12 because the fact that Γ is a group of isometries implies bu Γ(r) = bΓ(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof (Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Observe that the closest point projection pY : Z → Y defined by z �→ zy for some point in the closed ball zy ∈ B � z, u(|z|) � admits: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' |yz| ≤ |z| + u(|z|) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' d � yz, yz′� ≥ d(z, z′) − 2u(max{|z|, |z′|}) The first item follows from triangle inequality: |yz| ≤ d(yz, z) + d(z, x0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The second item follows from the quadrilateral inequality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', using triangle inequality twice along the quadrilateral [z, z′, yz′, yz].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The above properties allow me to use Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 with constant L = 1 and function u′ = 2u to get bY (r) ≥ bZ � r − u′(r) � /bu Z � u′� r − u′(r) �� Since I assume bu Z = bZ, I can omit the superscript u in the last expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recalling the definition δ(W) = lim supr→∞ bW (r) r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' it remains to prove: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='lim sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r→∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r · log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r − u′(r) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='/bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='u′� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r − u′(r) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='≥ (1 − 4ε) · δ(Z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='The proof of this inequality involves nothing more than log rules and arithmetic of limits: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='lim sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r→∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r · log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r − u′(r) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='/bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='u′� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r − u′(r) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='= lim sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r→∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r − u′(r) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='− log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='u′� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r − u′(r) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='≥ lim sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r→∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r · log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r − u′(r) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='− lim sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='s→∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='�1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='s log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='u′� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='s − u′(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='����� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='= lim sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r→∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r · log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r − u′(r) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='− lim sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='s→∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='s log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='u′� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='s − u′(s) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='��� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='≥ lim sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r→∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r · log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r − 2εr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='− lim sup ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='s→∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='s log ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='bZ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2εs ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='= (1 − 2ε)δ(Z) − 2εδ(Z) = (1 − 4ε)δ(Z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='(5) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='Below I justify the steps in the above inequalities: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='41 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' First equality is by rules of log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Second and third inequalities are by arithmetic of limits: let (an)n, (bn)n be two sequences of positive numbers, and A = lim supn an, B = lim supn bn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then lim sup(an − bn) ≥ lim supn(an − B) = A − B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fourth inequality: u′(r) < 2ε(r) for all large enough r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fifth equality: definition of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This completes the proof in the general case, which is what is needed for the proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For the more refined statement in the case u is sublinear, one has to show a bit more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From inequality 5 (specifically from the fourth line of the inequality) it is clearly enough to prove: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' lim supr→∞ 1 r · log � bZ � r − u′(r) �� = δ(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' lim sups→∞ 1 s log � bZ � u′� s − u′(s) ��� = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Starting from the second item, indeed it holds that 1 s log � bZ � u′� s − u′(s) ��� = u′� s − u′(s) � s log � bZ � u′� s − u′(s) ��� u′� s − u′(s) � Clearly lim sup of the right factor in the above product is bounded by δ(Z), and in particular it is uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand sublinearity of u′ implies that the left factor tends to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that this product tends to 0 as s tends to ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It remains to prove lim supr→∞ 1 r · log � bZ � r − u′(r) �� = δ(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In a similar fashion, 1 r log � bZ � r − u′(r) �� = r − u′(r) r log � bZ � r − u′(r) �� r − u′(r) The left factor limits to 1 by sublinearity of u′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The right factor is nearly the expression in the definition of δ(Z), and I want to prove that indeed taking lim sup of it equals δ(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A priori {r − u′(r)}r∈R>0 is just a subset of R>0, so changing variable and writing t := r−u′(r) requires a justification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' But there is no harm in assuming that u′ is a non-decreasing continuous function, hence R≥R ⊂ {r − u′(r)}r∈R>0 for some R ∈ R>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore for any sequence rn → ∞ there is a sequence r′ n with rn = r′ n − u′(r′ n) for all large enough n (note that in particular r′ n → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the other direction, for every sequence r′ n → ∞ there is clearly a sequence rn → ∞ for which rn = r′ n − u′(r′ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude lim sup r→∞ log � bZ � r − u′(r) �� r − u′(r) = lim sup r→∞ 1 r · log � bZ(r) � = δ(Z) This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Define ε(G) = c∗(G) 4·2∥ρ∥, and assume u(r) ⪯∞ ε(G) · r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Notice that ε(G) < 1 2, and since δ(Γ) = 2∥ρ∥ Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 gives δ(Λ) ≥ � 1 − 4ε(G) � 2∥ρ∥ = 2∥ρ∥ − 4ε(G) · 2∥ρ∥ ≥ 2∥ρ∥ − c∗(G) By Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6, Λ is a lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 42 Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The question of existence of interesting groups that coarsely cover a lattice is a key question that arises naturally in the context of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The first question that comes to mind is whether there exist groups that are not commensurable to a lattice but that sublinearly, or even ε-linearly, cover one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Perhaps the growth rate point of view could be used to rule out groups that cover a lattice ε-linearly but not sublinearly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6 SBE Rigidity for Lattices of Higher Q-Rank 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 Sublinear Distortion and SBE Maps Denote a ∨ b := max(a, b) for a, b ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a pointed metric space (X, x0, dX) and x, x1, x2 ∈ X, denote |x|X := dX(x, x0) and |x1 −x2|X := dX(x1, x2) (or simply |x| and |x1 −x2| when there is no ambiguity about the space X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Following Cornulier [14], Pallier [44] makes the following definition: Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A function u : R≥0 → R is admissible if it satisfies the following conditions: u is non-decreasing u grows sublinearly: lim sup r�→∞ u(r) r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' u is doubling: u(tr) u(r) is bounded above for all t > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' u ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The focus in this paper is condition 2, namely that the function u is strictly sublinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I moreover require it to be subadditive, resulting in the following terminology which I use from now on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A function u : R≥0 → R is sublinear if it is admissible and subadditive, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' u(t + s) ≤ u(t) + u(s) for all t, s > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From now on by an SBE I mean an (L, u)-SBE where u is sublinear in the sense of Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 SBE Rigidity Two finitely generated groups Γ and Λ are said to be SBE if they are SBE when viewed as metric spaces with some word metrics and base points eΓ, eΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Observe that every quasi-isometry is an SBE, and in particular the word metric is an SBE-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' An SBE admits an SBE inverse, defined as quasi-inverse maps are defined for quasi-isometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A class of groups A is said to be SBE complete if, for every finitely generated group Λ that is SBE with some group Γ ∈ A, there is a short exact sequence 1 → F → Λ → Λ1 → 1 for a finite group F ≤ Λ and some Λ1 ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In this chapter I prove: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real centre-free semisimple Lie group without compact or R-rank 1 factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The class of uniform lattices of G is SBE complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The class of non-uniform lattices of G is SBE complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 43 Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof I present is quite indifferent to whether the lattice Γ is uniform or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In order to have a unified proof and to ease notation, I fix the convention that for both uniform and non-uniform lattices, X0 denotes the compact core of the lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This just means that X = X0 in case Γ is uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' My works heavily relies on that of Drut¸u in [16], where the theorems are stated for non-uniform lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Nonetheless one readily sees that her proofs work perfectly well for uniform lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Indeed the arguments of [16] are only much simpler in the uniform case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 The Quasi-Isometry Case The outline of the proof I present for Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 is identical to that of quasi-isometric rigidity, which I now describe briefly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The main step is that for any quasi-isometry f : X0 → X0 of the compact core of Γ, there exists an isometry g : X → X such that f, g are boundedly close, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' there is some D > 0 for which d � f(x), g(x) � < D for all x ∈ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let Λ be an abstract group with a quasi-isometry q : Λ → Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Using Lubotzki-Mozes-Raghunathan ([38], see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='19 above), Γ is quasi-isometrically embedded in X as the compact core X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One can thus extend q to a quasi-isometry q0 : Λ → X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A conjugation trick allows to associate to each λ ∈ Λ a quasi-isometry fλ : X0 → X0 defined by fλ := q ◦ Lλ ◦ q−1 (Lλ : Λ → Λ is the left multiplication by λ in Λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By the first paragraph, there exists gλ ∈ Isom(X) that is boundedly close to fλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, the proof also shows that the bound D depends only on the quasi-isometry constants of fλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' These could be seen to depend only on q and not on any specific λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From this one concludes that the map λ �→ gλ is a group homomorphism Φ : Λ → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is then straightforward to show that Φ has finite kernel and that Γ ⊂ ND � Im(Φ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One then uses Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 (for higher rank groups, see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3) to deduce that Im(Φ) is a non-uniform lattice in G that is commensurable to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 The SBE Case Moving to SBE rigidity, one starts with an SBE q : Λ → Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The first step is to find an isometry of X that is close to an SBE of X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Drut¸u’s proof is preformed in the asymptotic cone of X, which allows for a smooth transition to the SBE setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Indeed, given an SBE f : X0 → X0, one can find an isometry g : X → X that is close to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The difference is that in the SBE setting, these maps are only sublinearly close.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let (X, x0) be a pointed metric space, and f, g : X → X be two maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Maps f, g are said to be sublinearly close maps on X if there is a sublinear function u such that d � f(x), g(x) � ≤ u(|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 (Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let f : X0 → X0 be an SBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then there exists a unique isometry g ∈ Isom(X) that is sublinearly close to f (in X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From this point on, one would like to continue as in the quasi-isometry case: define the map Φ : Λ → G in a similar fashion and show that Γ ⊂ Nu � Im(Φ) � for some sublinear function u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' That Im(Φ) is a lattice is then a result of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6, proving Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is however one additional obstacle that is unique to the SBE setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Namely the SBE constants of fλ do depend on λ, and the resulting sublinear bound on d � fλ(x), gλ(x) � in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 is not enough to define Φ properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As far as I can see, one needs to get some uniform control on that bound in terms of the SBE constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The following statement is enough: Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='8 (Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let {fr}r∈R>0 be a family of (L′, vr)-SBE maps fr : X0 → X0, where vr(s) = L′·v(s)+v(r) for some sublinear function v ∈ O(u) and a constant L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let gr be the associated isometry given by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then for any x ∈ X0, there is a sublinear function ux ∈ O(u) such that d � fr(x), gr(x) � ≤ ux(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This type of uniform control is often needed when working with SBE maps, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Section I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 in Pallier’s thesis [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Using it, I am able to complete the argument as in the quasi-isometry case and prove: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be as in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the notations described above, the map Φ : Λ → G is a group homomorphism with Ker(Φ) finite, and there is a sublinear function u such that for Λ1 := Im(Φ) it holds that Γ ⊂ Nu(Λ1) and Λ1 ⊂ Nu(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 44 Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 is an immediate corollary of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9 and Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 Outline This section is divided into two parts that correspond to the steps of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 deals with the task of finding an isometry that is sublinearly close to an SBE, and Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 establishes the properties of the map Φ : Λ → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I keep Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 slim and concise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The main reason for this choice is that the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 is merely a mimic of Drut¸u’s argument in [16], or an adaptation of it to the SBE setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' While these adaptations are somewhat delicate, giving a complete detailed proof would require reproducing Drut¸u’s argument more or less in full.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I felt that this is not desirable, and instead I only indicate the required adaptations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I believe that a reader who is familiar with Drut¸u’s argument and with asymptotic cones could easily produce a complete proof using these indications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, there is no preliminary section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I do not present buildings or dynamical results that go into Drut¸u’s argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I only shortly present asymptotic cones and some ideas from Drut¸u’s proof of the quasi-isometry version of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 is elementary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 SBE Maps are Close to Isometries In this section I indicate how to adapt Drut¸u’s arguments in [16] in order to prove: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There is a sublinear function v = v(L, u) such that for every � L, u � SBE f : X0 → X0, there exists a unique isometry g = g(f) ∈ Isom(X) such that d � f(x), g(x) � ≤ v(|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 requires some control on the sublinear distance between f and g, in terms of the sublinear constants of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is the meaning of the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let {fr}r∈R>0 be a family of (L′, vr)-SBE maps fr : X0 → X0, where vr = L′ · v + v(r) for some sublinear function v ∈ O(u) and a constant L′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let gr be the associated isometry given by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then for any x ∈ X0, there is a sublinear function ux ∈ O(u) such that d � fr(x), gr(x) � ≤ ux(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Combined with Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10, a different way to phrase the above statement is to say that the function D : Λ× X0 → R≥0 defined by D(λ, x) = d � fλ(x), gλ(x) � is sublinear in each variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', there is a function u : R≥0 × R≥0 → R≥1 such that u is sublinear in each variable and D(λ, x) ≤ u(|λ|, |x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Outline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I begin with a short presentation of asymptotic cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I then give an account of the original proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 when f : X0 → X0 is a quasi-isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I present the routines required to modify the proof for the SBE setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I exemplify the modification procedure in a specific representative example, and finish with a road map for proving Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11 using the aforementioned routines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 Asymptotic Cones Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let (X, d) be a metric space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix an ultrafilter ω, a sequence of points xn ∈ X and a sequence of scaling factors ın −→ ω 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The asymptotic cone of X w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' xn, ın, denoted C(X), is the metric ω-ultralimit of the sequence of pointed metric spaces (X, 1 ın · d, xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The metric on C(X) is denoted dω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' See Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 in [16] for an elaborate account, including the definitions of ultrafilters and ultralimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The strength of SBE maps is that they induce bi-Lipschitz maps between the respective asymptotic cones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='14 (See e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Cornulier [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let f : X → Y be an (L, u)-SBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then f induces an L-bi-Lipschitz map C(f) : C(X) → C(Y ) between the corresponding asymptotic cones with the same scaling factors C(X) = (X, 1 ın dX, x0 n) and C(Y ) = (Y, 1 ın dY , y0 n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 45 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 The Argument A High-Level Description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The core of the argument lies in elevating an SBE f0 : X0 → X0 to an isometry g0 ∈ G = Isom(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There are two gaps to fill: first, Γ is non-uniform and so f0 is not even defined on the whole space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' And obviously, f0 is just an SBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume for a moment that Γ is uniform and that f is defined on the whole space X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Elevating f : X → X to an isometry is done by considering the map C(f) : C(X) → C(X) that f induces on an asymptotic cone C(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This map is bi-Lipschitz, and the work of Kleiner and Leeb [32] allows one to conclude that C(f) is, up to a scalar, an isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In turn, this isometry induces an isometry ∂g on the spherical building structure of X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is done by the relation between the Euclidean building structure of C(X) and the spherical building structure of ∂∞X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A theorem of Tits [56] associates to ∂q a unique isometry g ∈ Isom(X) that induces ∂f as its boundary map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By construction, it is then not too difficult to see that g and f are ‘close’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In case Γ is non-uniform, an SBE f : Γ → Γ does not readily yield a cone map on C(X), but only on C(X0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Overcoming this difficulty requires substantial work and is the heart of Drut¸u’s proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In short, she uses dynamical results stating that the vast majority of flats in X are close enough to X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As mentioned in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2, the building structure on X(∞) is determined by the boundaries of flats F ⊂ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The same holds for the (Euclidean) building structure of C(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore the fact that the majority of flats in X are ‘close enough’ to X0 results in the fact that C(X0) composes the majority of C(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is a very rough sketch of the logic behind Drut¸u’s argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The procedure described above results in an isometry g ∈ Isom(X) associated to f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' To complete the argument one needs to verify that the map f0 �→ g is a group homomorphism between SBE(Γ) = SBE(X0) and Isom(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Composing this map with a representation of Λ into SBE(Γ) yields a map Λ → G by λ �→ fλ �→ gλ := gfλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A computation then shows that this map has finite kernel and that Γ lies in a sublinear neighbourhood of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Flat Rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The adaptations that are required for the SBE setting lie mainly in the part of Drut¸u’s work that concerns flat rigidity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' That is, the proof that the quasi-isometry q0 maps a flat F ⊂ X0 to within a uniformly bounded neighbourhood of another flat F ′ ⊂ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is proved by passing to the cone map, using an analogous result for bi-Lipschitz maps between Euclidean buildings, which translates back down to the space X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Drut¸u’s argument requires many geometric and combinatorial definitions - some classical and widely known (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', Weyl chambers of a symmetric space X) some less known (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' an asymptotic cone with respect to an ultrafilter ω) and some new (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', the horizon of a set A ⊂ X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I use her definitions, terminology and notations freely without giving the proper preliminaries or even the definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I assume most readers who are interested in the question of SBE rigidity are familiar to some extent with most of these objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For the new definitions, I try to say as little as needed to allow the reader to follow the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof consists of 6 steps: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The horizon of an image of a Weyl chamber is contained in the horizon of a finite union of Weyl chambers, and the number of chambers in this union depend only on the Lipschitz constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' (Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 in [16], consult Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='16 below for a sketchy definition of horizon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The horizon of an image of a flat coincides with the horizon of a finite union of Weyl chambers, and the number of chambers depends only on the Lipschitz constant of the quasi-isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The union of Weyl chambers in the previous step limits to an apartment in the Tits building at X(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Such a union is called a fan over an apartment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For each Weyl chamber W there corresponds a unique chamber W ′ such that q0(W) and W ′ have the same horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This amounts to an induced map on the Weyl chambers of the Tits building at X(∞) (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 in [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 46 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Given a flat F through a point x, the unique flat F ′ asymptotic to the union of Weyl chambers obtained in step 3 is at uniform bounded distance from f0(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The bound depends only on the quasi-isometry constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The flat F ′ is called the flat associated to F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If F1 and F2 are two flats through x which intersect along a hyperplane H, then the boundaries at X(∞) of the associated flats F ′ 1 and F ′ 2 intersect along a hyperplane of the same codimension as H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For a precise definition of horizon see [16], section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For now, it suffices to say the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The horizon of a set A ⊂ X is contained in the horizon of a set B ⊂ X if, looking far away at A from some point x ∈ X, A appears to be contained in an ε-neighbourhood of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This intuition is made precise by considering the angle at x that a point a ∈ A makes with the set B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Two sets have the same horizon if each set’s horizon is contained in the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the case A and B have the same horizon, an important aspect is the distance R starting from which A and B seem to be ε-contained in one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Call this distance the horizon radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It depends on x and ε.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proofs for most of these steps have similar flavour: in any asymptotic cone C(X), f0 induces a bi-Lipschitz map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Kleiner and Leeb [32] proved many results about such maps between cones of higher rank symmetric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One assumes towards contradiction that some assertion fails (say, in step 5, assume that there is no bound on the distance between f0(xn) and the associated flat F ′ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This gives an unbounded sequence of scalars (say, d � f0(xn), F ′ n � = ın → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' These scalars are used to define a cone in which one obtains a contradiction to some fact about bi-Lipschitz maps (say, that the point [q0(xn)]ω is at dω-distance 1 from [F ′ n]ω, while it should lie in [F ′ n]ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Typically, the bounds obtained this way depend on the quasi-isometry constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A priori, they also depend on the specific point x ∈ X or flat F ⊂ X in which you work (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' the horizon radius for the chambers in step 3 or the bound on d � q(x), F ′� in step 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' However, it is easy to see that in the quasi-isometry setting, the bounds are actually independent of the choice of point/flat/chamber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This independence stems from the fact that one can pre-compose f0 with an isometry translating any given point/flat/chamber to a fixed point/flat/chamber (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' ), without changing the quasi-isometry constants (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11 in [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that these bounds depend only on the quasi-isometry constants is essential for the proof that the map Λ → Isom(X) has the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moving to the SBE setting, the essential difference is exactly that the bounds one obtains depend on the specific point, Weyl chamber or flat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Indeed it is clear that these bounds should depend on the size |x|, as they depend on the additive constant in the quasi-isometry case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is sensible to guess though that the bounds only grow sublinearly in |x|, which is enough in order to push the argument forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the next section I show how to elevate a typical cone argument from the quasi-isometry setting to the SBE setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I focus on showing that the bound one obtains depend only on the SBE constants (L, u) and sublinearly |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3 Generalization to SBE: Adapting Cone Arguments To adapt for the SBE setting, split each step into three sub-steps: the first two amount to proving Theo- rem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10, and the third step amounts to proving Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Sub-Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Repeat the argument of the quasi-isometry setting verbatim, to obtain a bound c = c(x) which depend on the point x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Sub-Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume towards contradiction that there is a sequence of points xn for which limω c(xn) |xn| ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This means limω |xn| c(xn) ̸= ∞, and so the point (xn)n lies in the cone C(X) = Cone � X, x0, c(xn) � , and one may proceed as in the corresponding quasi-isometry setting to obtain a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Sub-step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix x ∈ X and a sequence of SBE maps as in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' {fn}n∈N with the same Lipschitz constant and with sublinear constants vn(s) = v(s) + u(n), for some sublinear functions u, v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote by cn(x) the constants that were achieved in the previous steps for x and the SBE map fn, and assume towards 47 contradiction that |cn(x)| is not bounded above by any function sublinear in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This means in particular that limω u(n) cn(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One concludes that the cone map C(frn) is bi-Lipschitz, and gets a contradiction in the same manner as in the first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In order to give the reader a sense of what is actually required, I now demonstrate this procedure in full in a specific claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I chose to do this for proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 of [16], which is complicated enough to require some attention to details, but not too much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The statement is as follows: Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='18 (SBE version of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7 in [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let f : X → X be an (L, u)-SBE, and F ⊂ X a flat through x to which f associates a fan over an apartment, ∪p i=0Wi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If F ′ is the maximal flat asymptotic to the fan, then d � f(x), F ′� ≤ c(x) where c(x) = c(|x|) is sublinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, let fn : X → X be a sequence of (L, vn) SBE maps for vn = v + u′(n) for v, u′ some sublinear functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The constant cn(x) associated to x and the SBE fn achieved in the first part of the proposition admits cn(x) ≤ ux(n) for some sublinear function ux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' F ′ is then said to be the associated flat to F by f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proceed in two (sub-)steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I show that for a given x, there exists such a constant c(x) independent of the flat F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This is done exactly as in [16], but I repeat the proof here because it contains the terminology and necessary preparation for the second step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Fix x ∈ X and assume towards contradiction that there exists a sequence Fn of flats through x and a sequence fn : X → X of (L, u)-SBE maps such that cn := d � fn(x), F ′ n � → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In Cone(X, x, c−1 n ) one can show that [∪p i=0W n i ], the union of Weyl chambers associated to fn(Fn), is a maximal flat (see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 in [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote Fω := [∪p i=0W n i ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Furthermore, since the bi-Lipschitz flat [fn(Fn)] ⊂ Cone(X, x, c−1 n ) is contained in it, it coincides with it: [fn(Fn)] = Fω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand, since the Hausdorff distance between ∪p i=0W n i and F ′ n is by assumption cn = d(x, F ′ n), in the cone the maximal flats Fω and F ′ ω := [F ′ n] are at Hausdorff distance 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This implies that Fω = F ′ ω (see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 in [32]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' But since d � q(x), F ′ n � = cn the limit point yω := Q(xω) = [q(x)] , which is contained in Fω, is at distance 1 from F ′ ω - a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume c(x) is taken to be the smallest possible for each x, and then modify the function c so that c(x) = maxy:|y|=|x| c(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The function c : X → R now only depends on |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I wish to show that c(|x|) = O(u(|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume towards contradiction that there exists a sequence xn with |xn| → ∞ such that limω c(xn) u(|xn|) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote cn = c(xn) and consider the cone Cone(X, xn, c−1 n ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The assumption limω c(xn) u(|xn|) = ∞ implies (x0)ω = (xn)ω hence Cone(X, xn, cn) = Cone(X, x0, cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By the definition of c(x) this means that there is a sequence of flats Fn through xn such that d � q(xn), F ′ n � = cn, so one may proceed as in step 1 for a cone with a fixed base point (x0)ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In this cone the flat F ′ ω := [F ′ n] is at distance 1 from the point [q(xn)], which lies on the maximal flat [∪p i=0W n i ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The latter flat is, on the one hand, at Hausdorff distance 1 from F ′ ω (by the definition of the scaling factors cn), so they actually coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand, [∪p i=0W n i ] coincides with Fω := Q(Fω) = [q(Fn)], so Fω = F ′ ω, contradicting the fact that dω([q(xn)], F ′ ω) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Thus c(|x|) = O(u(|x|)), as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For the moreover part (uniform control on the growth of c(x) as a function of the sublinear constants), the proof is identical to Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This time, consider a sequence fn as in the statement, and denote by cn = cn(x) the constant obtained in step 1 w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' the SBE constants (L, vn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume towards contradiction that limω u(n) cn(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof goes exactly as in step 1, with the sole difference that now one might need convincing in the fact that in C(X), the cone with 1 cn as scaling factors, the cone map C(fn) is bi-Lipschitz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' But indeed for any two cone points (xn), (yn) ∈ C(X) it holds: dω � C(fn)(xn), C(fn)(yn) � = lim ω 1 cn d � fn(xn), fn(yn) � ≤ lim ω 1 cn L · d(xn, yn) + vn(|xn| ∨ |yn|) 48 By definition of the cone metric, limω 1 cn L · d(xn, yn) = L · dω � (xn), (yn) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It thus remains to show limω 1 cn vn(|xn| ∨ |yn|) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By definition of vn it amounts to proving limω 1 cn v(|xn|) = 0 = limω 1 cn v(|yn|) and limω 1 cn u(n) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The former follows from the fact that (xn), (yn) ∈ C(X) and therefore both limω 1 cn |xn| and limω 1 cn |xn| are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The latter follows from the assumption on the cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One obtains a contradiction identical to the one in Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Following the claims of Sections 3, 4, 5 in [16] carefully, and making the SBE adaptations as depicted in the above example, one obtains flat rigidity in the SBE setting, that is Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 together with the uniform control described in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Here is the complete list of claims involving cone arguments in [16] that should be modified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' All claims starting from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='5 through Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' All statements should consider, instead of a quasi-isometry, a general (L, u)-SBE f and, when relevant, a general point x ∈ X with f(x) = y (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', f(x) does not necessarily equal x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Also when relevant one should consider a family fn of (L, vn) SBE maps as depicted in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='7, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' When relevant, statements should be modified so that the distance between f(x) and an associated flat of it should be uniformly sublinear in |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Also when relevant one should consider a family fn of (L, vn) SBE maps as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 (D = D(x) should be uniformly linear in c = c(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='2 (the constant D = D(x) should be replaced be a sublinear function D(|x|)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' When relevant one should consider a family fn of (L, vn) SBE maps as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This yields a proof for uniform flat rigidity in the SBE setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The other major part of Drut¸u’s argument concerns the fact that f0 is defined only on X0 and not on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The considerations for this aspect are intertwined in the proof, but they all involve only Γ and the quasi-isometry between Γ and X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For this reason, the fact that f0 is an SBE to begin with does not effect any of these arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, this argument is indifferent to whether or not Γ is uniform or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' If Γ is uniform all that changes is that that part of Drut¸u’s argument dealing with extending the cone map from C(X0) to C(X) is not necessary since X = X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Her proof still works perfectly well also for uniform lattices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore the argument above proves Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 Some Remarks On R-rank 1 Factors Quasi-isometric rigidity holds for groups of R-rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is worth mentioning that Schwartz’s proof also relies on ‘flat rigidity’ - but in this case the flats are the horospheres of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' While these are not isometrically embedded flats, the induced metric on horospheres is flat and Schwartz uses that in order to construct the boundary map and find the associated isometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The same phenomena occurs in the SBE setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Considering the compact core X0 ⊂ X of Γ, one can use Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6 in Drut¸u-Sapir [18], in order to show that horospheres are mapped boundedly close to horospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In general, this work characterizes and explores a certain class of spaces they call asymptotically tree graded, a class that is very suitable for the setting of the compact core of a non-uniform lattice in R- rank 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A key ingredient in the proof is the fact that the boundary map ∂q induced by the quasi-isometry is quasi-conformal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This in particular implies that it is almost everywhere differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I spent some time trying to generalize the proof of Schwartz to the SBE setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One obstacle is that it is not clear that the boundary map is going to be differentiable almost everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Gabriel Pallier found that there are SBE maps of the hyperbolic space whose boundary maps are not quasi-conformal (see Appendix A in [45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For this reason Pallier develops the notion of quasi-conformality [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' While these maps may be differentiable, he told me of examples he constructed where the differential is almost everywhere 0 - a property which also nullifies Schwartz’s argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In the context of SBE rigidity, the maps I consider seem to indeed have ‘flat rigidity’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' to map a horosphere to within bounded distance of a unique horosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As in the higher rank flat rigidity, this 49 bound is not uniform but rather grows sublinearly with the distance of the horosphere to a fixed base point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' These are very specific maps, that coarsely preserve the compact core of that lattice X0 ⊂ X and basically map horospheres to horospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This means there might still be hope for these specific maps to induce boundary maps that admit the required analytic properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In a subsequent paper [53], Schwartz proves quasi-isometric rigidity for lattices in products of R-rank 1 groups, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' in Hilbert modular groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' His proof there is different, but it also makes use of the fact that horospheres are mapped to within uniformly bounded distance of horospheres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that in the SBE setting this bound is not uniform seems like a real obstruction to any attempt of generalizing his proof in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 From SBE to Sublinearly Close Groups In this section I prove Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I restate it here for convenience Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let G be a real centre-free semisimple Lie group without compact or R-rank 1 factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let Γ ≤ G be an irreducible lattice, and Λ an abstract finitely generated group that is SBE to Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then there is a group homomorphism Φ : Λ → G with finite kernel such that Γ ⊂ Nu � Φ(Λ) � and Φ(Λ) ⊂ Nu(Γ) for a sublinear function u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof is a sublinear adaptation of the classical arguments by Schwartz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The only difference is that some calculations are in order, but there is no essential difference from Section 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 in [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Before I start, I need one well known preliminary fact, namely that sublinearly close isometries are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X be a symmetric space of noncompact type and with no R-rank 1 factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let Γ ≤ Isom(X) be a non-uniform lattice, X0 its compact core with respect to x0 ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let g, h ∈ G = Isom(X) and u a sublinear function such that for every x ∈ X0, d � g(x), h(x) � ≤ u(|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then g = h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof is essentially just the fact that a sublinearly bounded convex function is uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Up to multiplying by h−1, one may assume h = idX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' First I show that the continuous map ∂g : X(∞) → X(∞) is the identity map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Recall that the space X(∞) can be represented by all geodesics emanating from the fixed point x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let η : [x0, ξ) be a Γ-periodic geodesic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By definition, there is some T > 0 and a sequence γn ∈ G for which η(nT ) = γnx0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular, xn := γnx0 ∈ X0 hence d � g(xn), xn � ≤ u(|xn|) = u(nT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand, the distance function d � η(t), g·η(t) � is convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' A convex sublinear function is bounded, and so by definition in X(∞) one has [η] = [g · η], for all Γ-periodic geodesics η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The manifold X is of non- positive curvature, hence ∂g is a homeomorphism of X(∞), and the density of Γ-periodic geodesics implies ∂g = idX(∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This implies that g = idX (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10 in [20] for a proof of this last implication).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof for the fact that ∂g = idX(∞) implies g = idX appears in [20] (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10) as part of the proof of the following important theorem of Tits: Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='22 ([56], see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='1 in [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let X, X′ be symmetric spaces of noncompact type and of higher R-rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume X has no R-rank 1 factors, and let φ : X(∞) → X′(∞) be a bijection that is a homeomorphism with respect to the cone topology and an isometry with respect to the Tits metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then, after multiplying the metric of X by positive constants on de Rham factors, there exists a unique isometry g : X → X′ such that φ = ∂g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' This theorem is actually a key ingredient in Drut¸u’s argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Much of her work is directed towards showing that the cone map C(q) will correspond to a map on X(∞) satisfying the above hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The restriction to X with no R-rank 1 factors in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 comes from this restriction in Tits’ Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='22 Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof that ∂g = idX(∞) ⇒ g = idX only uses the fact that X has no Euclidean de Rham factors (see pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 251 in [20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Since I only use it in the setting of no R-rank 1 factors, I added that assumption to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 50 The Map Φ : Λ → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The orbit map q0 : Γ → X0 defined by γ �→ γx0 is a quasi-isometric embedding: this is ˘Svarc-Milnor in case Γ is uniform and X0 = X, and Lubotzki-Mozes-Raghunathan (Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='19 above) if Γ is non-uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' An SBE f : Λ → Γ thus gives rise to an SBE Λ → X0, which I also denote by f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For each λ ∈ Λ let fλ := f ◦ Lλ ◦ f −1 : X0 → X0, where Lλ is the left multiplication by λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The left translation Lλ is an isometry, hence fλ is a self SBE of X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='10, there exists a unique isometry gλ ∈ Isom(X) that is sublinearly close to fλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Define the map Φ : Λ → G by λ �→ gλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The goal in this section is to prove Φ is a homomorphism with finite kernel, and that Γ and Φ(Λ) are each contained in a sublinear neighbourhood of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I begin by controlling the SBE constants of the fλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For each λ ∈ Λ, fλ is an (L2, vλ)-SBE, for vλ(|x|) := (L + 1)u(|x|) + u(|λ|) In particular vλ ∈ O(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Before the proof I state a corollary which follows immediately by combining Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='24 with Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Assume Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='11 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Then for any x ∈ X there is a sublinear function ux such that d � fλ(x), gλ(x) � ≤ ux(|λ|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The proof is a straightforward computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Up to an additive constant I may assume f −1 is an (L, u)-SBE with f −1(eΓ) = eΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let x1, x2 ∈ X0, and assume w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g |x2| ≤ |x1|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By the properties of an SBE, this also means that for i ∈ {1, 2}: |f −1(xi)| ≤ L|xi − x0| + u(|xi|) ≤ L|x1| + u(|x1|) (6) Notice that fλ(x) = f � λ · f −1(x) � , and f is an (L, u)-SBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The following inequalities, justified below, give the required upper bounds: ��fλ(x1) − fλ(x2) �� ≤ L · ��λf −1(x1) − λf −1(x2) �� + u � |λf −1(x1)| ∨ |λf −1(x2)| � ≤ L2|x1 − x2| + Lu � |x1| � + u � |λ|) + L|x1| + u(|x1|) � ≤ L2|x1 − x2| + (L + 1)u � |x1| � + u(|λ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' (7) From the first line to the second line I used: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For the first term: left multiplication in Λ is an isometry, and f −1 is an (L, u)-SBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' For the second term: triangle inequality, left multiplication in Λ is an isometry, and Equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From the second to the third line I used the properties of u as an admissible function, namely that it is sub-additive and doubling, so u � (L + 1)|x1| � ≤ (L + 1)u(|x1|) for all large enough x1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I remark that the proof of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='24 is the only place where I use the properties of an admissible function and not just the sublinearity of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Φ : Λ → G is a group homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let λ1, λ2 ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I begin with some notations: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' f1 = fλ1, f2 = fλ2, f12 = fλ1λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='24, these are all O(u) SBE maps with the same Lipschitz constant L′ := L2 and sublinear constants v1, v2, v12 ∈ O(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 51 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' g1 = Φ(λ1), g2 = Φ(λ2), g12 = Φ(λ1λ2) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' u1, u2, u12 the sublinear functions that bound the respective distances between any g and f, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' |g1(x) − f1(x)| ≤ u1(|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' One has to prove that gλ2 ◦ gλ1 = gλ1λ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In view of Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='20, it is enough to find a sublinear function v such that for all x ∈ X0 |g1g2(x) − g12(x)| ≤ v(|x|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By triangle inequality and the above definitions and notation, it is enough to show that each of the following four terms are bounded by a function sublinear in x: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' |g1g2(x) − g1f2(x)| = |g2(x) − f2(x)| ≤ u2(|x|) (g1 is an isometry).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' |g1f2(x) − f1f2(x)| ≤ u1 � |f2(x)| � ≤ u1 � L2|x| + v2(|x|) � 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' |f1f2(x) − f12(x)| 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' |f12(x) − g12(x)| ≤ u12(|x|) Clearly items 1, 2, 4 are bounded by a sublinear function in |x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It remains to bound |f1f2(x) − f12(x)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The map Λ → Aut(Λ) given by λ �→ Lλ is a group homomorphism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Lλ1λ2 = Lλ1Lλ2, so it remains to bound: |f1f2(x) − f12(x)| = |fLλ1f −1fLλ2f −1(x) − fLλ1Lλ2f −1(x)| f ◦ Lλ is a composition of an isometry with an SBE, so it is still an SBE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Denote the SBE constants of fLλ1 by L′, v (clearly one can take L′ = L and v ∈ O(u), but this is not needed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Writing y := Lλ2f −1(x), this shows |fLλ1f −1f(y) − fLλ1(y)| ≤ L|f −1fy − y| + v(|f −1fy| ∨ |y|) By definition of an SBE inverse it holds that |f −1f(y) − y| ≤ u(|y|) and in particular also |f −1f(y)| ≤ |y| + u(|y|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that |f1f2(x) − f12(x)| ≤ L · u(|y|) + v � |y| + u(|y|) � The right-hand side is a sublinear function in |y|, hence it only remains to show that |y| is bounded by a linear function in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Indeed |y| = |Lλ2f −1(x)| ≤ |λ2| + |f −1(x)| ≤ |λ2| + L|x| + u(|x|) This completes the proof, rendering Φ a group homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Φ has discrete image and finite kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I show that for any radius R > 0, there are finitely many λ ∈ Λ for which gλx0 ∈ B(x0, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=', that there is a finite number of Φ(Λ)-orbit points, with multiplicities, inside an R ball in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In particular the set {λ ∈ Λ | gλx0 = x0} is finite, and clearly contains Ker(Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' In addition, the actual number of Φ(Λ)-orbit points inside that R ball is finite, so Φ(Λ) is discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let R > 0, and λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' By the defining property of gλ and the definition of fλ, reverse triangle inequality gives d � x0, gλ(x0) � ≥ d � x0, fλ(x0) � − d � gλ(x0), fλ(x0) � ≥ |f(λ)| − uλ(|x0|) Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='25 gives d � gλ(x0), fλ(x0) � ≤ ux0(|λ|) for some sublinear function ux0 ∈ O(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' On the other hand f is an SBE, and so |f(λ)| grows close to linearly in λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Formally, |f(λ)| = d � f(λ), x0 � = d � f(λ), f(eΛ) � ≥ 1 Ld(λ, x0) − u(|λ| ∨ |eΛ|) ≥ 1 L|λ| − u(|λ|) 52 To conclude, one has d � x0, gλ(x0) � ≥ 1 L|λ| − u(|λ|) − ux0(|λ|) and both u, ux0 are sublinear in |λ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Therefore there is a bound S ∈ R>0 such that |λ| > S ⇒ 1 L|λ| − u(|λ|) − ux0(|λ|) > R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The group Λ is finitely generated and so only finitely many λ ∈ Λ admit |λ| ≤ S, hence gλ(x0) ∈ B(x0, R) only for finitely many λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There exists a sublinear function u′ : R≥0 → R≥1 such that Γ · x0 ⊂ Nu′� Φ(Λ) · x0 � Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I claim that there is a sublinear function u0, depending only on f and q0, such that for all γ ∈ G, d � gλ(x0), γ(x0) � ≤ u0(|γ|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' As before, I only have control on gλ via fλ, and so I use triangle inequality to get: d � gλ(x0), γ(x0) � ≤ d � gλ(x0), fλ(x0) � + d � fλ(x0), γ(x0) � By Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='25, d � gλ(x0), fλ(x0) � ≤ ux 0(|λ|) for a sublinear function ux0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' It is beneficial to distinguish between the SBE fΓ : Λ → Γ and the same SBE composed with the orbit quasi-isometry q0 : Γ → X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From now on I keep the notation fΓ : Λ → Γ for the SBE of the groups and f0 for the same SBE composed with the orbit quasi-isometry so f0 = q0 ◦ fΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Define λγ := f −1 Γ (γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I show that d � fλγ(x0), γ(x0) � is bounded by a function sublinear in γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Indeed, recall that I assumed without loss of generality q0 ◦f −1 Γ (x0) = eΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Moreover, Γ is assumed to be torsion-free, and so there is no ambiguity or trouble in defining the restriction of the map q−1 to the orbit Γ · x0 to be of the form q−1(γx0) = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' All together, this gives d � fλγ(x0), γ(x0) � = d � f0(λγ), γ(x0) � = d � q0 ◦ fΓf −1 Γ (γ), q0(γ) � Since fΓ is an SBE d � fΓf −1 Γ (γ), γ � ≤ u(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' The fact that q0 is an (L′, C)-quasi-isometry implies that d � q0 ◦ fΓf −1 Γ (γ), q(γ) � ≤ Ld � fΓf −1 Γ (γ), γ � + C ≤ L′u(|γ|) + C Combining everything, one has d � gλγ(x0), γ(x0) � ≤ ux0(|λγ|) + L′u(|γ|) + C As before, |λγ| = |f −1(γ)| ≤ |γ| + u(|γ|) ≤ 2|γ|, where the last inequality holds for all large enough γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' What matters is that |λγ| is linear in |γ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' I conclude that indeed Γ · x0 ⊂ Nu′� Φ(Λ) · x0 � for the sublinear function u′ = ux0 + L′u + C ∈ O(u), as wanted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' (To be pedantic, u′ = ux0 ◦ 2 + L′u + C ∈ O(u) where 2 is the ‘multiplication by 2’ function, r �→ 2r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' There exists a sublinear function u′ : R≥0 → R≥1 such that Φ(Λ) · x0 ⊂ Nu′(Γ · x0) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let λ ∈ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Let γ = f(λ) and consider the distance d(gλx0, γx0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' From triangle inequality and Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='25 one has d(gλx0, γx0) ≤ d(gλx0, fλx0) + d(fλx0, γx0) ≤ u′(|λ|) + d(fλx0, γx0) By definition of fλ it holds that fλ(x0) = f(λ) · x0 = γ · x0 and so d � fλ(x0), γx0 � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Claims 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 result in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='4 then follows from Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' 53 References [1] Paul Albuquerque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Patterson-Sullivan theory in higher rank symmetric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' GAFA Geom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Funct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/7tE4T4oBgHgl3EQfcwyw/content/2301.05086v1.pdf'} +page_content=' Anal.' metadata={'source': 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