diff --git "a/EtE3T4oBgHgl3EQfVQrm/content/tmp_files/load_file.txt" "b/EtE3T4oBgHgl3EQfVQrm/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/EtE3T4oBgHgl3EQfVQrm/content/tmp_files/load_file.txt" @@ -0,0 +1,1865 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf,len=1864 +page_content='ALGEBRAIC ACTIONS II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' GROUPOID RIGIDITY CHRIS BRUCE AND XIN LI Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We establish rigidity for partial transformation groupoids associated with algebraic actions of semigroups: If two such groupoids (satisfying appropriate conditions) are isomorphic, then the globalizations of the initial algebraic actions rationally embed in each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For specific example classes arising for instance from toral endomorphisms, actions from rings, or actions from commutative algebra, this mutual embedability can be improved in various ways to obtain surprisingly strong rigidity phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This is witnessed in a particularly striking fashion for actions arising from algebraic number theory: We prove that the groupoids associated with the action of the multiplicative monoid of non-zero elements in a ring of algebraic integers on the additive group of the ring remembers the initial algebraic action up to isomorphism, which in turn remembers the isomorphism class of the ring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This resolves an open problem about isomorphisms of Cartan pairs and leads to a dynamical analogue of the Neukirch–Uchida theorem using topological full groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Introduction Algebraic actions of groups form an interesting and important class of dynamical systems which provides a rich supply of actions of general groups (see, for instance, [49]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' On the one hand, this example class is interesting and exhibits new phenomena, and on the other hand, due to the particular structure of algebraic actions, a variety of tools is available, allowing for a systematic and detailed study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In [13], we initiated the study of one-sided or irreversible analogues, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', algebraic actions of semigroups, which have not been studied in detail before in general (but see [30, 22] and the references in [13] for special cases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' An interesting new phenomenon that arises in this new setting is that actions by non-invertible endomorphisms of a given group automatically produce a particular completion of the group, and the original action induces a system of partial homeomorphisms on this completion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The idea of [13] was to study the corresponding groupoids, which are interesting in their own right but also give access to analyzing properties of C*-algebras generated by natural representations of the initial algebraic action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Our goal now is to study the natural question of how much information the groupoids constructed in [13] remember about the original algebraic actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Surprisingly, we discover the phenomenon of groupoid rigidity for a variety of example classes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', our groupoids remember more information than expected – in special cases, they even remember everything.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us now formulate our rigidity results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' An algebraic action σ: S ↷ A consists of a monoid S, an Abelian group A, and a semigroup homomorphism from S to injective endomorphisms of A, denoted by S → End (A), s �→ σs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We will always assume our algebraic actions to be non-automorphic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', not all σs are automorphisms) and faithful (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', the map s �→ σs is injective).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let C be the collection of subgroups of A which are of the form σ−1 t1 σs1 · · · σ−1 tm σsmA, where σ−1 t X := {a ∈ A: σt(a) ∈ X} for a subset X ⊆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In this paper, we will always assume that σ has the finite index property (FI), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', #(A/C) < ∞ for all C ∈ C, or equivalently, #(A/σsA) < ∞ for all s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In this case, the completion of A mentioned above is given by A := lim ←−C∈C A/C, where C is partially ordered by inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The standing assumptions in this paper will furthermore include that σ: S ↷ A admits a globalization (which is always assumed to be minimal in the sense of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', an embedding of S into a group S and a group A containing A together with an algebraic action ˜σ: S ↷ A (necessarily by automorphisms) such that ˜σs|A = σs for all s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This allows us to form a partial action of S on A by letting s ∈ S act via the restriction A ∩ ˜σ−1 s A → ˜σsA ∩ A of ˜σs to A ∩ ˜σ−1 s A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Similarly, we Date: January 12, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Primary 37A20, 37B99, 22A22;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Secondary 20M18, 37A55, 46L05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Bruce was supported by a Banting Fellowship administered by the Natural Sciences and Engineering Research Council of Canada (NSERC) and has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 101022531.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Li has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 817597).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='04459v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='DS] 11 Jan 2023 also have the partial action of A on A by translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In [13], we identified a condition, called (JF), which ensures that these partial actions on A extend to a partial action of A ⋊ S on A by partial homeomorphisms (see § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1 and [13, § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3] for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In this paper, we will always assume that (JF) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The associated partial transformation groupoid Gσ := (A ⋊S )⋉A is the groupoid constructed in [13, § 3] arising from our algebraic action σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We can now state our main rigidity result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Theorem A (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let σ: S ↷ A and τ : T ↷ B be two algebraic actions of monoids as above, with globalizations ˜σ: S ↷ A , ˜τ : T ↷ B and groupoids Gσ, Gτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume that S and T are Abelian, that A and B are torsion-free and finite rank, and that there exist s ∈ S and t ∈ T such that idA − σs : A → A and idB − τt : B → B are injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If Gσ and Gτ are isomorphic as topological groupoids, then ˜σ: S ↷ A and ˜τ : T ↷ B embed rationally into each other, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', there exist injective homomorphisms t: S �→ T , b: Q⊗A �→ Q⊗B, s: T �→ S , and a: Q ⊗ B �→ Q ⊗ A such that b((idQ ⊗ ˜σs)(x)) = (idQ ⊗ ˜τt(s))(b(x)) and a((idQ ⊗ ˜τt)(y)) = (idQ ⊗ ˜σs(t))(a(y)) for all s ∈ S , x ∈ Q ⊗ A , t ∈ T , and y ∈ Q ⊗ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We refer the reader to § 3, in particular § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5, for more general results and further explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us now present a first class of algebraic actions where our general rigidity result applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Corollary B (see Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume that S and T are Abelian, torsion-free monoids, that A and B are torsion-free Abelian groups of finite rank, and that σ: S ↷ A and τ : T ↷ B are non- automorphic faithful algebraic actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Further suppose that the dual actions ˆσ and ˆτ are mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let ˜σ: S−1S ↷ S−1A and ˜τ : T −1T ↷ T −1B be the canonical globalizations as in [13, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4], and denote the groupoids attached to σ and τ by Gσ and Gτ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If Gσ and Gτ are isomorphic as topological groupoids, then there exist injective homomorphisms t: S−1S �→ T −1T and b: S−1A �→ T −1B such that b(˜σs(x)) = ˜τt(s)(b(x)) for all s ∈ S−1S and x ∈ S−1A, injective homomorphisms s: T −1T �→ S−1S and a: T −1B �→ S−1A such that a(˜τt(y)) = ˜σs(t)(a(y)) for all t ∈ T −1T and y ∈ T −1B, and the images of b and a are finite index subgroups of T −1B and S−1A, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The reader may consult § 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1 for more explanations and details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The concrete case of toral endomor- phisms is treated in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Another motivating example class is given by the action of the monoid of non-zerodivisors of a ring on the additive group of the ring by multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For instance, torsion-free commutative rings which are finitely generated as additive groups have received a great deal of attention because of Bhargava’s work [5, 6, 7, 8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In this setting, our rigidity result implies the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Theorem C (see Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let Ri, i = 1, 2, be finitely generated torsion-free commutative rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For i = 1, 2, let R× i be the monoid of non-zerodivisors in Ri and σi : R× i ↷ Ri the algebraic action given by multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If the corresponding groupoids Gσ1 and Gσ2 are isomorphic, then Q ⊗ R1 and Q ⊗ R2 are isomorphic as Q-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Somewhat surprisingly, using very different methods, we obtain a similar rigidity result for special classes of non-commutative rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Theorem D (see Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For i = 1, 2, let Ri be a ring whose additive group is finitely generated and torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let R× i be the monoid of left regular elements (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', non-left-zerodivisors) in Ri and σi : R× i ↷ Ri the algebraic action given by left multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose that Q ⊗ R1 and Q ⊗ R2 are semisimple Q-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If the corresponding groupoids Gσ1 and Gσ2 are isomorphic, then Q ⊗ R1 and Q ⊗ R2 are isomorphic as Q-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Examples of rings that are covered by Theorem D include integral group rings of finite groups and rings of matrices over orders in algebraic number fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us now apply Theorem C to rings of algebraic integers in number fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let R be such a ring, with quotient field K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Consider the algebraic action σ: R× ↷ R by multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The completion R is given in this case by the (additive group of the) integral adele ring, and the partial action from above is given by the canonical partial action K ⋊ K× ↷ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, the groupoid Gσ in this 2 case is the partial transformation groupoid (K ⋊ K×) ⋉ R, and its C*-algebra coincides with the ring C*-algebra A[R], which has been introduced and studied in [19, 21, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since (K ⋊ K×) ⋉ R is effective, A[R] contains a canonical Cartan subalgebra D[R].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, consider the topological full group F ((K ⋊ K×) ⋉ R) of (K ⋊ K×) ⋉ R given by the group of global bisections (see for instance [40, 42]) and its commutator subgroup D((K ⋊ K×) ⋉ R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Corollary E (see Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let Ri, i = 1, 2, be rings of algebraic integers in number fields Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' With the notation introduced above, the following are equivalent: (i) K1 and K2 are isomorphic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) (K1 ⋊ K× 1 ) ⋉ R1 and (K2 ⋊ K× 2 ) ⋉ R2 are isomorphic as topological groupoids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iii) K1 ⋊ K× 1 ↷ R1 and K2 ⋊ K× 2 ↷ R2 are continuously orbit equivalent in the sense of [37, 38];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iv) (A[R1], D[R1]) and (A[R2], D[R2]) are isomorphic as Cartan pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (v) F ((K1 ⋊ K× 1 ) ⋉ R1) and F ((K2 ⋊ K× 2 ) ⋉ R2) are isomorphic as abstract groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (vi) D((K1 ⋊ K× 1 ) ⋉ R1) and D((K2 ⋊ K× 2 ) ⋉ R2) are isomorphic as abstract groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The equivalences of (i), (v), and (vi) in Corollary E gives dynamical analogues of the Neukirch–Uchida theorem from anabelian geometry which says that the absolute Galois group of a number field remembers the field up to isomorphism [43, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' A different dynamical version of the Neukirch–Uchida theorem is given in [14] using completely different techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, the structure of our groups is much different from the absolute Galois groups or the topological full groups from [14] since, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', D((K ⋊ K×) ⋉ R) is simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The equivalence of (i) and (iv) in Corollary E is in stark contrast with [39, Corol- lary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3], which says that the ring C*-algebras A[R1] and A[R2] are always isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' One conse- quence of this is that A[Z] contains a family, parameterized by all number fields, of isomorphic but non-conjugate Cartan subalgebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Semigroup C*-algebras C∗(R ⋊ R×) of ax + b-semigroups R ⋊ R× were studied in [20, 35, 36] for rings of algebraic integers R in number fields K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' C∗(R ⋊R×) has a canonical groupoid model (see [21, 35]), and hence contains a canonical Cartan subalgebra D(R⋊R×).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It was shown in [36] how to recover the Dedekind zeta function and the ideal class group of K from the Cartan pair (C∗(R⋊R×), D(R⋊R×)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' However, the natural question of whether (C∗(R⋊R×), D(R⋊R×)) completely determines the number field K has been left open in [36] (see the question at the end of [36, § 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since (A[R], D[R]) can be recovered from (C∗(R ⋊ R×), D(R ⋊ R×)), we are now able to answer this question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Corollary F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let Ri, i = 1, 2, be rings of algebraic integers in number fields Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then K1 ∼= K2 if and only if (C∗(R1 ⋊ R× 1 ), D(R1 ⋊ R× 1 )) and (C∗(R2 ⋊ R× 2 ), D(R2 ⋊ R× 2 )) are isomorphic as Cartan pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In fact, we completely resolve the more general problem left open in [12, § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2], see Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Theorem D, applied to matrix algebras over rings of algebraic integers, yields the following analogous result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Corollary G (see Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For i = 1, 2, let Ri be the ring of integers in a number field Ki, and let ni be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Consider the algebraic action σi : Mni(Ri)× ↷ Mni(Ri) by multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The groupoids Gσ1 and Gσ2 are isomorphic if and only if n1 = n2 and K1 ∼= K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We also have further equivalent statements analogous to (iii) – (vi) in Corollary E, and the analogue of Corollary F holds as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We would like to point out that we obtain more general results than the ones presented in this introduction (see § 3 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, we can additionally treat the following example classes: (a) Semigroups of canonical endomorphisms of finite rank torsion-free Abelian groups (§ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (b) Actions form adding scalars to algebraic actions of subgroups of special linear groups (§ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (c) Arithmetical S-integer dynamical systems (§ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (d) Actions of congruence monoids on rings of algebraic integers (§ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (e) Nd-actions from zero-dimensional ideals in commutative algebra (§ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (f) Actions from integral group rings of finite groups (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='23);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (g) Actions from orders in central simple algebras over number fields (Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 3 The proofs of our rigidity results are inspired by continuous orbit equivalence rigidity for odometers (see [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given an algebraic action σ: S ↷ A satisfying our standing assumptions, the restriction of the partial action of A ⋊ S ↷ A to A yields an odometer action A ↷ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Our key insight is that a careful analysis allows us to identify situations where rigidity can be upgraded from these odometer actions to groupoid rigidity in our sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For Abelian acting monoids, we take advantage of algebraic identities in semidirect product groups arising from our algebraic actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For non-Abelian acting monoids, our rigidity results rely on the structure of nilpotent elements in certain matrix algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Preliminaries 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Standing assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We first explain the standing assumptions on algebraic actions that we will assume in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let S be a non-trivial left cancellative monoid and A an Abelian group, written additively, with identity element 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume σ: S ↷ A is an algebraic S-action, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', σ: S → End Z(A), s �→ σs is a monoid homomorphism such that σs is an injective endomorphism A → A for all s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Actions of this form are called algebraic actions (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' [13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Unless S is a group, we shall assume that σ: S ↷ A is non-automorphic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', there exists s ∈ S such that σsA ⊊ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This in particular implies that A is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We will also always assume that the action σ: S ↷ A is faithful, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', s �→ σs is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let C = CS↷A be the family of S-constructible subgroups of A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', C is the smallest collection of subgroups of A such that A ∈ C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' s ∈ S and C ∈ C implies σsC, σ−1 s C ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It follows that C is closed under taking finite intersections (see [13, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We say that σ: S ↷ A satisfies the finite index property (see [13, Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1]) if (FI) #(A/σsA) < ∞ for all s ∈ S, If σ: S ↷ A satisfies (FI), then [13, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2] implies that every member of C is a finite index subgroup of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In this case, we get a compact group A := lim ←−C∈C A/C, and the canonical homomorphism A → A has kernel � C∈C C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given C ∈ C, we denote by C the kernel of the canonical projection A ↠ A/C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', the preimage of C ∈ A/C in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We have a canonical homeomorphism C ∼= lim ←−D∈C, D⊆C C/D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It suffices to check (FI) for generators of S: Say S is generated by S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then we can proceed inductively on the word length with respect to S of an element in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose #(A/σsA) < ∞ for some s ∈ S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', A = R + σsA for some finite set R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, for t ∈ S, write A = F + σtA for some finite set F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then A = R + σsA = R + σs(F + σσA) = R + σsF + σstA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence it follows that #(A/σstA) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In this paper, we will only consider algebraic actions σ: S ↷ A which satisfy (FI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let ∂ �E be the compact space of characters on the semilattice E := {b + C : C ∈ C, b ∈ A} ∪ {∅} (see [13, § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Each x = (xC + C)C ∈ A determines an element χx of ∂ �E by χx(b + C) := � 1 if xC + C = b + C, 0 if xC + C ̸= b + C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since (FI) is satisfied, it is not hard to see that the map A → ∂ �E given by x �→ χx is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In addition, we will always assume that σ: S ↷ A has a globalization ˜σ: S ↷ A , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', S embeds into the group S , A is a subgroup of the group A , and ˜σ: S → Aut(A ) is an algebraic action such that ˜σs|A = σs for all s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then A ⋊ S acts on A by affine maps: (z, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='x = z + ˜σγ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Reducing to A ⊆ A , we get a partial action (in the sense of [25]) on the group A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Explicitly, g ∈ A ⋊ S acts by the partial bijection A ∩ g−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='A → (g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='A) ∩ A, x �→ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 4 Note that s ∈ S acts via σs (where we view both maps as partial bijections on A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Consider the condition (JF) C ⊆ ker (id − ˜σg) =⇒ g = 1 for all C ∈ C, g ∈ ⟨S⟩, where ⟨S⟩ is the subgroup of S generated by S (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' [13, § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, our standing assumptions in this paper include that (JF) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In this case, the partial action A ⋊ S ↷ A extends uniquely to a partial action A ⋊ S ↷ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given g ∈ A ⋊ S , let Ug−1 ⊆ A be the domain of g, and for x ∈ Ug−1, we let g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='x denote the image of x under g with respect to the action A ⋊ S ↷ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The associated transformation groupoid (A ⋊ S ) ⋉ A := {(g, x) ∈ (A ⋊ S ) × A : x ∈ Ug−1, g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='x ∈ A} is canonically isomorphic to the groupoid Gσ = Iσ ⋉ ∂ �E from [13, § 3] (see [13, § 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since (FI) is satisfied, (A ⋊ S ) ⋉ A is minimal by [13, Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By [13, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='14], (A ⋊ S ) ⋉ A is effective if and only if σ: S ↷ A is exact in the sense of [13, Definition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='11], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', � C∈C C = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If S is left Ore, then we can replace (JF) by the condition that C ⊆ ker (σs − σs′) for some C ∈ C implies that s = s′ (see [13, Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='17 (iii)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If S is left Ore, then there is a canonical partial action of G on A in general, without the assumption that (JF) holds: We first construct the enveloping action as in [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' σ extends to an action of S of A, also denoted by σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then set S−1A := lim −→S � A, σ � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' this is a locally compact (non- compact) topological group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Extend σ to σ : ⟨S⟩ ↷ S−1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This way, we obtain a global dynamical system S−1A ⋊ ⟨S⟩ ↷ S−1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then A is a clopen subset of S−1A, so that we obtain the desired partial dynamical system by restricting S−1A ⋊ ⟨S⟩ ↷ S−1A to A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' However, note that (JF) holds automatically in this setting if A is torsion-free by [13, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We can and will always assume that S is generated by S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', S = ⟨S⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, by [13, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='7], if S embeds into the group S , then we can always take A = ZS ⊗ZS A, and the map A → ZS ⊗ZS A, a �→ 1 ⊗ a will always be injective if σ admits a globalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In this case, we have A = ⟨� s∈S s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='(1 ⊗ A)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence, for any globalization ˜σ : S ↷ A , we may and will always assume that (1) A = ⟨� s∈S ˜σs(A)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If S is left Ore, then we can always take S = S−1S, and in this case, we can and will always arrange that A = � r∈S ˜σ−1 r (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Further properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us now discuss a few properties which are not part of our standing assumptions, but which we will assume for some of our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The principal S-constructible subgroups are cofinal in C if (PC) for every C ∈ C, there exists s ∈ S such that σsA ⊆ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that (PC) is satisfied if S is left reversible (see [13, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Consider the following condition on the algebraic action S ↷ A : (F) For all 1 ̸= s ∈ S , 1 − ˜σs := id − ˜σs : A → A is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Condition (F) is a freeness condition, modulo the fact that in the linear setting 0 will always be a fixed point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' From cocycles to embeddings Let σ: S ↷ A and τ : T ↷ B be algebraic actions with globalizations ˜σ: S ↷ A and ˜τ : T ↷ B, respectively, which satisfy all our standing assumptions from § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1 (and we use the same notation as in § 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We will often view S as a submonoid of A ⋊ S via S → A ⋊ S , s �→ (0, s) and A as a subgroup of A ⋊ S via A → A ⋊ S , a �→ (a, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, we will use multiplicative notation for A ⋊ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 5 Suppose c: (A ⋊ S ) ⋉ A → B ⋊ T is a continuous cocycle (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', a groupoid homomorphism) such that c−1(0, 1) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that c satisfies the cocycle identity c(gh, x) = c(g, h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='x)c(h, x) whenever these expressions make sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We denote the map A ⋊ S → B ⋊ T , p �→ c(p, 0) again by c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For each p ∈ A ⋊ S, there exists C(p) ∈ C such that c(p, −): A → B ⋊ T is constant on x + C(p) for every x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since A is compact and c is continuous, the image of c(p, −) must be a finite set, which we shall denote by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For each f ∈ F, let Uf := c(g, −)−1({f}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then each Uf is compact open, and we have a partition A = � f∈F Uf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since the collection {x + C : C ∈ C, x ∈ A} forms a base consisting of compact open sets for the topology of A, we can write Uf as a finite disjoint union Uf = � i xi + Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If we now set Cf := � i Ci, then since Cf is a finite index subgroup of each Ci, we can even write Uf as a disjoint union of the form � i yi + Cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now set C(p) := � f∈F Cf, so that A is a finite disjoint union A = � k xk + C(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For all x ∈ A, there exists k such that x + C(p) = xk + C(p), and c(g, −) is constant on xk + C(p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For every finitely generated subgroup A of A, there exists CA ∈ C such that c(x, −) is constant on a + CA for all a ∈ A and x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let a ∈ A, and let {xi}i be a finite collection of elements in A that generate A as an additive monoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Set CA := � i C(xi, 1) where the subgroups C(xi, 1) are provided by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We now show that c((x, 1), −) is constant on a + CA by induction on the word length ℓ(x) of x with respect to {xi}i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The induction base case ℓ(x) = 1 follows from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now suppose the claim is true for all x ∈ A with ℓ(x) = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given x ∈ A with ℓ(x) = l + 1, we can write x = xix′ for some index i and x′ ∈ A with ℓ(x′) = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now we have for all a + x, a + y ∈ a + CA, c((x, 1), a + x) = c((xi, 1)(x′, 1), a + x) = c((xi, 1), (x′, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (a + x)) c((x′, 1), a + x) = c((xi, 1), (x′, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (a + x)) c((x′, 1), a + y) = c((xi, 1), x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (a + x)) c((x′, 1), a + y) = c((xi, 1), x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (a + y)) c((x′, 1), a + y) = c((x, 1), a + y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Here, we used the induction hypothesis for the third equality and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1 for the fifth equality (x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (a + x) and x′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (a + y) both lie in x′ + a + CA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2 can also be derived from the general results in [16], but we chose to give a direct proof in our special setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For every finitely generated subgroup A of A, the map A ∩ CA → B ⋊ T given by x �→ c(x, 0) is an injective group homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Additivity is easy to see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now suppose we have x, y ∈ A ∩ CA with c(x, 0) = c(y, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Consider the element (x, 0)(y, 0)−1 of the groupoid (A ⋊ S ) ⋉ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since c is a groupoid homomorphism, c((x, 0)(y, 0)−1) = (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence, using the assumption c−1(0, 1) = A, we conclude that (xy−1, 0) = (x, 0)(y, 0)−1 ∈ A, which implies x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Note that A ∩ CA is a finite index subgroup of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In particular, if A is finitely generated, then we get an injective group homomorphism from a finite index subgroup of A into B ⋊ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The additive homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given a finitely generated subgroup A of A, we now want to find l ∈ Z>0 and C ∈ C together with a homomorphism b: C := l(A∩C) → B such that c(x) = (b(x), 1) for all x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose that A ⊆ A is a finite rank subgroup and that s ∈ S satisfies σs(A) ⊆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let d be the degree of the polynomial det(z − ˙σs), where ˙σs := idQ ⊗ (σs|A): Q ⊗ A → Q ⊗ A, and let κd ∈ Z>0 be the smallest positive integer such that p(z) := κd det(z − ˙σs) has integer coefficients, and write p(z) = κdzd − κd−1zd−1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' − κ1z − κ0 (for some κ• ∈ Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then for every finitely generated subgroup A of A, there exists ˇC ∈ C depending on s such that (i) The restriction of c to ˇC := A ∩ ˇC is an injective group homomorphism ˇC → B ⋊ T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) c(s)dc(x)κd = c(x)κ0c(s) · · · c(x)κd−1c(s) for all x ∈ ˇC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iii) The following holds for all x ∈ ˇC: If c(x) = (β, α) and c(s) = (δ, γ), then (2) γdακd = ακ0γακ1γ · · · ακd−1γ in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 6 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us denote the restriction of σs to A again by σs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For all x ∈ A, the following holds in A⋊S as κdσd s(x) = κd−1σd−1 s (x) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' + κ0x in A: sd(κdx) = κdσd s(x)sd = (κ0x)s(κ1x)s · · · (κd−1x)s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given a finitely generated subgroup A of A, choose CA as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Set ˇC := C(s) ∩ σ−1 s C(s) ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' ∩ σ−(d−1) s C(s) ∩ CA ∩ σ−1 s CA ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' ∩ σ−(d−1) s CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then (i) is satisfied because of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We have for all x ∈ ˇC: c(sd(κdx)) = c(sd(κdx), 0) = c(sd, κdx)c(κdx, 0) = c(sd−1, σs(κdx))c(s, κdx)c(κdx, 0) = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' = c(s, σd−1 s (κdx))c(s, σd−2 s (κdx)) · · · c(s, κdx)c(κdx, 0) = c(s, 0)c(s, 0) · · · c(s, 0)c(κdx, 0) = c(s)dc(κdx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Here we are allowed to replace σ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' s(x) by 0 because x lies in C(s) ∩ σ−1 s C(s) ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' ∩ σ−(d−1) s C(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We also have for all x ∈ A ∩ ˇC: c((κ0x)s(κ1x)s · · · (κd−1x)s) = c((κ0x)s(κ1x)s · · · (κd−1x)s, 0) = c((κ0x)s(κ1x)s · · · (κd−1x), 0)c(s, 0) = c((κ0x)s(κ1x)s · · · , κd−1x)c(κd−1x, 0)c(s, 0) = c(κ0x, σs(κ1x) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' + σd−1 s (κd−1x))c(s, (κ1x) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' + σd−2 s (κd−1x)) · · c(κd−2x, σs(κd−1x))c(s, κd−1x) c(κd−1x, 0)c(s, 0) = c(κ0x, 0)c(s, 0) · · c(κd−2x, 0)c(s, 0) c(κd−1x, 0)c(s, 0) = c(κ0x)c(s) · · · c(κd−1x)c(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Here we are allowed to replace the second arguments of c by 0 because x lies in C(s)∩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='∩σ−(d−1) s C(s)∩ A ∩ CA ∩ σ−1 s CA ∩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' ∩ σ−(d−1) s CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, we have for all x ∈ ˇC and κ ∈ Z that c(κx) = c(x)κ because x lies in A ∩ CA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' So all in all, we have c(s)dc(x)κd = c(x)κ0c(s) · · · c(x)κd−1c(s) for all x ∈ ˇC = A ∩ ˇC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This shows (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now let us prove (iii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose that c(x) = (β, α) and c(s) = (δ, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Comparing T -components, we obtain γdακd = ακ0γακ1γ · · · ακd−1γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose we are in the setting of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5, and put ϵ := κd − κd−1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' κ1 − κ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If T is Abelian, then (2) is equivalent to αϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If p(z) = zd − κ0, then (2) is γdαγ−d = ακ0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', α and γd satisfy the defining relation for the Baumslag–Solitar group BS(1, κ0) ∼= Z[1/κ0] ⋊ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If γd and α both have infinite order, then ⟨γd, α⟩ ∼= BS(1, κ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' With the same notation as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5, set ϵ := κd − κd−1 − .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' κ1 − κ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In addition to the assumptions in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5, assume that 1 − σs is injective on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then we have ϵ ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If, in addition, every 2-generated subgroup of T is free or Abelian, then αϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The first claim follows from κd det(1 − ˙σs) = ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For the second claim, our assumption implies that the subgroup ⟨α, γ⟩ ⊆ ⟨T⟩ is free or Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We claim that α and γ must commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Indeed, it suffices to treat the case that the subgroup is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Because α and γ satisfy the non-trivial relation (2), the group ⟨α, γ⟩ is either trivial or infinite cyclic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' if it is trivial, we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose ⟨α, γ⟩ is cyclic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then, in particular, αγ = γα, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now (2) becomes γdακd = γdακ0+κ1+···κd−1, which implies that αϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us mention two classes of groups whose 2-generated subgroups are either free or Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' A group is called semifree if it has a presentation where the only relators are of the form 7 [s, t], where s and t are generators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If s, t are elements in a semifree group and [s, t] ̸= 1, then {s, t} is a basis for a free group by [1, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' A group is 2-free if every subgroup generated by 2 elements is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given a non-empty set of prime numbers ω, a Dω-free group is a group whose elements each have exactly one p-th root for all primes p ∈ ω (see the introduction in [2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By [3, § 8], every Dω-free group from [2] is 2-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us introduce the following conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let A ⊆ A be a subgroup of finite rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Consider the following conditions: (i1) There exists s ∈ S such that σs(A) ⊆ A, 1 − σs is injective on A, and every 2-generated subgroup of T is free or Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i2) There exists s ∈ S with σs = κ idA for some κ ∈ Z\\{0, 1}, and for all α ∈ T , if α is conjugate to ακ in T , then α must be torsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If A ⊆ A is a subgroup of finite rank and (i1) or (i2) holds, then for all finite generated subgroups A ⊆ A, there exist l ∈ Z>0 and C ∈ C such that, with C := l(A ∩ C), there exists an injective homomorphism b : C → B such that c(x) = (b(x), 1) for all x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let s ∈ S be as in (i1) or (i2), and ˇC, ˇC as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let cT be the composition A → B ⋊ T ↠ T , where the first map is c and the second map is the canonical projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now cT (ˇC) is finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, if (i1) holds, then Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='7 implies that cT (ˇC) is torsion, hence finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If (i2) holds, then Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5 (iii) implies that cT (ˇC) is torsion, hence finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Set l := #cT (ˇC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then for all x ∈ lˇC, cT (x) = 1, so that our claim follows (Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4 gives injectivity of b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let A ⊆ A be a subgroup of finite rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Consider the following conditions: (ii1) For all torsion orders l > 1 of elements of T , there exists 1 ̸= sl ∈ S and coprime integers µ, ν ∈ Z>0 such that σsl(A) ⊆ A and µ det(z− ˙σsl) = µzδ−ν with gcd(l, µ) > 1 or gcd(l, ν) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii2) Condition (F) holds for ˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that, in particular, (ii1) holds if T is torsion-free or if for all κ ∈ Z>0 there exists sκ ∈ S with σsκ = κ idA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We say that condition (III) holds if there exists s ∈ S such that σs = κ idA for some κ ∈ Z \\ {0, 1}, B is of finite rank, and condition (F) holds for ˜τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let A ⊆ A be a subgroup of finite rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume that one of the following is true: (i1) or (i2) holds, and (ii1) is satisfied or (ii2) holds and A is torsion-free, (III) holds and A is torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then for all finitely generated subgroup A ⊆ A, there exists C ∈ C such that, with C := A ∩ C, there exists an injective homomorphism b : C → B such that c(x) = (b(x), 1) for all x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' To prove the first item, let s ∈ S be as in (i1) or (i2), and ˇC, ˇC as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' First assume that (ii1) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let cT be the composition A → B ⋊ T ↠ T , where the first map is c and second map is the canonical projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' As cT (ˇC) is finitely generated, the set L of possible non-trivial torsion orders of elements in cT (ˇC) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For each l ∈ L, choose sl as in (ii1) and set C := ˇC ∩ � l∈L ˇC(sl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Applying Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5 to sl, we obtain that for all x ∈ A ∩ C with c(x) = (β, α), we have αµ and αν are conjugate in T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If α ̸= 1, then the torsion order of α is a number l ∈ L, but (ii1) implies that αµ and αν have different torsion orders, which is absurd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now suppose that (ii2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Take x ∈ ˇC and write c(x) = (β, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='7 implies that αϵ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then (β, α)ϵ = (β + ˜τα(β) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' + ˜τ ϵ−1 α (β), αϵ) = (β + ˜τα(β) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' + ˜τ ϵ−1 α (β), 1), and (1 − ˜τα)(β + ˜τα(β) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' + ˜τ ϵ−1 α (β)) = 0, which implies β + ˜τα(β) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' + ˜τ ϵ−1 α (β) = 0 if α ̸= 1 by (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' So c(x)ϵ = 1 and hence c(ϵx) = 1 and thus ϵx = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' As A is torsion-free, this implies x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' So if x ̸= 0, we must have α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now we prove the second item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' As above, let s ∈ S be as in (III), and ˇC, ˇC as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given x ∈ ˇC, write c(x) = (β, α) and c(s) = (δ, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Equation (2) implies that γα = ακγ, and therefore α = γ−1ακγ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It follows that every eigenvalue of ˙τα is a root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Indeed, take λ1 ∈ Sp ( ˙τα).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then 8 α = γ−1ακγ implies that there exists λ2 ∈ Sp ( ˙τα) with λ1 = λκ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Similarly, there exist λ3, λ4, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' ∈ Sp ( ˙τα) such that λi = λκ i+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' As Sp ( ˙τα) is finite, we must have λi = λi+p for some i and p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It follows that λi = λκp i and hence that λi is a root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' But then λ1 = λκi i implies that λ1 must be a root of unity as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence there exists m ∈ Z>0 such that 1 is an eigenvalue of ˙τ m α .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This implies that 1 − ˙τ m α is not injective, so that ˙τ m α = 1 by (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now argue as for the first item that this – together with (F) – implies α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The difference between Corollaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='10 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13 is that in the latter, we may choose l = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume that A = � n An for an increasing family of finite rank subgroups An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Consider the following conditions: (I) Condition (i1) holds for An for all n, or condition (i2) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (II) Condition (ii1) holds for An for all n, or condition (ii2) holds and A is torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose we can write A = � n An for an increasing family An ⊆ A of finite rank subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If (I) holds, then (a) there is an increasing family of finitely generated subgroups Ak ⊆ A with A = � k Ak, and for any such Ak, there are lk ∈ Z>0 and Ck ∈ C such that, with Ck := lk(Ak ∩ Ck), there exists an injective homomorphism bk : Ck → B such that c(x) = (b(x), 1) for all x ∈ Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If (I) and (II) hold, or if (III) is satisfied and A is torsion-free, then (a*) there is an increasing family of finitely generated subgroups Ak ⊆ A with A = � k Ak, and for any such Ak, there is Ck ∈ C such that, with Ck := Ak ∩ Ck, there exists an injective homomorphism bk : Ck → B such that c(x) = (bk(x), 1) for all x ∈ Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since A = � n An, we can find Ak with the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, for each k, since Ak is finitely generated, and A = � n An, we can find n such that Ak ⊆ An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now apply Corollaries 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='10 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' With the notation from Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='16, we have Ak/Ck �→ A/Ck, so that #(Ak/Ck) is finite and divides #(A/Ck).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The multiplicative homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Define t: S → T to be the composition S → B⋊T ↠ T , where the first arrow is given by c(−, 0), and the second arrow is the canonical projection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Clearly, t is a homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i) If A is torsion-free and (a) holds, then t is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) Suppose that condition (F) is satisfied for ˜σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If there exist l ∈ Z>0, a non-zero, finitely generated subgroup A ⊆ A, C ∈ C and an injective homomorphism b : C := l(A ∩ C) → B such that c(x) = (b(x), 1) for all x ∈ C, then t is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose t(s) = t(s′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then c(s−1s′, 0) = (χ, t(s−1s′)) = (χ, 1) for some χ ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Set ε := s−1s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since 0 lies in the domain of ˜σε, and since c(ε, −) is locally constant, there must exist C(ε) ∈ C such that C(ε) is contained in the domain of ˜σε and c(ε, −) is constant on C(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now suppose that C ⊆ A is a non-zero subgroup such that there exists an injective homomorphism b : C → B with c(x) = (b(x), 1) for all x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then c(ε, 0) commutes with c(x, 0) for all x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We have (0, ε)(x, 1) = (˜σϵ(x), ϵ) = (˜σε(x), 1)(0, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' So if 0 ̸= x ∈ C ∩ C(ε), then c((˜σε(x), ε), 0) = c(˜σε(x)ε, 0) = c(˜σε(x), 0)c(ε, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' At the same time, c((˜σε(x), ϵ), 0) = c(εx, 0) = c(ε, x)c(x, 0) = c(ε, 0)c(x, 0) = c(x, 0)c(ε, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Here we are allowed to replace x by 0 because x lies in C(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, we used that c(ε, 0) commutes with c(x, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Therefore, a comparison yields c(˜σε(x), 0) = c(x, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It follows that ˜σε(x) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For (i), write A = � k Ak as in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Applying the above to C = Ck from (a), we obtain ˜σε(x) = x for all x ∈ Ck ∩ C(ε) for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now let y ∈ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' then there exists k with y ∈ Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Ck ∩ C(ε) is finite 9 index in Ak, there exists N ∈ Z>0 such that Ny ∈ Ck ∩ C(ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We have N ˜σε(y) = ˜σε(Ny) = Ny, so that, because A is torsion-free, ˜σε(y) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now (JF) implies ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For (ii), if ˜σε(x) = x for any x ̸= 0, then condition (F) implies that ε = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Equivariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We set I := {#(A/C): C ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume that B is torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If (a) holds, then for all s ∈ S, x ∈ Ck, σs(x) ∈ Al (k ≤ l) there exists n ∈ Z>0 such that nσs(x) = σs(nx) ∈ Cl, and we have b(σs(nx)) = ˜τt(s)(b(nx)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If (a*) holds, then we may take n ∈ I in the statement above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Cl is of finite index in Al (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='17), there exists n ∈ Z>0 such that nσs(x) = σs(nx) ∈ Cl (since l depends on s, n also depends on s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Ck ∩ C(s) is of finite index in Ck, we can find N such that Nnx ∈ Ck ∩ C(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Set y := Nnx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let c(s) = (δ, t(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We have (b(σs(y)) + δ, t(s)) = (b(σs(y)), 1)(δ, t(s)) = c(σs(y), 0)c(s, 0) = c((σs(y), s), 0) = c((0, s)(y, 0), 0) = c((0, s), y)c((y, 1), 0) = c((0, s), 0)c((y, 1), 0) = (δ, t(s))(b(y), 1) = (δ + ˜τt(s)(b(y)), t(s)), where the sixth equality uses that y ∈ C(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence b(σs(y)) = ˜τt(s)(b(y)), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', Nb(σs(nx)) = N ˜τt(s)(b(nx)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since B is torsion-free, we obtain b(σs(nx)) = ˜τt(s)(b(nx)), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Conclusion: The embedding theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume that σ: S ↷ A and τ : T ↷ B are algebraic actions satisfying our standing assumptions from § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let Z[I−1] be the subring of Q generated by Z together with � 1 n: n ∈ I � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We start with the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We have Z[I−1] ⊗ A = Z[I−1] ⊗ A , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', the canonical map Z[I−1] ⊗ A → Z[I−1] ⊗ A , 1 ⊗ x �→ 1 ⊗ x is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Recall that we always assume (1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', A = ⟨� s∈S ˜σs(A)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Otherwise, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='21 would not be true in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Because of (1), it suffices to prove that for all x ∈ ˜σ−1 t1 ˜σs1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' ˜σ−1 tl ˜σslA (where t1, s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' , tl, sl ∈ S) there exists N in the multiplicative submonoid ⟨I⟩+ of Z× generated by I with Nx ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We proceed inductively on l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For the case l = 1, observe that [˜σ−1 t A : A] < ∞ for all t ∈ S because ˜σt induces a bijection ˜σ−1 t A/A ∼= A/σtA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now suppose that x = ˜σ−1 t ˜σs(y) for some y ∈ A with Ny ∈ A for some N ∈ ⟨I⟩+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since [˜σ−1 t A : A] < ∞, there exists M ∈ ⟨I⟩+ with M ˜σ−1 t (σs(Ny)) ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence MNx ∈ A, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We say that condition (∗) is satisfied if A = � n An for an increasing family of finite rank subgroups An, conditions (I) and (II) hold, or condition (III) holds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' in addition, there exists an increasing family of finitely generated subgroups Ak ⊆ A with A = � k Ak such that σs(Ak) ⊆ Ak for all s ∈ S and all k, S and T are right reversible, A = S−1A, S = S−1S, B = T −1B, T = T −1T, and σ: S ↷ A satisfies (PC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now suppose c: (A ⋊ S ) ⋉ A → B ⋊ T is a continuous cocycle such that c−1(0, 1) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose A and B are torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i) If (I) holds, then there exist injective homomorphisms t : S �→ T and ˙b: Q ⊗ A �→ Q ⊗ B such that ˙b( ˙σs(x)) = ˙τt(s)(˙b(x)) for all s ∈ S and x ∈ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) If (∗) holds, then there exist injective homomorphisms t : S �→ T and b′ : S−1A → T −1B such that b′(σs(x)) = ˜τt(s)(b′(x)) for all x ∈ S−1A and s ∈ S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i): We proceed in several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' First, we claim that for each k, bk : Ck → B has a unique extension to an injective homomorphism ˜bk : Ak → Q ⊗ B, and that moreover, ˜bk satisfies (3) ˜bk(x) = m−1bk(mx), for any m ∈ Z>0 with mAk ⊆ Ck.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' To see this, choose m ∈ Z>0 such that mAk ⊆ Ck, and define ˜bk(x) := m−1b(mx) ∈ Q ⊗ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It is easy to check that ˜bk is a group homomorphism (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', additive) 10 and that ˜b is injective (here we need that Ak is torsion-free).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This is independent of the choice of m, because if m′ ∈ Z>0 with m′Ak ⊆ Ck, then for x ∈ Ak, we have m−1b(mx) = m−1(m′)−1b(m′mx) = (m′)−1m−1mb(m′x) = (m′)−1b(m′x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Next, we claim that the maps ˜bk are compatible in the sense that ˜bl|Ak∩Al = ˜bk|Ak∩Al for all k, l ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Choose m large enough so that mAk ⊆ Ck and mAl ⊆ Cl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then, using (3), we have for all x ∈ Ak ∩Al, that ˜bl(x) = m−1bl(mx) = m−1bk(mx) = ˜bk(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It follows that we get a well-defined injective homomorphism ˜b: A = � k Ak → Q ⊗ B such that ˜b|Ak = ˜bk for all k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us show that ˜b is equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let x ∈ A and s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then x ∈ Ak for some k, so by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20, we can find n ∈ Z>0 large enough so that nx ∈ Ck, nσs(x) = σs(nx) ∈ Cl, and b(σs(nx)) = ˜τt(s)(b(nx)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now we have ˜b(σs(x)) = n−1b(σs(nx)) = n−1˜τt(s)(b(nx)) = ˙τt(s)(n−1b(nx)) = ˙τt(s)(˜b(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lastly, we extend ˜b to Q⊗A as follows: Given p q ⊗x ∈ Q⊗A for p ∈ Z and q ∈ Z×, there exists a unique element y ∈ B such that qy = ˜b(px) in B, and we set ˙b( p q ⊗x) := y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now it is straightforward to check that this is independent of p, q and x, and that ˙b is an injective homomorphism Q ⊗ A �→ Q ⊗ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, equivariance of ˜b with respect to ˜σ and ˙τ implies equivariance of ˙b with respect to ˙σ and ˙τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii): First of all, every element of S−1A is of the form s−1a = ˜σ−1 s a for some a ∈ Ck (for some k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Indeed, the statement is clear if we just ask for a ∈ Ak for some k because of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By (PC), there exists ˙s ∈ S such that σ ˙s(a) ∈ Ck, so that σ ˙s(a) ∈ Ck because σ ˙s(Ak) ⊆ Ak, and we have ˜σ−1 s a = ˜σ−1 s ˜σ−1 ˙s (˜σ ˙sa), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now given an element of S−1A of the form ˜σ−1 s a for some a ∈ Ck (for some k), we claim that b′(˜σ−1 s a) := ˜τ −1 t(s)(b(a)) is well defined and has the desired properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' To prove that it is well-defined, assume that ˜σ−1 s a = ˜σ−1 t ˙a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since S is right reversible, there exist u, v ∈ S with us = vt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence ˜σ−1 s ˜σ−1 u ˜σua = ˜σ−1 s a = ˜σ−1 t ˙a = ˜σ−1 t ˜σ−1 v ˜σv ˙a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It follows that ˜σua = ˜σv ˙a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now choose an integer m such that mσu(a), mσv(a) ∈ Ck and that equivariance holds (here, we use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then we have m˜τ −1 t(s)b(a) = ˜τ −1 t(s)b(ma) = ˜τ −1 t(s)˜τ −1 t(u)b(σu(ma)) = ˜τ −1 t(t)˜τ −1 t(v)b(σv(m˙a)) = ˜τ −1 t(t)b(m˙a) = m˜τ −1 t(t)b(˙a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' As B is torsion-free, we conclude that ˜τ −1 t(s)b(a) = ˜τ −1 t(t)b(˙a), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It is easy to see that b′ is additive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' To show equivariance, take r, s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since S is right reversible, there exist u, v ∈ S with ur = vs and thus rs−1 = u−1v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given a ∈ Ck, choose an integer m such that ˜σv(ma) ∈ Ck and equivariance holds (here, we use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then we have mb′(˜σr˜σ−1 s (a)) = b′(˜σ−1 u ˜σv(ma)) = ˜τ −1 t(u)b(˜σv(ma)) = ˜τ −1 t(u)˜τt(v)b(ma) = m˜τt(r)˜τ −1 t(s)b(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' As B is torsion-free, we deduce that b′(˜σr˜σ−1 s (a)) = ˜τt(r)(˜τ −1 t(s)b(a)) = ˜τt(r)(b′(˜σ−1 s a)), as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Actually we obtain an equivariant embedding Z[I−1] ⊗ A �→ Q ⊗ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' And if (I) and (II) are satisfied, or (III) holds, then we even obtain an embedding Z[I−1] ⊗ A �→ Z[I−1] ⊗ B, using Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20, because (a*) holds by Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us formulate symmetrized versions of our rigidity results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Schmidt defined finite (algebraic) equivalence for algebraic actions of a fixed group (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', [50, Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The dual version of Schmidt’s notion will appear naturally in our setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In order to explain this, let us introduce some terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i) An (algebraic) embedding of ˜σ: S ↷ A into ˜τ : T ↷ B consists of a pair (t, b), where t: S �→ T and b: A �→ B are injective homomorphisms such that b(˜σs(x)) = ˜τt(s)(b(x)) for all s ��� S and x ∈ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The embedding (t, b) is called finite index if the image of b has finite index in B, full if b is surjective, and strict if t is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 11 (ii) We say that ˜σ: S ↷ A and ˜τ : T ↷ B are mutually embeddable (written ˜σ: S ↷ A ∼ME ˜τ : T ↷ B) if each can be embedded into the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If, in addition, the embeddings can be chosen to be of finite index, then we write ˜σ: S ↷ A ∼MEF I ˜τ : T ↷ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given an Abelian group Q, we write ˜σ: S ↷ A ∼MEQ ˜τ : T ↷ B if ˙σ: S ↷ Q ⊗ A ∼ME ˙τ : T ↷ Q ⊗ B, where ˙σs = idQ ⊗ ˜σs and ˙τ := idQ ⊗ ˜τs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We write ˜σ: S ↷ A ∼MEQ∼ = ˜τ : T ↷ B if there exist full embeddings of ˙σ: S ↷ Q ⊗ A into ˙τ : T ↷ Q ⊗ B and of ˙τ : T ↷ Q ⊗ B into ˙σ: S ↷ Q ⊗ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We say that ˜σ: S ↷ A and ˜τ : T ↷ B are strictly mutually embeddable (and we write ˜σ: S ↷ A ∼sME ˜τ : T ↷ B) if each can be strictly embedded into the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If, in addition, the strict embeddings can be chosen to be of finite index, then we write ˜σ: S ↷ A ∼sMEF I ˜τ : T ↷ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given an Abelian group Q, we write ˜σ: S ↷ A ∼sMEQ ˜τ : T ↷ B if ˙σ: S ↷ Q ⊗ A ∼sME ˙τ : T ↷ Q ⊗ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We write ˜σ: S ↷ A ∼=Q ˜τ : T ↷ B, and call ˜σ: S ↷ A and ˜τ : T ↷ B isomorphic over Q, if there exists a full and strict embedding of ˙σ: S ↷ Q ⊗ A into ˙τ : T ↷ Q ⊗ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iii) We say that the algebraic actions σ: S ↷ A and τ : T ↷ B are isomorphic if there is a pair (t, b), where t: S → T is an isomorphism of semigroups and b: A → B is a group isomorphism such that b(σs(x)) = τt(s)(b(x)) for all s ∈ S and x ∈ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The dual notion of a strict (algebraic) embedding in our sense is an algebraic factor map in the sense of [50, Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If (t, b) is a finite index embedding of ˜σ: S ↷ A into ˜τ : T ↷ B and A and B are torsion-free, then A and B are quasi-isomorphic in the sense of [51, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If ˜σ: S ↷ A ∼sMEF I ˜τ : T ↷ B, then we also call ˜σ: S ↷ A and ˜τ : T ↷ B finitely algebraically equivalent (compare [50, Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We will mostly be interested in our notions involving an Abelian group Q when Q = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If A and B are torsion-free, then ∼MEF I implies ∼MEQ∼ = and ∼sMEF I implies ∼=Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If A and B are torsion-free and of finite rank, then ∼ME implies ∼MEF I and ∼sME implies ∼sMEF I (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', [27, Exercise 5 in § 92]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given algebraic actions σ: S ↷ A and τ : T ↷ B, let (Is) be the symmetrized version of condition (I) from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='15, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', condition (I) holds and the analogue of (I) with reversed roles for σ and τ holds as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Similarly, let (IIs), (IIIs) and (∗s) be the symmetrized versions of (II), (III) and (∗) from Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='15, Definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='12), and Definition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose A is torsion-free and of finite rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We say that ˜σ: S ↷ A is strongly faithful if (SF) ˙σs = ρ ˙σtρ−1 implies s = t for all s, t ∈ S and ρ ∈ Aut(Q ⊗ A ), where ˙σs := idQ ⊗ ˜σs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If det ◦ ˙σ: S → Q× is injective, then ˜σ: S ↷ A satisfies (SF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We are now ready for the main result of this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume that σ: S ↷ A and τ : T ↷ B are algebraic actions satisfying our standing assumptions from § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1, with globalizations ˜σ: S ↷ A and ˜τ : T ↷ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose that A and B are torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i) If (Is) holds and there exists an isomorphism of topological groupoids (A ⋊ S ) ⋉ A ∼= (B ⋊ T ) ⋉ B, then ˜σ: S ↷ A ∼MEQ ˜τ : T ↷ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) If (∗s) holds and there exists an isomorphism of topological groupoids (S−1A ⋊ S−1S) ⋉ A ∼= (T −1B ⋊ T −1T) ⋉ B, then ˜σ: S−1S ↷ S−1A ∼ME ˜τ : T −1T ↷ T −1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If, in addition, A and B have finite rank, then we obtain ˜σ: S ↷ A ∼MEQ∼ = ˜τ : T ↷ B in (i) and ˜σ: S−1S ↷ S−1A ∼MEF I ˜τ : T −1T ↷ T −1B in (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If, in addition, A and B have finite rank and ˜σ: S ↷ A , ˜τ : T ↷ B both satisfy (SF), then we obtain ˜σ: S ↷ A ∼=Q ˜τ : T ↷ B in (i) and ˜σ: S−1S ↷ S−1A ∼sMEF I ˜τ : T −1T ↷ T −1B (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ˜σ: S−1S ↷ S−1A and ˜τ : T −1T ↷ T −1B are finitely algebraically equivalent) in (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 12 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Everything except the last claim follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For the last claim, suppose that (t, b) is an embedding of ˙σ: S ↷ Q ⊗ A into ˙τ : T ↷ Q ⊗ B and (s, a) is an embedding of ˙τ : T ↷ Q ⊗ B into ˙σ: S ↷ Q ⊗ A .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For s ∈ S , we have (a ◦ b) ˙σt(a ◦ b)−1 = a ˙τt(s)a−1 = ˙τs◦t(s), so that (SF) implies (s ◦ t)(s) = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence, s ◦ t = idS , so by symmetry, t ◦ s = idT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Our assumptions imply that Q ⊗ A and Q ⊗ B have the same dimension as rational vector spaces, so that the injective maps a and b are invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The second part of the last claim is similar using that any injective endomorphism of a torsion-free finite rank Abelian group necessarily has finite index image (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', [27, Exercise 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In some of our examples, A will be finitely generated, in which case we take Ak = A for all k in condition (∗), so that the requirement σs(Ak) ⊆ Ak in (∗) is automatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose that both σ: S ↷ A and τ : T ↷ B are exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then the corresponding groupoids are effective and minimal by [13, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='14] (see also [33, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='23]) and [13, Corol- lary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence the following are equivalent: (i) (A ⋊ S ) ⋉ A and (B ⋊ T ) ⋉ B are isomorphic as topological groupoids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) (Aσ, Dσ) and (Aτ, Dτ) are isomorphic as Cartan pairs, where Aσ and Aτ are as in [13, Defi- nition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1], and Dσ and Dτ are as in [13, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='30];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iii) F ((A ⋊ S ) ⋉ A) and F ((B ⋊ T ) ⋉ B) are isomorphic as abstract groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iv) D((A ⋊ S ) ⋉ A) and D((B ⋊ T ) ⋉ B) are isomorphic as abstract groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This follows from [47] (see also [45]) and [48, Theorems 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3] (or [41, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Algebraic actions on finite rank torsion-free Abelian groups In this section, we apply our rigidity results to example classes of algebraic actions on finite rank torsion-free Abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Algebraic actions of torsion-free Abelian monoids whose dual actions are mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let σ: S ↷ A be an algebraic action, with A Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let ˆσ: S ↷ �A be the dual action as in [13, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2] and denote by µ the normalized Haar measure on �A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Recall (see, for instance, [49, § 1] or [53, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5]) that ˆσ is (strongly) mixing (with respect to µ) if for all Borel subsets X, Y of �A we have lim s→∞ µ(X ∩ ˆσs(Y )) = µ(X)µ(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If S has no non-trivial finite subsemigroups, we have the following relation between the mixing property of ˆσ and condition (F) for σ: Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume that S has no non-trivial finite subsemigroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then ˆσ is mixing if and only if we have, for all 0 ̸= a ∈ A and 1 ̸= s ∈ S, that σs(a) ̸= a, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', the analogue of condition (F) holds for σ: S ↷ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Indeed, in general (without our assumption on S), ˆσ is mixing if and only if for all infinite sub- semigroups S′ ⊆ S and 0 ̸= a ∈ A, we have that # {σs′(a): s′ ∈ S′} = ∞ (see [49, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='6] and also [4], in particular [4, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It is now straightforward to see that, if S has no non- trivial finite subsemigroups, the latter statement is equivalent to the condition that for all 0 ̸= a ∈ A, we have # {s ∈ S: σs(a) = a} < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This condition, in turn, is equivalent to the statement that we have σs(a) ̸= a for all 0 ̸= a ∈ A and 1 ̸= s ∈ S (again assuming that S has no non-trivial finite subsemigroups), as desired, because {s ∈ S: σs(a) = a} is always a subsemigroup of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that the condition that S has no non-trivial finite subsemigroups is in particular satisfied if S is torsion-free, in the sense that for all s1, s2 ∈ S and i ∈ Z>0, si 1 = si 2 implies that s1 = s2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' With the help of this observation, let us now present the first example class to which we can apply our general rigidity results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume that S and T are non-trivial, Abelian, cancellative and torsion-free monoids, that A and B are torsion-free Abelian groups of finite rank, and that σ: S ↷ A and τ : T ↷ B are non-automorphic faithful algebraic actions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Further suppose that the dual actions ˆσ and ˆτ are mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let ˜σ: S−1S ↷ S−1A and ˜τ : T −1T ↷ T −1B be the canonical globalizations as in [13, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 13 If there exists an isomorphism of topological groupoids (S−1A ⋊ S−1S) ⋉ A ∼= (T −1B ⋊ T −1T) ⋉ B, then ˜σ: S−1S ↷ S−1A ∼MEF I ˜τ : T −1T ↷ T −1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' First of all, note that condition (JF) is satisfied because of [13, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5] and (FI) holds by [13, Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus σ and τ satisfy our standing assumptions from § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, it is straightforward to check that S−1S and T −1T are torsion-free and that rkZS−1A = rkZA, rkZT −1B = rkZB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now our statement follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='30 (ii) for the finite rank case because of Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Let us briefly explain the conclusion of our results for the case of (duals of) toral endomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let a ∈ Mn(Z) and b ∈ Mm(Z) with | det(a)|, | det(b)| > 1, where n, m ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If a and b both have no roots of unity as eigenvalues, then the duals of the N-actions σ: N ↷ Zn and τ : N ↷ Zm given by σk(v) = akv and τk(w) = bkw for k ∈ N, v ∈ Zn, and w ∈ Zm are mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose that the corresponding groupoids are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then, since (SF) holds in this case, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='30 implies that n = m and that the matrices a and b must be conjugate over Q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', there exists c ∈ GLn(Q) such that a = cbc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Canonical endomorphisms of torsion-free finite rank Abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let A ⊆ Qn be a torsion-free Abelian group of rank n ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The multiplicative monoid Z× := Z \\ {0} acts on A by multiplication: Each s ∈ Z× gives rise to the endomorphism σs : A → A given by σs(x) = sx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For any submonoid M ⊆ Z×, the associated algebraic action σM : M ↷ A, where σM := σ|M, is faithful and satisfies (FI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It is easy to see that σM : M ↷ A is non-automorphic if and only if there exists m ∈ M such that A is not m-divisible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', mA ⊊ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since det( ˙σs) = sn, we see that det ◦ ˙σ: Z>0 → Q× is injective, so that σM : M ↷ A satisfies (SF) (see Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since M is Abelian, we obtain a globalization ˜σM : ⟨M⟩ ↷ M−1A, where ⟨M⟩ := M−1M ⊆ Q× acts on M−1A := � s∈M 1 sA by multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It is easy to see that ˜σM : ⟨M⟩ ↷ M−1A satisfies (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For s, s′ ∈ Z>0, we have sA ∩ s′A = lcm(s, s′)A (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', [26, § 20]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence, the M-constructible subgroups for M ↷ A are given by {sA : s ∈ M}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' From this, we see that ˜σM : ⟨M⟩ ↷ M−1A satisfies (JF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', xn ∈ A be rationally independent, and put A := spanZ({x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', xn}) ∼= Zn ⊆ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For each k ∈ Z>0, let Ak := {x ∈ A : (k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' )x ∈ A} = A ∩ (k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' )−1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Each Ak is an Z×-invariant finitely generated subgroup of A, and we have A = � k Ak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We can now apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='30 to obtain the following result: Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let A and B be torsion-free finite rank Abelian groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let M, N ⊆ Z× be submonoids such that there exist m ∈ M and n ∈ N with mA ⊊ A and nB ⊊ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If there is an isomorphism of topological groupoids (M−1A ⋊ ⟨M⟩) ⋉ A ∼= (N−1B ⋊ ⟨N⟩) ⋉ B, then the actions ⟨M⟩ ↷ M−1A and ⟨N⟩ ↷ N−1B are finitely algebraically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If Z>0 ⊆ M, then M ↷ A is exact if and only if the Ulm subgroup of A vanishes (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' [26, § 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Actions adding scalars to algebraic actions of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let Γ ⊆ SLn(Z) be any subgroup, and let M ⊆ Z>0 a submonoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then, MΓ := {aγ : a ∈ M, γ ∈ Γ} is a submonoid of Mn(Z)× := {x ∈ Mn(Z) : det(x) ̸= 0}, where we view M as a submonoid of Mn(Z) via the diagonal embedding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Mn(Z)× acts canonically on Zn, we obtain a faithful algebraic action MΓ ↷ Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It is easy to see that MΓ ↷ Zn is exact if and only if M is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that ⟨M⟩Γ ↷ (M−1Z)n is a globalization for MΓ ↷ Zn that satisfies (JF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Γ acts by automorphisms on Zn that commute with the action of M, we have CMΓ↷Zn = CM↷Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let Z n M denote the completion of Zn with respect to the family CM↷Zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The globalization ⟨M⟩Γ ↷ (M−1Z)n often will not satisfy (F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For instance, take M = Z>0 and Γ = SL2(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then id − γ is not injective on M−1Z2 = Q2, where γ = � 1 1 0 1 � ∈ SL2(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 14 In order to apply our rigidity result in this setting, we need an observation on subgroups of SLn(Z), which comes from [23, Example 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='8 & Lemma 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let Γ ⊆ SLn(Z) be a subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If γαγ−1 = ακ for α, γ ∈ Γ and κ ∈ Z>1, then α ∈ tor(Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose γαγ−1 = ακ for α, γ ∈ Γ and κ ∈ Z>1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let p be a prime divisor of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For l ≥ 1, consider the congruence subgroup Γ(pl) := {a ∈ SLn(Z) : a ≡ In mod pl}, and put Γpl := Γ ∩ Γ(pl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Γp is a finite index subgroup of Γ, we can find m ∈ Z>0 such that αm ∈ Γp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If αm ̸= In, then since � l Γpl = {In}, we can find l ≥ 1 with αm /∈ Γpl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now αmΓpl and ακmΓpl have the same order in the p-group Γp/Γpl because γαγ−1 = ακ, which is a contradiction since p | κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let Γ, Λ ⊆ SLn(Z) be subgroups and M, N ⊆ Z>0 nontrivial submonoids such that for every γ ∈ tor(Γ), there exists s ∈ M with gcd(ord(γ), s) > 1, and for every λ ∈ tor(Λ), there exists t ∈ N with gcd(ord(λ), t) > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If there is an isomorphism of topological groupoids (M−1Zn ⋊ ⟨M⟩Γ) ⋉ Z n M ∼= (N−1Zm ⋊ ⟨N⟩Λ) ⋉ Z m N, then M = N and there exist g, h ∈ Mn(N−1Z) ∩ GLn(Q) such that Γ ⊆ gΛg−1 and hΛh−1 ⊆ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='7, condition (∗s) holds, so this follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The assumptions on Γ, Λ and M, N in the statement Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='8 are satisfied, for instance, if M = N = Z>0 or if Γ and Λ are torsion-free (and M, N ̸= {1}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If in the statement of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='8, Γ and Λ are not conjugate via an element in GLn(Q) to any of their proper subgroups, then the conclusion can be strengthened to the following: M = N and there exists g ∈ Mn(N−1Z) ∩ GLn(Q) such that Γ = gΛg−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This holds, for instance, if Γ and Λ are co-Hopfian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Arithmetic dynamical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us consider the algebraic N-actions studied by Chothi, Everest, and Ward in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let K be a number field with ring of integers R, and let PK denote the set of non-zero prime ideals of R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For p ∈ PK, let vp and | · |p denote the associated additive and multiplicative p-adic valuations on K, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given a subset S ⊆ PK, the corresponding ring of S-integers is RS := {x ∈ K : |x|p ≤ 1 for every p ∈ PK \\ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' That is, RS consists of the elements of K that are p-adic integers for every p ∈ PK \\ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For ξ ∈ R× S = RS \\ {0}, the map mξ : RS → RS given by mξ(x) = ξx is an injective endomorphism of the additive group of RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The dual action �mξ : N ↷ � RS is called an arithmetic S-integer dynamical system, see [15, § 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The group of units (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', invertible elements) in RS is R∗ S = {x ∈ K∗ : |x|p = 1 for every p ∈ PK \\ S}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that RS is a proper subring of K if and only if S ⊊ PK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Also note that R∗ S ⊊ R× S whenever RS ⊊ K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us record some basic observations about the algebraic action mξ : N ↷ RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let ˜mξ : RS[1/ξ] → RS[1/ξ] be given by ˜mξ(x) = ξx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then ˜mξ : Z ↷ RS[1/ξ] is a globalization of mξ : N ↷ RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i) mξ : N ↷ RS is faithful if and only if ξ is not a root of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) mξ : N ↷ RS is exact if and only if it is non-automorphic if and only if ξ is a non-unit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iii) ˜mξ : Z ↷ RS[1/ξ] satisfies (JF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i) and (iii) are obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For (ii), let ξ ∈ R× S \\ R∗ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then there exists p ∈ PK \\ S such that |ξ|p < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If x ∈ RS lies in � n≥0 ξnRS, then for each n ≥ 0, we can write x = ξnyn for some yn ∈ RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now we have |x|p = |ξ|n p |yn|p ≤ |ξ|n p for every n, so that x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The other implications in (ii) are easy to see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' An element x ∈ K is integral over Z if and only if Z[x] is finitely generated as a Z-module, so the additive group of RS is not finitely generated whenever S ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For each k ∈ Z>0, let Ak := RS ∩ (k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' )−1R = {x ∈ RS : (k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' )x ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then RS = � k Ak, and every Ak is finitely generated and invariant under R×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let K1 and K2 be number fields with rings of algebraic integers R1 and R2, respec- tively, let S ⊊ PK and T ⊊ PL by proper subsets of primes, and let ξ ∈ R× 1 \\ R∗ 1,S and η ∈ R× 2 \\ R∗ 2,T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If there is an isomorphism of topological groupoids (R1,S[1/ξ] ⋊ ⟨ξ⟩) ⋉ R1,S ∼= (R2,T [1/η] ⋊ ⟨η⟩) ⋉ R2,T , 15 then ˜mξ ↷ R1,S[1/ξ] and ˜mη ↷ R2,T [1/η] are finitely algebraically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Condition (∗s) is satisfied, so Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='30 yields the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given any ξ ∈ R× S \\R∗ S, there exists l ∈ Z>0 such that lξ ∈ R, and then the pair (id, ml) is a strict, finite index embedding of mξ : N ↷ RS into mlξ : N ↷ RS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' in particular, mξ : N ↷ RS and mlξ : N ↷ RS are isomorphic over Z[l−1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, up to inverting an integer, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13 applies to all (faithful, exact) actions of the form mξ : N ↷ RS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Algebraic actions from rings The rank of a ring R is defined to be the rank of the additive group of R, that is, the dimension of Q ⊗Z R as a vector space over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We shall say that R is torsion-free if the additive group of R is a torsion-free (Abelian) group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Examples of torsion-free rings of finite rank include integral group rings of finite groups and Rn or Mn(R), where R an order in a central simple algebra over an algebraic number field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' General preparations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let R be a unital torsion-free ring of finite rank n ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then Q⊗Z R is an n-dimensional Q-algebra containing R as a full subring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The sum of two elementary tensors in Q ⊗Z R is again an elementary tensor, so that Q ⊗Z R = QR := {q ⊗ x : q ∈ Q, x ∈ R}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, if L ⊆ R is a full rank subgroup, then Q ⊗Z L = QL, and QL = QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Each a ∈ QR gives rise to a Q-linear map ˙σa : QR → QR given by x �→ ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We let χa(t) denote the characteristic polynomial of this map, and put N(a) := | det( ˙σa)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For a ∈ R×, put σa := ˙σa|R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Following the notation from [34], we let R× denote the multiplicative monoid of left regular elements in R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', R× consists of those a ∈ R such that σa is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since R is a torsion-free ring of finite rank, the element a ∈ R is left regular if and only if N(a) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The action of any submonoid M ⊆ R× on (the additive group of) R by left multiplication is faithful, by injective endomorphisms, and satisfies (FI) (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', [27, Exercise 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If L is a full rank additive subgroup of R that is invariant under the action of M, then M also acts faithfully on L by injective endomorphisms and the action M ↷ L satisfies (FI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Under the canonical inclusion R ⊆ QR, R× is carried into (QR)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If M ⊆ R× is a submonoid, then we let ⟨M⟩ denote the subgroup of (QR)∗ generated by M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since L is of full rank, we have L ∩ R× ̸= ∅ and thus M ↷ L is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For i = 1, 2, let Ri be a torsion-free ring of rank n and Mi ⊆ R× i a submonoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let L be a rank n subgroup of R1, and assume that spanZ(M1) has finite index in R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If there is an injective additive group homomorphism b: QL → QR2 and a group homomorphism t: ⟨M1⟩ → ⟨M2⟩ such that b(ax) = t(a)b(x) for all a ∈ M1 and x ∈ QL, then there exists a unital Q-algebra isomorphism ϕ: QR1 → QR2 such that ϕ|⟨M1⟩ = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' First, we show that b(1) is invertible in QR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For every a ∈ M1, we have b(a) = b(a1) = t(a)b(1), so that b(x) lies in the Q-vector space (QR2)b(1) for all x ∈ spanZ(M1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since b is injective and rkZ(spanZ(M1)) = n, we have n = rkZ(im(b)) ≤ dimQ((QR2)b(1)) ≤ dimQQR2 = n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence, dimQ((QR2)˜β(1)) = dimQ(QR2), which implies that b(1) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We now define ϕ: QR1 → QR2 by ϕ(x) := b(x)b(1)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Clearly, ϕ is additive and ϕ(1) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since b is injective and b−1 is invertible, we see that ϕ is also injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let a ∈ M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We obtain t(a) = b(a)b(1)−1 = ϕ(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, for a ∈ M1 and x ∈ R1, we have ϕ(ax) = b(ax)b(1)−1 = t(a)b(x)b(1)−1 = t(a)ϕ(x) = ϕ(a)ϕ(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Set Mult(ϕ) := {a ∈ R1 : ϕ(ax) = ϕ(a)ϕ(x) for all x ∈ R1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It is straightforward to see that Mult(ϕ) is a subring of R1 containing Z and M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since spanZ(M1) is of finite index in R1, it follows that Mult(ϕ) = R1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence ϕ is a ring homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus ϕ is a Q-algebra isomorphism QR1 → QR2 satisfying ϕ|⟨M1⟩ = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Given a torsion-free ring of finite rank R, we let O denote the integral closure of Z in QR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If R is finitely generated, then R ⊆ O by [46, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' However, O may not be a subring if R is non-commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Given a submonoid M ⊆ R×, let � M := ⟨M⟩ ∩ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that � M need not be closed under multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 16 The following Corollary demonstrates criteria under which we can deduce rigidity results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For i = 1, 2, suppose Ri is a finitely generated torsion-free ring of finite rank and that Mi ⊆ R× i is submonoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume there exist Q-algebra isomorphisms ϕ1 : QR1 → QR2 and ϕ2 : QR2 → QR1 such that ϕ(⟨M1⟩) ⊆ ⟨M2⟩ and ϕ2(⟨M2⟩) ⊆ ⟨M1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If (S’) ψ(⟨M1⟩) ⊆ ⟨M1⟩ =⇒ ψ(⟨M1⟩) = ⟨M1⟩ for every ψ ∈ AutQ-alg(QR1), then ϕ1(⟨M1⟩) = ⟨M2⟩ and ϕ2(⟨M2⟩) = ⟨M1⟩, so that ⟨M1⟩ ↷ QR1 and ⟨M2⟩ ↷ QR2 are isomor- phic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If (N) Mi = � Mi (for i = 1, 2), and (S) ψ(M1) ⊆ M1 =⇒ ψ(M1) = M1 for every ψ ∈ AutQ-alg(QR1), then ϕ1(M1) = M2 and ϕ2(M2) = M1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' therefore, M1 ↷ spanZ(M1) and M2 ↷ spanZ(M2) are isomorphic, and, if each Oi is closed under addition and Mi-invariant, then M1 ↷ O1 and M2 ↷ O2 are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let ψ := ϕ2 ◦ ϕ1 ∈ AutQ-alg(QR1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then ψ(⟨M1⟩) = ϕ2(ϕ1(⟨M1⟩)) ⊆ ϕ2(⟨M2⟩) ⊆ ⟨M1⟩, so that ψ(⟨M1⟩) = ⟨M1⟩ by (S’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now we have ⟨M1⟩ = ψ(⟨M1⟩) = ϕ2(ϕ1(⟨M1⟩)) ⊆ ϕ2(⟨M2⟩) ⊆ ⟨M1⟩, so that ϕ2(⟨M2⟩) = ⟨M1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now assume that (N) and (S) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Q-algebra homomorphisms preserve integrality, we have ϕ1(O1) = O2 and ϕ2(O2) = O1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' moreover, we have Mi ⊆ Oi for i = 1, 2, so our assumption that ϕ1(M1) ⊆ ⟨M2⟩ and ϕ2(M2) ⊆ ⟨M1⟩ forces ϕ1(M1) ⊆ O2 ∩ ⟨M2⟩ and ϕ2(M2) ⊆ O1 ∩ ⟨M1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We have ψ(M1) = ϕ2(ϕ1(M1)) ⊆ ϕ2(O2 ∩ ⟨M2⟩) (N) = ϕ2(M2) ⊆ O2 ∩ ⟨M1⟩ (N) = M1, so that condition (S) forces ψ(M1) = M1, so that ϕ1(M1) = M2 and ϕ2(M2) = M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Groupoid rigidity when the acting monoid is Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In this section, we specialise to the case where the acting monoids are Abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The following is an immediate consequence of Theo- rem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='23 and Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For i = 1, 2, suppose Ri is a torsion-free finitely generated ring, Mi ⊆ R× i an Abelian submonoid such that spanZ(Mi) has finite index in Ri, and Li ⊆ Ri an Mi-invariant full rank subgroup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume that there exists a ∈ M1 such that L1 → L1, x �→ (1 − a)x is injective, and similarly for M2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If there is an isomorphism of topological groupoids (M−1 1 L1⋊⟨M1⟩)⋉L1 ∼= (M−1 2 L2⋊⟨M2⟩)⋉L2, then there exist Q-algebra isomorphisms ϕ1 : QR1 ∼ = → QR2 and ϕ2 : QR2 ∼ = → QR1 such that ϕ1(M1) ⊆ � M2 and ϕ2(M2) ⊆ � M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We obtain the following rigidity results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose that, in addition to the assumptions in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3, conditions (N) and (S) from Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2 hold and that Li = spanZ(Mi) = Ri or Li = Oi, then the following statements are equivalent: (i) the algebraic actions M1 ↷ R1 and M2 ↷ R2 are isomorphic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) (M−1 1 R1 ⋊ ⟨M1⟩) ⋉ L1 and (M−1 2 R2 ⋊ ⟨M2⟩) ⋉ L2 are isomorphic as topological groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that Li = Oi requires that Oi is closed under addition and Mi-invariant, neither of which is automatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Connection to Bhargava’s work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Torsion-free commutative rings whose additive groups are finitely generated have received a great deal of attention recently [5, 6, 7, 8, 9, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let Ri, i = 1, 2, be finitely generated torsion-free rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If the groupoids (QR1 ⋊ (QR1)∗) ⋉ R1 and (QR2 ⋊ (QR2)∗) ⋉ R2 are isomorphic, then QR1 ∼= QR2 as Q-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' This follows from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3, applied to Mi = R× i and Li = Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Z ⊆ Ri, we just need to show that spanZ(Mi) = Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Indeed, for every a ∈ Ri, there exists κ ∈ Z× such that a + κ ∈ R× i : As sp( ˙σa) is finite, we have 0 /∈ sp( ˙σa+κ) = sp( ˙σa + κ id) = sp( ˙σa) + κ for sufficiently big κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ 17 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Actions of congruence monoids on rings of algebraic integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let K be a number field with ring of integers R, and let Rm,Γ ⊆ R× = R \\ {0} be a congruence monoid as in [11, § 3], where m = m∞m0 is a modulus for K and Γ is a group of residues modulo m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let C∗ λ(R ⋊ Rm,Γ) denote the left regular C*-algebra of the monoid R ⋊ Rm,Γ and Dλ(R ⋊ Rm,Γ) the canonical Cartan subalgebra of C∗ λ(R ⋊ Rm,Γ) (see [12, § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Using the results from [11, § 2], it is not difficult to show that the family of constructible subgroups for the multiplication action Rm,Γ ↷ R is given by CRm,Γ↷R = {(0) ̸= I � R : I is coprime with m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In particular, the completion R of R with respect to CRm,Γ↷R depends only on the prime divisors of m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The ring spanZ(Rm,Γ) is an order in R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', it is of finite index in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Rm,Γ contains (1 + m0)+, it follows that the subring of R generated by Rm,Γ contains (m0)+, the set of totally positive elements in the ideal m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If x ∈ m0, choose k ∈ N× ∩ m0 such that x + k is totally positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since x = (x + k) − k, we see that every element of m0 is a difference of totally positive elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence, the ring generated by Rm,Γ contains m0, which implies that it is of finite index in R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i) The monoid Rm,Γ satisfies conditions (N) and (S) from Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) the action Rm,Γ ↷ R is exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We shall use the notation from [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i): First, let us show condition (N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By Proposition [11, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2], we have ⟨Rm,Γ⟩ = {x ∈ K× : vp(x) = 0 for all p | m0, [x]m ∈ Γ}, from which we see that ⟨Rm,Γ⟩ ∩ R = Rm,Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now we verify that condition (S) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let ψ ∈ Gal(K/Q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We have ψ(Rm,Γ) = Rψ(m),ψ(Γ) where ψ(m) is the modulus defined by w | ψ(m)∞ if and only if w ◦ ψ | m∞ and ψ(m)0 := ψ(m0), and ψ(Γ) is the image of Γ under the isomorphism (R/m)∗ ∼= (R/ψ(m))∗ given by [a]m �→ [ψ(a)]ψ(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Rψ(m),ψ(Γ) = ψ(Rm,Γ) ⊆ Rm,Γ, [11, Proposition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2(1)] implies that ψ(m) | m, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ψ(m)0 | m0 and ψ(m)∞(w) = 1 =⇒ m∞(w) = 1, and πm,ψ(m)(ψ(Γ)) ⊆ Γ, where πm,ψ(m) : (R/m)∗ → (R/ψ(m))∗ is the canonical quotient map arising from the divisibility condition ψ(m) | m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since ψ(m0) and m0 have the same norm, ψ(m0) | m0 forces ψ(m0) = m0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' since the finite sets supp(ψ(m)∞) and supp(m∞) have the same cardinality, supp(ψ(m)∞) ⊆ supp(m∞) forces supp(ψ(m)∞) = supp(m∞), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ψ(m)∞ = m∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Therefore, ψ(m) = m which implies that πm,ψ(m) = id, so that πm,ψ(m)(ψ(Γ)) ⊆ Γ becomes ψ(Γ) ⊆ Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since #ψ(Γ) = #Γ, we must have ψ(Γ) = Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, we have Rψ(m),ψ(Γ) = Rm,Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii): It is enough to show that Rm,Γ contains a non-unit a since then � n≥0 anR = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Observe that Rm,Γ contains the set (1 + m0)+ of totally positive elements in 1 + m0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since (1 + m0)+ contains infinitely many positive integers, we see that Rm,Γ contains non-units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ In this setting, we have the following complete rigidity theorem: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For i = 1, 2, let Ki be a number field with ring of integers Ri, and suppose (Ri)mi,Γi ⊆ R× i is a congruence monoid as in [11, § 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The following statements are equivalent: (i) the algebraic actions (R1)m1,Γ1 ↷ R1 and (R2)m2,Γ2 ↷ R2 are isomorphic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) ((R1)−1 m1,Γ1R1 ⋊ ⟨(R1)m1,Γ1⟩) ⋉ R1 and ((R2)−1 m2,Γ2R2 ⋊ ⟨(R2)m2,Γ2⟩) ⋉ R2 are isomorphic as topological groupoids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iii) (R2)−1 m2,Γ2R2 ⋊ ⟨(R2)m2,Γ2⟩ ↷ R1 and (R2)−1 m2,Γ2R2 ⋊ ⟨(R2)m2,Γ2⟩× ↷ R2 are continuously orbit equivalent in the sense of [37, 38];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iv) (A(R1)m1,Γ1↷R1, D(R1)m1,Γ1↷R1) and (A(R2)m2,Γ2↷R2, D(R2)m2,Γ2↷R2) are isomorphic as Cartan pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (v) (C∗ λ(R1 ⋊ (R1)m1,Γ1), Dλ(R1 ⋊ (R1)m1,Γ1)) and (C∗ λ(R2 ⋊ (R2)m2,Γ2), Dλ(R2 ⋊ (R2)m2,Γ2)) are isomorphic as Cartan pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (vi) F (((R1)−1 m1,Γ1R1 ⋊ ⟨(R1)m1,Γ1⟩) ⋉ R1) and F (((R2)−1 m2,Γ2R2 ⋊ ⟨(R2)m2,Γ2⟩) ⋉ R2) are isomorphic as abstract groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (vii) D(((R1)−1 m1,Γ1R1 ⋊ ⟨(R1)m1,Γ1⟩) ⋉ R1) and D(((R2)−1 m2,Γ2R2 ⋊ ⟨(R2)m2,Γ2⟩) ⋉ R2) are isomorphic as abstract groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i)⇔(ii): Let 1 ̸= a ∈ Rmi,Γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then multiplication by 1 − a is injective on Ki = Q ⊗Z Ri, and thus also injective on R−1 mi,ΓiRi ⊆ Ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='7(i) and Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='6, we can apply Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4 to obtain the desired equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 18 Equivalence of (ii), (iii), and (iv) follows from [38, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (v)⇒(iv) follows from the description of the primitive ideals of C∗ λ(Ri ⋊(Ri)mi,Γi) in [11, Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1] combined with the observation that A(Ri)mi,Γi↷Ri is the unique simple quotient of C∗ λ(Ri ⋊ (Ri)mi,Γi) and the quotient map C∗ λ(Ri ⋊(Ri)mi,Γi) → A(Ri)mi,Γi↷Ri carries Dλ(Ri ⋊(Ri)mi,Γi) onto D(Ri)mi,Γi↷Ri (compare [11, § 8]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Clearly, (i)⇒(iv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Equivalence of (vi), (vii), and (ii) follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='7(ii) and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Specalizing to the case where the moduli are trivial, and observing that the algebraic actions R× 1 ↷ R1 and R× 2 ↷ R2 are isomorphic if and only if K1 ∼= K2, we obtain: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For i = 1, 2, let Ki be a number field with rings of integers Ri, denote the corre- sponding ring C*-algebras by A[Ri], and their canonical Cartan subalgebras by D[Ri].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The following are equivalent: (i) K1 and K2 are isomorphic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) (K1 ⋊ K× 1 ) ⋉ R1 and (K2 ⋊ K× 2 ) ⋉ R2 are isomorphic as topological groupoids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iii) K1 ⋊ K× 1 ↷ R1 and K2 ⋊ K× 2 ↷ R2 are continuously orbit equivalent in the sense of [37, 38];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iv) (A[R1], D[R1]) and (A[R2], D[R2]) are isomorphic as Cartan pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (v) (C∗ λ(R1 ⋊ R× 1 ), Dλ(R1 ⋊ R× 1 )) and (C∗ λ(R2 ⋊ R× 2 ), Dλ(R2 ⋊ R× 2 )) are isomorphic as Cartan pairs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (vi) F ((K1 ⋊ K× 1 ) ⋉ R1) and F ((K2 ⋊ K× 2 ) ⋉ R2) are isomorphic as abstract groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (vii) D((K1 ⋊ K× 1 ) ⋉ R1) and D((K2 ⋊ K× 2 ) ⋉ R2) are isomorphic as abstract groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The equivalence of (i) and (v) in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='8 completely answers the natural problem left open in [12, § 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2]: The Cartan pair (C∗ λ(R ⋊ Rm,Γ), Dλ(R ⋊ Rm,Γ)) remembers the isomorphism class of the semigroup R ⋊ Rm,Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The equivalences of (i), (iii), and (v) in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='9, completely answers the natural question left open in [36, § 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Algebraic actions from commutative algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In this subsection, we analyze a class of algebraic Nd-actions that are irreversible analogues of the algebraic Zd-actions studied in [49, Chapter II].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We need the following observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If R is a commutative finitely generated torsion-free ring of rank n, then integral closure O of Z in QR is then a ring by [46, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since R ⊆ O, for each element a ∈ R, the map ˙σa : QR → QR, ˙σa(x) = ax, leaves O invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, N(a) = | det( ˙σa)| lies in Z>0 for every a ∈ R×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let R be a commutative finitely generated torsion-free ring of rank n and a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ak ∈ R× \\ R∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In addition, assume that for every 1 ≤ i ≤ k, there exists a prime p such that p | N(ai) and p ∤ N(aj) for j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i) If ψ ∈ AutQ-alg(QR) is such that ψ(⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ak⟩+) ⊆ ⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ak⟩+, then ψ = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) ⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ak⟩ ∩ O = ⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ak⟩+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iii) a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ak are multiplicatively independent, so that the canonical map Nk ↠ ⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ak⟩+ is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i): Suppose ψ ∈ AutQ-alg(QR) is such that ψ(⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ak⟩+) ⊆ ⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ak⟩+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then for each 1 ≤ i ≤ k, there exist n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', nk ∈ N such that ψ(ai) = an1 1 · · · ank k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now, for a ∈ R×, we have ψ ◦ σa = σψ(a)◦ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In particular, det(σψ(a)) = det(σa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, we have N(ai) = N(ψ(ai)) = N(a1)n1 · · · N(ak)nk, which, together with our assumption on N(ai), shows that ψ(ai) = ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii): Let x ∈ ⟨a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ak⟩ ∩ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then there exists n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', nk ∈ Z with x = an1 1 · · · ank k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We need to show that each ni is non-negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By assumption, for each 1 ≤ i ≤ k, there exists a rational prime p dividing N(ai) such that p ∤ N(aj) for j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, 0 ≤ vp(N(x)) = �k j=1 njvp(N(aj)) = nivp(N(ai)), which shows ni ≥ 0 (here, the first inequality uses that x lies in O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The containment “⊇” is obvious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iii): Suppose an1 1 · · · ank k = 1 for some n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', nk ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Fix 1 ≤ i ≤ k, and choose a rational prime p such that vp(N(ai)) > 0 and vp(N(aj)) = 0 for j ̸= i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now 0 = vp(N(a1)n1 · · · N(ak)nk) = nivp(N(ai)), so that ni = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ 19 Let d ∈ Z>0 and denote by R+ d := Z[u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ud] the ring of polynomials with integer coefficients in the d variables u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let I �R+ d be a non-zero ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By the Hilbert Basis Theorem (see, for instance, [24, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='2]), R+ d is Noetherian, so that there exists f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', fm ∈ R+ d such that I is generated by {f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', fm}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since we are only interested in the quotient ring R+ d /I, let us assume that ui /∈ I for all 1 ≤ i ≤ d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let V (I) := {z ∈ Cd : f(z) = 0 for every f ∈ I} ⊆ Cd be the complex variety defined by I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It follows from [18, Chapter 5, Theorem 6] that dimQQ ⊗Z R+ d /I < ∞ if and only if V (I) is a finite set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If #V (I) < ∞, then C ⊗Z I is said to be zero- dimensional, in which case there exists a basis for C ⊗Z R+ d /I consisting of (cosets of) monomials (see [18, Chapter 5, Proposition 4]), so that R+ d /I is finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For the remainder of this section, we shall assume #V (I) < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For f ∈ R+ d , let σf denote the endomorphism of R given by left multiplication with the coset f + I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let χf(t) denote the characteristic polynomial of σf viewed as an endomorphism of C ⊗Z R+ d .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us record the following properties of these endomorphisms: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For f ∈ R+ d , we have (i) χf(t) = � z∈V (I)(t−f(z))µ(z), where µ(z) := dimCOz/(C⊗ZI)Oz, where Oz is the localisation of C ⊗Z R at the maximal ideal mz := {g ∈ C ⊗Z R : g(z) = 0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) σf is injective if and only if f(z) ̸= 0 for every z ∈ V (I);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iii) id − σf = σ1−f is injective if and only if f(z) ̸= 1 for every z ∈ V (I);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (iv) if for all F ⊆ V (I), � z∈F f(z) ̸= ±1, then χf is not divisible by any unimodular polynomial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i) follows from [17, Chapter 4, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='7], and the other parts are consequence of this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ If N ∈ Z>0, then it follows from part (ii) of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='12 that the endomorphism σN is injective, so we see that R+ d /I is torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We let ˙ui denote the image of ui modulo I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that ˙ui ̸= 0 by our assumption that ui /∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If zi ̸= 0 for all z ∈ V (I), then σui is an injective endomorphism of R+ d /I by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='12, and we obtain an algebraic action ⟨ ˙u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ˙ud⟩+ ↷ R+ d /I, which satisfies (FI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let Rd := Z[u±1 1 , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', u±1 d ] be the ring of Laurent polynomials in the variables u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ud;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' then R+ d /I embeds in Rd/IRd, and the multiplicative group ⟨ ˙u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ˙ud⟩ acts on Rd/IRd by multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It is easy to see that ⟨ ˙u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ˙ud⟩ ↷ Rd/IRd is a globalization of ⟨ ˙u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ˙ud⟩+ ↷ R+ d /I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let ΩI denote the completion of R+ d /I with respect to the family of ⟨ ˙u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ˙ud⟩+-constructible subgroups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For i = 1, 2, let di ∈ Z>0 and let Ii be a non-zero ideal of Z[u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', udi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume that for i = 1, 2, (a) #V (Ii) < ∞ and uk /∈ Ii for all k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', di;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (b) zk ̸= 0 for every z ∈ V (Ii) and k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', di;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (c) there exists a monomial f in u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' , ud such that f(z) ̸= 1 for all z ∈ V (Ii);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (d) for each 1 ≤ j ≤ di, there exists a rational prime p with p | N( ˙uj) and p ∤ N( ˙uk) for k ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then the following statements are equivalent: (i) the algebraic actions Nd1 ↷ R+ d1/I1 and Nd2 ↷ R+ d2/I2 are isomorphic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (ii) (Rd1/I1Rd1 ⋊ Zd1) ⋉ ΩI1 and (Rd2/I2Rd2 ⋊ Zd2) ⋉ ΩI2 are isomorphic as topological groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' First, note that (d) implies |N( ˙uj)| > 1, so that ˙uj is a non-unit for all 1 ≤ j ≤ di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Conditions (a) and (b) ensure that R+ di/Ii is a finitely generated torsion-free ring and that the action Ndi ↷ R+ di/Ii is by injective group endomorphisms whose images all have finite index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We are in the situation of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4, so we only need to show that (N) and (S) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' That (N) and (S) are satisfied follows from parts (ii) and (i) of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='11, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In the situation of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13, Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='11(iii) implies that Ndi ∼= ⟨ ˙u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ˙udi⟩+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Every finitely generated commutative ring of finite rank is isomorphic to a ring of the form Z[u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ud]/I, where I is zero-dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' However, the isomorphism will typically not be canonical, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', if R is the ring of algebraic integers in a number field K, then any choice of Z-basis {x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', xd} for R gives rise to a surjective homomorphism Z[u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ud] → R, whose kernel must be a zero-dimensional ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 20 Let us explain two concrete example classes that are covered by Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='16 (Principal algebraic N-actions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' A proper ideal I � Z[u] satisfies #V (I) < ∞ if and only if I = Z[u]f for a non-constant monic polynomial f ∈ Z[u] (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', [24, Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='a]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The action σu : N ↷ Z[u]/Z[u]f is called a principal algebraic N-action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' When f is non-constant and monic, the cosets of 1, u, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', un−1 form a Z-basis for Z[u]/Z[u]f, where n = deg(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The matrix for σu with respect to this basis is equal to the companion matrix Cf of f, so σu is injective if and only if f(0) = ± det(Cf) is non-zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' All in all, since V (Z[u]f) is the set of zeros of f, we see that σu : N ↷ Z[u]/Z[u]f satisfies conditions (a)-(d) in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13 if and if f is non-contant, monic, and |f(0)| > 1, and f(1) ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It follows from [32, Theorem] that σu : N ↷ Z[u]/Z[u]f is exact if and only if no unimodular polynomial divides f (in Q[u]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='18 (Algebraic Nd-actions defined by a point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' A special class of Nd-actions arises from d-tuples of algebraic integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' These are the irreversible analogues of the algebraic Zd-actions from [49, § 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose that c = (c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', cd) ∈ (Z ×)d, where Z denotes the ring of all algebraic integers, and let pc denote the kernel of the evaluation at c map R+ d → Z[c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', cd] ⊆ Q(c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', cd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then pc is a prime ideal of R+ d , and we can characterize when the action Nd ↷ R+ d /pc satisfies conditions (a)-(d) in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13 in terms of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' First, identify V (pc) with the set Hom (Q(c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', cd), Q) of field embeddings of Q(c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', cd) into Q, the algebraic closure of Q in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Explicitly, this identification is given by sending z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', zd) ∈ V (pc) to the embedding Q(c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', cd) �→ Q determined by ci �→ zi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' From this, it is easy to see that conditions (a) and (b) from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13 are satisfied if and only if ci ̸= 0 for all i, and condition (c) from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13 is satisfied if and only if there exists a finite non- empty set F ⊆ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', d} such that � i∈F ci ̸= 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' to see this, note that for any z = (z1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', zd) ∈ V (pc), we have � i∈F zi ̸= 1 if and only if � i∈F ci ̸= 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Condition (d) from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='13 is satisfied if and only if for each 1 ≤ j ≤ di, there exists a rational prime p with p | N(cj) and p ∤ N(ck) for k ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Algebraic actions from rings: The non-commutative case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For i = 1, 2, let Ri be a ring whose additive group is finitely generated and torsion-free, let Li ⊆ Ri be full rank subgroup, and let Mi ⊆ R× i a submonoid such that Li is Mi-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then, Mi ↷ Li is faithful since Li has finite index in Ri and Ri is torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let Li denote the completion of Li with respect to the family Ci of Mi-constructible subgroups of Li.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Our goal now is to establish the following rigidity result: Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Continue with the notation and assumptions above, with the additional assumptions that, for i = 1, 2, spanZ(Mi) has finite index in Ri, that there exists κi ∈ Mi for some κi ∈ Z \\ {0, 1}, and that QRi is a semisimple Q-algebra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', the (Jacobson) radical of QRi is trivial (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', [31, Part II, § 1]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If there is an isomorphism of topological groupoids (QR1 ⋊ ⟨M1⟩) ⋉ L1 ∼= (QR2 ⋊ ⟨M2⟩) ⋉ L2, then there are Q-algebra isomorphisms ϕ1 : QR1 ∼ = → QR2 and ϕ2 : QR2 ∼ = → QR1 such that ϕ1(⟨M1⟩) ⊆ ⟨M2⟩ and ϕ2(⟨M2⟩) ⊆ ⟨M1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Taking Mi = R× i and Li = Ri in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='19 yields the following: Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If QR1 and QR2 are semisimple Q-algebras and there is an isomorphism of topo- logical groupoids (QR1 ⋊ (QR1)∗) ⋉ R1 ∼= (QR2 ⋊ (QR2)∗) ⋉ R2, then there is a Q-algebra isomorphism QR1 ∼= QR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Observe that spanZ(R× i ) = Ri as shown in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The claim now follows from Theo- rem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The actions R× i ↷ Ri in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20 are exact, so Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='32 applies here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Before proceeding to the proof, let us explain several example classes to which our results apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='22 (Matrices over orders in number fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let n ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let R be an order in a number field K, and let I�R be a nonzero ideal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then, Mn(I) ⊆ Mn(R) is invariant under the canonical action Mn(R)× ↷ Mn(R), so we get an algebraic action Mn(R)× ↷ Mn(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We have QMn(I) = Mn(K), so Mn(I) has full rank in Mn(R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We have Z1n ⊆ Mn(R)×, and spanZ(Mn(R)×) has finite index 21 in Mn(R) by the proof of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, the hypotheses of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='19 are satisfied (for L = Mn(I) and M = Mn(R)×).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Moreover, ⟨Mn(R)×⟩ = GLn(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Therefore, if K1 and K2 are number fields with rings of algebraic integers R1 and R2, respectively, I1 � R1, I2 � R2 are non-zero ideals, n1, n2 ∈ Z>0, and if (Mn1(K1) ��� GLn1(K1)) ⋉ Mn(I1) ∼= (Mn2(K2) ⋊ GLn2(K2)) ⋉ Mn(I2) as topological groupoids, then Mn1(K1) ∼= Mn2(K2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In particular, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='19 implies that the groupoids (Mn1(K) ⋊ GLn1(K)) ⋉ Mn1(R1) and (Mn2(K2) ⋊ GLn2(K2)) ⋉ Mn2(R2) are isomorphic if and only if n1 = n2 and K1 ∼= K2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='23 (Group rings of finite groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let F1 and F2 be finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By Maschke’s Theorem (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', [31, Theorem 25]), QFi is a semisimple Q-algebra (i = 1, 2), so Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20 implies the following: If there is an isomorphism (QF1 ⋊ (QF1)∗) ⋉ ZF1 ∼= (QF2 ⋊ (QF2)∗) ⋉ ZF2, then QF1 ∼= QF2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that if F1, F2 are Abelian, then QF1 ∼= QF2 if and only if F1 ∼= F2 by [44, Corollary 1 & Theorem 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It is a non-trivial result that there exist finite non-Abelian groups F1, F2 with ZF1 ∼= ZF2 and F1 ̸∼= F2 (see [29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='24 (Central simple algebras over number fields).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let A be a central simple algebra over the number fields K, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', A is a finite-dimensional simple K-algebra whose centre is precisely K, and let O be an order in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By the Wedderburn Structure Theorem, there exists a (central) division algebra D over K and µ ∈ Z>0 such that A ∼= Mµ(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, the algebraic action O× ↷ O fits into the setting of Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus if (A1 ⋊ A∗ 1) ⋉ O1 ∼= (A2 ⋊ A∗ 2) ⋉ O2 as topological groupoids, then A1 ∼= A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now our goal is to prove Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For the remainder of this section, we shall work with the assumptions and notation from Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We need some preparations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let R be ring whose additive group is torsion-free and of rank n, and let M ⊆ R× be a submonoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose that α ∈ ⟨M⟩ satisfies α = γακγ−1 for some γ ∈ (QR)∗ and κ ∈ Z>1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Set m := κ(dimQQR)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then there exists a nilpotent element ηα ∈ QR such that αm = 1 + ηα (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', αm is a unipotent element of the Q-algebra QR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The map π: QR → End C(CR) ∼= Mn(C) given by π(q ⊗ a)(z ⊗ b) = qz ⊗ ab is an injective Q-algebra homomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The equation α = γακγ−1 implies that π(α) = π(γ)π(α)κπ(γ)−1 in Mn(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It follows that sp(π(α)) = sp(π(α)κ) = sp(π(α))κ := {λκ : λ ∈ sp(π(α))}, where sp(π(α)) is the spectrum of π(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, the map sp(π(α)) → sp(π(α)) given by λ �→ λk is bijective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' write sp(π(α)) = {λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', λj}, and let ρ be the permutation of sp(π(α)) determined by λi = λκ ρ(i) for all 1 ≤ i ≤ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since j ≤ dimQQR, we have ρ(dimQQR)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' = id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, λi = λκ(dimQQR!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=') ρ(dimQQR)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (i) = λκ(dimQQR)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' i , so that λκ(dimQQR)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='−1 i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We now see that 1 is the only eigenvalue of π(α)m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By considering the Jordan Normal Form of π(α)m, it follows that there exists a nilpotent matrix Nα ∈ Mn(C) such that π(α)m = 1 + Nα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Nα = π(α)m − 1 = π(αm − 1), we see, by injectivity of π, that ηα := αm − 1 is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Given a division algebra D, we shall regard Dn as an Mn(D)-D-bimodule in the usual way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, a basis for Dn will always mean a right D-basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We shall use a subscript D on the right to indicate that we are viewing something as a right D-vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let D be a finite dimensional division algebra over Q and n ∈ Z>0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Suppose that Σ is a non-trivial finitely generated Abelian subgroup of GLn(D) consisting of unipotent matrices, so that every α ∈ Σ is of the form α = 1 + ηα, where ηα ∈ Mn(D) is a nilpotent matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let k := � α∈Σ ker ηα and k := dimkD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We have rkZΣ < n · (n − k)[D : Q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let N := spanQ{ηα : α ∈ Σ} ⊆ Mn(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For α, α′ ∈ Σ, we have 1 + ηαα′ = αα′ = (1 + ηα)(1 + ηα′) = 1 + ηα + ηα′ + ηαηα′, so that ηαηα′ = ηαα′ − ηα − ηα′ lies in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' From this, we see that N is a non-unital commutative sub-Q-algebra of Mn(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For α ∈ Σ, let log α := �∞ i=1(−1)i−1 (1−α)i i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' this is a finite sum because 1 − α is nilpotent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that log α lies in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Σ is Abelian, log defines an injective group homomorphism (with inverse given 22 by exp, see for instance [28, § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='10, Exercise 8]) from Σ into N, so that rkZΣ ≤ dimQN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, we will be done once we show that dimQN < n · (n − k)[D : Q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since N is closed under multiplication and consists of nilpotent elements, [31, Part II, § 5, Theorem 35] asserts that there exists a right D-basis w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', wn for Dn such that N is strictly upper triangular with respect to this basis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' this means that if we define Wi := span{w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', wi}D for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='., n, then ηw1 = 0 and for each i ≥ 2, ηWi ⊆ Wi−1 for all η ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let W ⊆ {w1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', wn} be a subset such that W together with a basis for k is a basis for Dn, and put W = span(W)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since η|k ≡ 0 for all η ∈ N, the map N → Hom (W, Dn)D, η �→ η|W is injective;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' moreover, since wn ̸∈ NDn, we see that it is not surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence, dimQN < dimQHom (W, Dn)D = dimQMn×(n−k)(D) = n · (n − k)[D : Q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let c be the cocycle defined by the composition (QR1 ⋊ ⟨M1⟩) ⋉ L1 ∼= (QR2 ⋊ ⟨M2⟩) ⋉ L2 (h,y)�→h −→ QR2 ⋊ ⟨M2⟩, where the second map is the canonical cocycle obtained by projecting onto the group component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='4 produces a (finite index) subgroup C ∈ C1 such that g(x) := c(x, 0) defines an injective group homomorphism from C into QR2⋊⟨M2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let T ⊆ ⟨M2⟩ be the image of C under the composition (4) C g→ QR2 ⋊ ⟨M2⟩ (b,t)�→t → ⟨M2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By assumption, there exists κ ∈ M1 with κ ∈ Z \\ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let m := κ(dimQQR2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then we claim that for every α ∈ Tm, there exists a nilpotent element ηα ∈ QR2 such that α = 1 + ηα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Indeed, as κ ∈ M2, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='5 implies that there exists γ ∈ ⟨M2⟩ such that α = γακγ−1 for all α ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The result now follows from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' In particular, Tm is torsion-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The subgroup mC ⊆ C is mapped onto Tm under the projection in (4), so Tm is moreover finitely generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We will show that Tm is trivial, which will imply that there exists an injective group homomorphism b: mC → QR2 such that g(x) = (b(x), 1) for all x ∈ mC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since Tm is free abelian, there exists a subgroup CT ⊆ mC such that mC = C′ ⊕ CT, where C′ = {x ∈ mC : g(x) = (y, 1) for some y ∈ QR2}, and CT is mapped isomorphically onto Tm under the map in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let B be the be the image of C′ under the composition mC g→ QR2 ⋊ ⟨M2⟩ (b,t)�→b → QR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Note that B is a subgroup of QR2 because the second map is a homomorphism on QR2 ⋊ {1}, which contains the image of C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let us now show that the group Tm is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since QR2 is a semisimple Q-algebra, the Artin– Wedderburn theorem implies that there exists a decomposition of Q-algebras QR2 = �r i=1 Mni(Di), where r, n1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', nr ∈ Z>0 and each Di is a finite-dimensional division algebra over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, ⟨M2⟩ ⊆ �r i=1 GLni(Di).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', r, let Bi be the image of B under the canonical projection QR2 → Mni(Di).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' For each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', r, let Ti be the image of Tm under the canonical projection ⟨M2⟩ → GLni(Di).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then Ti = {1 + ηi α : α ∈ Tm}, where ηi α denotes the i-th coordinate of ηα (which come from Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='25);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' in particular, the group Ti consists of unipotent elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let ki := � α∈Tm ker (ηi α) and ki := dim(ki)D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Take x′ ∈ C′ and write g(x′) = (β′, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' If x ∈ CT, we can write g(x) = (β, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Then g(x′)g(x) = (β′ + β, α), whereas g(x)g(x′) = (β + αβ′, α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, αβ′ = β′, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ηαβ′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It follows that ηi αβ = 0 for all α ∈ Tm and β ∈ Bi, so that im(β) ⊆ ki for all β ∈ Bi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' From this, we see that im(β) ⊆ ki for all β ∈ span(Bi)Di := {� j βjdj : βj ∈ Bi, dj ∈ Di}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence, dim(span(Bi)Di)Di ≤ ni · ki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now we have rkZBi = dimQ(Q ⊗ Bi) ≤ dimQ(span(Bi)Di) ≤ ni · ki · [Di : Q].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Assume for the sake of contradiction that Tm is non-trivial, so that Ti is non-trivial for some i (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=', ki < ni).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='26, we have rkZTi < ni · (ni − ki)[Di : Q], where ki := dim(ki)Di.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Since rkZTm ≤ �r i=1 rkZTi and rkZB ≤ �r i=1 rkZBi, we obtain dimQQR1 = rkZC = rkZC′ + rkZCT = rkZB + rkZTm ≤ r � i=1 rkZTi + r � i=1 rkZBi < r � i=1 n2 i · [Di : Q] = dimQR2, 23 where the strict inequality uses our assumption that ki < ni for some i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' By symmetry, we also get dimQQR2 < dimQQR1, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Thus, ki = ni for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence Tm is indeed trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' We conclude that there exists a group homomorphism b: mC → QR2 such that g(x) = (b(x), 1) for every x ∈ mC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Let t be the composition ⟨M1⟩ g→ QR2 ⋊ ⟨M2⟩ ↠ ⟨M2⟩, where the second arrow is the canonical projection map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' It is easy to check that t is a group homomorphism, and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='18 shows that t is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Now Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='20 shows that for all s ∈ M1 and x ∈ mC, we have b((σ1)s(x)) = (˜σ2)t(s)(b(x)), where σ1 is the algebraic action M1 ↷ L1 and ˜σ2 is the algebraic action ⟨M2⟩ ↷ QR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hence the same proof as for Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='23 (i) produces an injective homomorphism ˙b: QR1 �→ QR2 which is equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Applying Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='1 yields the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' □ References [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Baudisch, Subgroups of semifree groups, Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Acad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Sci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Hungar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 38 (1981), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 1-4, 19–28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' [2] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Baumslag, Some aspects of groups with unique roots, Acta Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 104 (1960), 217–303.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' [3] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Baumslag, Generalized free products whose two-generator subgroups are free, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' London Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 43 (1968), 601–606.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' [4] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Berend, Ergodic semigroups of epimorphisms, Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 289 (1985), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 1, 393–407.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' [5] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Bhargava, Higher composition laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' A new view on Gauss composition, and quadratic generalizations, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (2) 159 (2004), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 1, 217–250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' [6] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Bhargava, Higher composition laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' On cubic analogues of Gauss composition, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (2) 159 (2004), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 2, 865–886.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' [7] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Bhargava, Higher composition laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The parametrization of quartic rings, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (2) 159 (2004), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 3, 1329–1360.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' [8] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Bhargava, The density of discriminants of quartic rings and fields, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (2) 162 (2005), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 2, 1031–1063.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' [9] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Bhargava, Higher composition laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' The parametrization of quintic rings, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (2) 167 (2008), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 1, 53–94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' [10] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Bhargava, The density of discriminants of quintic rings and fields, Ann.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' (2) 172 (2010), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' 3, 1559–1591.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' [11] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content=' Bruce, C*-algebras from actions of congruence monoids on rings of algebraic integers, Trans.' metadata={'source': 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Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, United Kingdom Email address: Xin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='Li@glasgow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'} +page_content='uk 25' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/EtE3T4oBgHgl3EQfVQrm/content/2301.04459v1.pdf'}