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+arXiv:2301.05086v1  [math.GR]  12 Jan 2023
+Sublinear Rigidity of Lattices in Semisimple Lie Groups
+Ido Grayevsky
+Abstract
+Let G be a real centre-free semisimple Lie group without compact factors. I prove that irreducible
+lattices in G are rigid under two types of sublinear distortions.
+I show that if Λ ≤ G is a discrete
+subgroup that sublinearly covers a lattice, then Λ is itself a lattice. I use this result to prove that the
+class of lattices in groups that do not admit R-rank 1 factors is SBE complete: if Λ is an abstract
+finitely generated group that is Sublinearly BiLipschitz Equivalent (SBE) to a lattice in G, then Λ can
+be homomorphically mapped into G with finite kernel and image a lattice in G. This generalizes the well
+known quasi-isometric completeness of lattices in semisimple Lie groups.
+1
+Introduction
+The quasi-isometric rigidity and classification of irreducible lattices in semisimple Lie groups was established
+in the 1990’s by the accumulated work of many authors - Pansu [48], Schwartz [52], Kleiner-Leeb [32],
+Eskin [22], Drut¸u [16], to name a few which are closely related to this paper. See [23] for a concise survey of
+the following result.
+Theorem 1.1 (Quasi-Isometric Completeness, Theorem I in [23]). Let G be a real finite-centre semisimple
+Lie group without compact factors, Γ ≤ G an irreducible lattice and Λ an abstract finitely generated group.
+If Γ and Λ are quasi-isometric, then there is a group homomorphism Φ : Λ → G with finite kernel whose
+image Λ′ := Φ(Λ) is a lattice in G. Put differently, there is a lattice Λ′ ≤ G and a finite subgroup F ≤ Λ
+such that the sequence
+1 → F → Λ → Λ′ → 1
+is exact. Moreover, Λ′ is uniform if and only if Γ is uniform.
+In the standard terminology of metric rigidity, Theorem 1.1 states that the classes of uniform and non-
+uniform lattices in a group G as in the statement are quasi-isometrically complete. The main goal of the
+current work is to generalize this result to a sublinear setting. Sublinear BiLipschitz Equivalences (SBE) are
+a sublinear generalization of quasi-isometries, brought forward by Cornulier [13] in the past decade or so. A
+function u : R≥0 → R≥1 is sublinear if limr→∞
+u(r)
+r
+= 0. For a, b ∈ R≥0 denote a ∨ b : max{a, b}, and for a
+pointed metric space (X, x0, dX) and x, x1, x2 ∈ X, let |x|X := dX(x, x0), |x1 − x2|X := dX(x1, x2).
+Definition 1.2. Let (X, dX, x0), (Y, dY , y0) be pointed metric spaces, L ∈ R>0 a constant, u : R≥0 → R≥1
+a sublinear function. A map f : X → Y is an (L, u)-SBE if the following conditions are satisfied:
+1. f(x0) = y0,
+2. ∀x1, x2 ∈ X,
+1
+L|x1 − x2|X − u
+�
+|x1|X ∨ |x2|X
+�
+≤ |f(x1) − f(x2)|Y ≤ L|x1 − x2|X + u
+�
+|x1|X ∨ |x2|X
+�
+,
+3. ∀y ∈ Y ∃x ∈ X such that |y − f(x)|Y ≤ u(|y|Y ).
+1
+
+A map is an SBE if it is an (L, u)-SBE for some L and u as above.
+I prove SBE-completeness for irreducible lattices in groups without R-rank 1 factors.
+Theorem 1.3 (SBE-Completeness). Let G be a real centre-free semisimple Lie group without compact or
+R-rank 1 factors.
+Let Γ ≤ G be an irreducible lattice and Λ an abstract finitely generated group, both
+considered as metric spaces with some word metric. Assume there is an (L, u)-SBE f : Λ → Γ with u a
+subadditive sublinear function. Then there is a group homomorphism Φ : Λ → G with finite kernel whose
+image Λ′ := Φ(Λ) is a lattice in G. Moreover, Λ′ is uniform if and only if Γ is uniform.
+The main ingredients in the proof are of independent interest: the first is geometric rigidity for the
+corresponding symmetric space, stating that every self SBE of such a space is sublinearly close to an isom-
+etry. This generalizes Kleiner and Leeb’s result [32] on self quasi-isometries of symmetric spaces, as well as
+Eskin’s [22] and Drutu’s [16] results on self quasi-isometries of non-uniform lattices. For the definition of the
+‘compact core’ of a lattice see Section 2.2.2.
+Theorem 1.4 (Sublinear Geometric Rigidity). Let X be a symmetric space of noncompact type without
+R-rank 1 factors, Γ ≤ Isom(X) an irreducible lattice and X0 ⊂ X the compact core of Γ in X. For any
+(L, u)-SBE map f : X0 → X0 there is a sublinear function v = v(L, u) and an isometry g : X → X such
+that d
+�
+q(x), g(x)
+�
+≤ v(|x|) for all x ∈ X0.
+The proof of Theorem 1.3 actually requires a stronger version of Theorem 1.4, formulated in Lemma 6.24.
+Notice that Theorem 1.3 and Theorem 1.4 both hold for uniform as well as non-uniform lattices. I remark
+that ‘generalized quasi-isometries’ already appeared in the context of geometric rigidity, as a technical tool
+in Eskin and Farb’s work [21] [22] on quasi-isometries. Indeed much of their work is carried for maps which
+are even more general than SBE. It seems however that their approach cannot yield a sublinear bound as in
+Theorem 1.4, which is necessary for the proof of Theorem 1.3.
+The second ingredient is a property I call sublinear rigidity, stating that a discrete subgroup Λ ≤ G which
+sublinearly covers a lattice is itself a lattice. Sublinear rigidity holds for groups of any R-rank, and is the
+cornerstone of this work. Its proof contains the bulk of the original ideas that appear in this paper.
+Definition 1.5. For a function u : R≥0 → R>0 and a subset Y ⊂ X, define the u-neighbourhood of Y to be
+Nu(Y ) := {x ∈ X | d(x, Y ) ≤ u(|x|)}
+A subset Y ⊂ X is said to sublinearly cover Z ⊂ X if Z ⊂ Nu(Y ) for some sublinear function u.
+For the definition of a Q-rank 1 lattice, see Section 2.1 and Theorem 2.20.
+Theorem 1.6 (Sublinear Rigidity). Let G be a real centre-free semisimple Lie group without compact factors.
+Let Γ ≤ G be an irreducible lattice, Λ ≤ G a discrete subgroup that sublinearly covers Γ. If Γ is of Q-rank 1,
+assume further that Λ is irreducible. Then Λ is a lattice in G.
+The notion of irreducibility in the non-standard context of a general (non-lattice) subgroup is explained
+Section 4.4.2 (see Definition 4.47). Sublinear neighbourhoods arise naturally in the presence of SBE maps:
+the essential difference between a quasi-isometry and an SBE is that ‘far away in the space’, the ‘additive’
+error term of an SBE gets larger and larger. One is led to consider metric neighbourhoods that grow -
+sublinearly, yet unboundedly - with the distance to some (arbitrary) fixed base point.
+The hypothesis that u is sublinear is optimal in the sense that u could not be taken to be an arbitrary
+linear function. Indeed, the geometric meaning of Γ ⊂ Nu(Λ) is that for every element γ ∈ Γ, the ball
+BG
+�
+γ, u(|γ|)
+�
+‘of sublinear radius about γ’ must intersect Λ.
+Observe that if f is the identity function
+f(r) = r, then by definition f(|g|) = f
+�
+d(g, eG)
+�
+= d(g, eG). In particular G lies in the f-neighbourhood of
+the trivial subgroup which is, after all, not a lattice in G. For uniform lattices and for lattices with Kazhdan’s
+property (T) one can however relax the assumption of sublinearity:
+2
+
+Theorem 1.7 (Theorems 3.3 and 5.2). Let G be a real centre-free semisimple Lie group without compact
+factors, Γ ≤ G an irreducible lattice, Λ ≤ G a discrete subgroup. Fix ε > 0 and assume Γ ⊂ Nu(Λ) for the
+function u(r) = ε · r.
+1. If Γ is uniform and ε < 1, then Λ is a uniform lattice.
+2. If Γ has Kazhdan’s property (T) then there is a constant ε(G) such that if ε < ε(G) then Λ is a lattice.
+Theorem 1.6 generalizes the case where Γ lies in a bounded neighbourhood ND(Λ) for some D > 0, proved
+by Eskin and Schwartz in a slightly modified version (see Section 4.3 below). In the bounded case, the result
+is much stronger and states that Λ must be commensurable to Γ (except in groups locally isomorphic to
+SL2(R), see [52]). In the sublinear setting I can only prove a limited commensurability result, which stems
+from a reduction to the bounded setting:
+Theorem 1.8. Let G, Γ and Λ be as in Theorem 1.6. If Γ is uniform then so is Λ. If Γ is of Q-rank 1 then:
+1. If Γ ̸⊂ ND(Λ) for any D > 0, then also Λ is of Q-rank 1. .
+2. If Γ ⊂ ND(Λ) for some D > 0 and in addition Γ sublinearly covers Λ, then Λ is commensurable to Γ.
+I stress that the case where Γ and Λ each sublinearly covers the other arises naturally in the context of
+SBE-completeness, see Theorem 6.4.
+The most interesting case in the proof of Theorem 1.6 is when Γ is of Q-rank 1. In that case, the proof
+is entirely geometric, relying on the following key proposition which might be of independent interest:
+Proposition 1.9. In the setting of Theorem 1.6, assume that Γ is a Q-rank 1 lattice which does not lie
+in any bounded neighbourhood of Λ. Then there exists a horosphere H based at the rational Tits building
+associated to Γ such that
+�
+Λ ∩ StabG(H)
+�
+· x0 intersects H in a cocompact metric lattice. Moreover, the
+bounded horoball HB does not intersect the orbit Λ · x0.
+1.1
+Outline of Proof
+My proofs rely and draw on the works on the quasi-isometric rigidity for non-uniform lattices, due to
+Schwartz [52] in R-rank 1, and to Drut¸u [16] and Eskin [22] independently in groups of R-rank greater than
+1, often called higher rank groups. The geometric proof of Theorem 1.6 for Q-rank 1 lattices is quite delicate
+and involved. For this reason I give here a detailed sketch of the arguments and of the ideas one should have
+in mind when reading the proof. What is written here is a good enough account if one wishes to understand
+the main ideas while avoiding the technical details. I end the section with a brief sketch of the proofs for
+SBE-completeness and for sublinear rigidity in the case of property (T) groups.
+Strategy for Q-rank 1 Lattices.
+Denote dγ := d(γ, Λ). The novel case is when {dγ}γ∈Γ is unbounded.
+The rationale for the proof comes from a conjecture of Margulis, recently proved in full generality by Benoist
+and Miquel ([5], see Theorem 4.4 below). Their result states that a discrete subgroup in a higher rank Lie
+group is a lattice as soon as it intersects a horospherical subgroup in a lattice. This result could be seen
+as an algebraic converse to the geometric structure of a Q-rank 1 lattice, whose orbit in X intersects some
+parabolic horospheres in a cocompact (metric) lattice (see Section 2.2 for details). Proposition 1.9 is the
+geometric analogue of the Benoist-Miquel criterion, and basically completes the proof in the higher R-rank
+case (some non-trivial translation work is needed, see Section 4.4). Also, it easily follows from Proposition 1.9
+that every Γ-conical limit point is also Λ-conical (Corollary 4.27). The proof for R-rank 1 groups is then a
+simple use of a criterion of Kapovich-Liu for geometrically finite groups ([30], see Theorem 4.6). I now give
+a detailed description of the proof of Proposition 1.9.
+3
+
+The ABC of Sublinear Constraints.
+Fix a point x0 ∈ X = G/K, identify Γ and Λ with Γ · x0 and
+Λ · x0 respectively. Observe that by definition of dγ the interior of balls of the form B(γx0, dγ) does not
+intersect Λ · x0. I call such balls (or general metric sets) Λ-free. Moreover, these balls intersect Λ · x0 (only)
+in the bounding sphere: call such balls (sets) tangent to Λ. The Λ-free and, respectively, Γ-free regions in X
+are the main objects of interest in this work. Since Theorem 1.6 is known in the case of bounded {dγ}γ∈Γ,
+it makes sense to think about large Λ-free regions as ‘problematic’. The state of mind of the proof relies on
+two easy observations that complete each other.
+1. The sublinear constraint implies that dγn → ∞ forces |γn| → ∞, suggesting that ‘problematic’ Λ-free
+regions should appear only ‘far away’ in the space.
+2. On the other hand Λ is a group, and being Λ-free is a Λ-invariant property. In particular any metric
+situation that can be described in terms of the Λ-orbit (e.g. B(γ, dγ) is a Λ-free ball tangent to Λ) can
+be translated back to x0. This means that ‘problematic’ regions could actually be found near x0.
+The moral of these observations can be formulated into a general principle that lies in the heart of the
+argument. The sublinear constraint dγ ≤ u(|γ|) gives rise to many other constraints of ‘sublinear’ nature.
+Each such constraint actually yields a uniform constraint inside any fixed bounded neighbourhood of x0.
+Since Λ and Γ are groups, many of these uniform bounds which are produced ‘near’ x0 turn out to be global
+bounds that depend only on the group and not on a specific orbit point. Put differently: the trick is to
+describe metric situations in terms of the Γ and Λ orbits. One then uses the group invariance in order to
+move these metric situations around the space to a place where the sublinear constraint can be exploited.
+The Argument.
+The above paragraph should more or less suffice the reader to produce a complete proof
+for uniform lattices. For non-uniform lattices, denote by λγ the closest Λ-orbit point to the point γ ∈ Γ. For
+H a cusp horosphere of Γ, let ΓH := {γ ∈ Γ | γx0 ∈ H}. The ultimate goal is to show that the metric lattice
+ΓH · x0 yields a metric lattice that is more or less {λγx0}γ∈ΓH. One proceeds by the following steps.
+1. Finding Λ-free horoballs (Section 4.2.1): The arbitrarily large Λ-free balls B(γn, u(|γn|)) are translated
+to x0, and the compactness of the unit tangent space at x0 yields a converging direction which is the
+base point at infinity of a Λ-free horoball. Translating by Λ, this yields Λ-free horoballs tangent to
+every Λ-orbit point.
+2. Controlling angles (Section 4.2.2): For every γ ∈ Γ with dγ uniformly large enough, one associates a
+point ξ at X(∞) such that ξ is the base point of a Λ-free horoball tangent to λγx0. The angle between
+the geodesics [λγx0, γx0] and [λγX0, ξ) is shown to be small as dγ grows large. This is used to show
+that arbitrarily large dγ give rise to arbitrarily deep Γ-orbit points inside Λ-free horoballs. A key step
+is Lemma 3.8, producing a Γ-free Euclidean cylinder between λγx0 and γx0.
+3. Λ-cocompact horospheres (Section 4.2.3): One uses uniform bounds near x0 to prove that every Λ-free
+horoball that is (almost) tangent to Λ must lie in a uniformly bounded neighbourhood of Λ · x0. If
+dγ is large enough for some γ that lies on a horosphere HΓ of a cusp of Γ, then any γ′ ∈ ΓHΓ also
+admits large dγ′. Since the bounds from the previous steps only depend on dγ′, all λγ′ are forced to
+lie on the same horosphere HΛ parallel to HΓ. One concludes that Λ · x0 intersects HΛ on the nose in
+a cocompact metric lattice.
+Property (T).
+Sublinear rigidity for groups with property (T) is established by the criterion that a discrete
+subgroup there is a lattice if and only if it has the same exponential growth rate as a lattice (Leuzinger [36]).
+It is quite straightforward that sublinear distortion cannot affect this growth rate (Corollary 5.10).
+SBE Rigidity.
+The general scheme for SBE-completeness is parallel to the quasi-isometry case: each
+λ ∈ Λ naturally gives rise to an SBE X0 → X0 of the compact core of Γ. Each self SBE is close to an
+isometry by Theorem 1.4, allowing to embed Λ as a discrete subgroup of isometries in G that sublinearly
+4
+
+covers Γ. The proof for Theorem 1.4 heavily relies on Drut¸u’s argument for quasi-isometries [16], which uses
+the properties of the induced biLipschitz map on the asymptotic cone. As SBE also induce such a biLipschitz
+map - indeed that was a main motivation for Cornulier to study SBE [13] - it is possible to generally follow
+Drut¸u’s argument also in the SBE setting.
+1.2
+Possible Improvements, Related and Future Work
+The proof suggests three natural improvements to the statement of Theorem 1.6. One could probably relax
+the assumption of trivial centre and allow finite centre, if the same relaxation is applicable in Leuzinger’s
+work on property (T) groups [36] and in Prasad’s work [49]. In particular, the proofs for uniform lattices and
+for Q-rank 1 lattices hold also for groups G with finite centre. The irreducibility of Λ may be derived directly
+from the irreducibility of Γ. Lastly, in view of the geometric characterization of Q-rank (see Corollary D in
+[35]), it is reasonable that the Q-rank of Λ should equal that of Γ.
+There are problems that arise naturally from this work which seem to require new ideas:
+Question. Let G be a real finite-centre semisimple Lie group without compact factors that admits a R-
+rank 1 factor. Are the classes of uniform and non-uniform lattices of G SBE-complete? In particular, is this
+true when G is of R-rank 1?
+See Section 6.3.4 below for a discussion on the case of R-rank 1 factors. SBE of R-rank 1 symmetric spaces
+is the main focus in Pallier’s work [44]. He investigated the sublinearly large scale geometry of hyperbolic
+spaces, and proved that two R-rank 1 symmetric spaces that are SBE are homothetic, answering a question
+of Drut¸u (see Remarks 1.16 and 1.17 in [13]). Also in this context Pallier and Qing [47] recently showed that
+the sublinear Morse boundary is an SBE invariant.
+Another problem is to find a non-trivial example of the setting of Theorem 1.6:
+Question. Let G be a real finite-centre semisimple Lie group without compact factors, Γ ≤ G a lattice.
+1. Does there exist a finitely generated group that is SBE to Γ but not quasi-isometric to it? Or, at least,
+not known to be quasi-isometric to one?
+2. Does there exist Λ ≤ G discrete that ε-linearly covers Γ but which does not sublinearly cover it?
+On the side of the proof, it would be very interesting if the geometric ideas that prove Theorem 1.6 for
+Q-rank 1 lattices could be applied to any Q-rank. While there are apparent places where the proof uses the
+unique geometry of Q-rank 1 lattices, most of the geometric arguments leading to Proposition 1.9 seem to
+be susceptible to the higher Q-rank setting. Such a generalization of the proof would definitely shed more
+light on the mysterious lattice arising from growth considerations in property (T) groups, and in particular
+on the question of commensurability of Λ and Γ in that case. It would also be interesting to see whether
+one can push the geometric argument forward in order to establish a complete geometric analogue of the
+Benoist-Miquel criterion. Namely, can one find a direct geometric proof that Λ admits finite co-volume
+(perhaps similarly to Schwartz’s argument in the bounded case, see Section 4.3 below).
+Lastly, one could possibly relate this work to the work of Fraczyk and Gelander [24], who proved that
+a discrete subgroup (of a higher rank simple Lie group) is a lattice if and only if it has bounded injectivity
+radius. While their result seem very much related to the condition Γ ⊂ Nu(Λ), the nature of their work does
+not give explicit bounds on the injectivity radius. Specifically, given r > 0 one cannot tell directly from their
+results how ‘far’ one must wander in X in order to find a point with injectivity radius r. Perhaps one could
+use the sublinear results of this work to say something about the relation of |x|X and InjRad(x).
+1.3
+Acknowledgments
+This paper is based on my DPhil thesis, supervised by Cornelia Drut¸u.
+I thank her for suggesting the
+question of SBE-completeness and for guiding me in my first steps in the theory of Lie groups. I thank Uri
+Bader, Tsachik Gelander and the Midrasha on groups at the Weizmann institute, where I learned the basics
+5
+
+of symmetric spaces. I thank my thesis examiners Emmanuel Breuillard and Yves Cornulier for their careful
+inspection and numerous remarks. I thank Elon Lindenstrauss for telling me about Leuzinger’s result [36]
+on property (T), and Or Landesberg, Omri Solan, Elyasheev Leibtag, Itamar Vigdorovich and Tal Cohen
+for many discussions on different aspects of this paper. Finally I thank Gabriel Pallier for explaining his
+examples of some unusual SBE in R-rank 1, and for his interest in this work.
+2
+Preliminaries
+For standard definitions and facts about fundamental domains, see Chapter 5.6.4 in [17]. The facts about
+fundamental domains for Q-rank 1 lattices appear in Raghunathan’s book [50] and in Prasad’s work on
+rigidity of Q-rank 1 lattices [49]. In notations and generalities I follow: Borel’s book on algebraic groups [6],
+Helgason’s books on Lie groups and symmetric spaces [27, 28], and Eberline’s book on the geometry of
+symmetric spaces of noncompact type [20].
+2.1
+Generalities on Semisimple Lie Groups and their Lattices
+Let G be a real centre-free semisimple Lie group without compact factors. A discrete subgroup Γ ≤ G is
+a lattice if Γ\G carries a finite volume G-invariant measure. Equivalently, Γ is a lattice if Γ\X is a finite
+volume Riemannian manifold, where X = G/K is the symmetric space of noncompact type corresponding
+to G. A lattice is irreducible if its projection to every simple factor of G is dense. The group G can be
+viewed as an algebraic group via the adjoint representation. If G is of R-rank greater than 1, then by the
+Margulis arithmeticity theorem every irreducible lattice of G is arithmetic. The Q-rank Γ is the Q-rank of
+the Q-structure associated to (G, Γ) be the arithmeticity theorem. A result of Prasad [49] states that if G
+admits a R-rank 1 factor, then a non-uniform irreducible lattice of G is of Q-rank 1 .
+The group G has Kazhdan’s property (T) if and only if it does not admit an SO(n, 1) or an SU(n, 1)
+factor, and an irreducible lattice Γ ≤ G has property (T) if and only if G has property (T). Together with
+Prasad’s result, I may conclude:
+Lemma 2.1. Let G be a real centre-free semisimple Lie group without compact factors, and Γ ≤ G an
+irreducible lattice. Then at least one of the following occurs: (a) G has property (T) (b) Γ is a non-uniform
+Q-rank 1 lattice (c) Γ is uniform.
+Theorem 1.6 is therefore an immediate result of Theorem 3.1, Theorem 4.1 and Theorem 5.1.
+2.2
+Cusps and the Rational Tits Building
+The facts about symmetric spaces of noncompact type can be found in Eberline’s book [20].
+Since the
+geometry of Q-rank 1 lattices resembles that of lattices in R-rank 1, the reader could for the most part
+simply have the image of the hyperbolic plane in mind. If one wishes to see flats that are not geodesics, then
+a product of two hyperbolic planes is enough. Even the product of the hyperbolic plane and R is helpful,
+albeit this space has a Euclidean factor.
+2.2.1
+Basic Geometry of Symmetric Spaces of Noncompact Type
+Visual Boundary.
+The visual boundary X(∞) of X is the set of equivalence classes of geodesic rays,
+where two geodesic rays are equivalent if their Hausdorff distance is finite. For a ray η : [0, ∞) → X, η(∞)
+denotes the equivalence class of η in X(∞). There are two natural topologies on X(∞) that will be of use.
+The cone topology is the one given by viewing X(∞) as the set of all geodesic rays emanating from some
+fixed base point x0, with topology induced by the unit tangent space at x0. There is a natural topology on
+X := X ∪ X(∞) such that X is the compactification of X and the induced topology on X(∞) is the cone
+topology. A well known fact about geodesic rays in nonpositively curved spaces, stating that two ‘close’
+geodesic rays fellow travel:
+6
+
+Lemma 2.2. Given time T and ε > 0, there is an angle α = α(T, ε) so that if η1, η2 are two geodesic rays
+with η1(0) = η2(0) = x for some x ∈ X and ∡x(η1, η2) ≤ α then dX
+�
+η1(t), η2(t)
+�
+< ε for all t ≤ T .
+The Tits metric on X(∞) is defined as follows.
+Given a totally geodesic submanifold Y ⊂ X, let
+Y (∞) ⊂ X(∞) be the subset of all points that admit a geodesic ray η lying inside Y (or, equivalently, those
+points that admit a ray lying at bounded Hausdorff distance to Y ). For any ξ1, ξ2 ∈ X(∞) there exists a
+flat F ⊂ X such that ξ1, ξ2 ∈ F(∞). Define dT (ξ1, ξ2) ∈ [0, π] to be the angle between two geodesic rays
+η1, η2 ⊂ F emanating from some point x ∈ F and with η1(∞) = ξ1, η2(∞) = ξ2. This is a well defined metric
+on X(∞), called the Tits metric. The pair
+�
+X(∞), dT
+�
+is a geodesic metric space, and isometries of X act
+on it by isometries. I will use the following relation between the cone and the Tits topologies:
+Proposition 2.3 (Section 3.1 in [20]). Let X be a symmetric space of noncompact type. The Tits metric on
+X(∞) is semicontinuous with respect to the cone topology: for any ξ, ζ ∈ X(∞) and every ε > 0, there exists
+neighbourhoods of the cone topology U, V ⊂ X(∞) of ξ and ζ respectively such that for all ξ′ ∈ U, ζ′ ∈ V one
+has
+∡(ξ′, ζ′) ≥ ∡(ξ, ζ) − ε
+Moreover, for any flat F ⊂ X, the cone topology and the Tits topology coincide on F(∞).
+Busemann Functions, Horoballs and Horospheres.
+Horoballs and horospheres play a crucial role in
+the proof, a role which stems from their role in the geometric description of the compact core of non-uniform
+lattices (see Theorem 2.20 below). A Busemann function on X is any function of the form
+fη(x) = lim
+t→∞ d
+�
+x, η(t)
+�
+− t
+for some geodesic ray η of X. A horoball HB ⊂ X is an open sublevel set of a Busemann function. A
+horosphere H ⊂ X is a level set of a Busemann function. Two equivalent geodesic rays η1, η2 give rise to
+Busemann function which differ by a constant, i.e. fη1 − fη2 = C for some C ∈ R. If HB is the sublevel set
+of fη, then η(∞) is called the base point of the horoball HB (and respectively of the horosphere H = ∂HB).
+The base point of a horoball is well defined, i.e. it depends only on η(∞) and not on η. For every choice of
+x ∈ X, ξ ∈ X(∞) there is a unique horosphere H based at ξ with x ∈ H. I denote this horosphere by H(x, ξ)
+and the bounded horoball HB(x, ξ). The following proposition collects some basic properties that will be of
+use.
+Proposition 2.4 (Proposition 1.10.5 in [20]). Let x ∈ X, ξ ∈ X(∞), and let H = H(x, ξ), HB the horoball
+bounded by H and f the Busemann function based at ξ with f(x) = 0.
+1. For any point y ∈ X, let η be the bi-infinite geodesic determined by the geodesic [y, ξ). Then PH(y) =
+η ∩ H, where PH(y) is the unique point closest to y on H.
+2. For any point y ∈ X, f(y) = ±d
+�
+y, PH(y)
+�
+. Moreover, f(y) is negative if and only if y ∈ HB.
+3. If x′ ∈ X, then the horospheres H = H(x, ξ) and H′ = H(x′, ξ) are equidistant: if y ∈ H, y′ ∈ H′, then
+d(y, H′) = d(y′, H). Such horospheres are called parallel.
+A Busemann function fη thus naturally determines a filtration of X by the co-dimension 1 manifolds
+{Ht}t ∈ R. By convention I usually assume that Ht := {x ∈ X | fη(x) = −t}.
+Remark 2.5. The stabilizer of a point ξ ∈ X(∞) acts transitively on the set of horospheres based at ξ, so
+every two such horospheres are isometric. Moreover, there is a close relation between the induced metrics on
+horospheres with the same base point. Briefly, if dH denotes the induced distance on a horosphere H ⊂ X,
+then dH
+�
+PH(x), PH(y)
+�
+for any two points x, y ∈ H′ can be bounded uniformly below and above as a function
+of the distance dH′(x, y) and the curvature bounds on X. See Heintze-Im hof [26] for precise statements.
+7
+
+Using the above properties one can show that two horospheres are parallel if and only if they are based
+at the same point. In particular for every x ∈ X, ξ ∈ X(∞) it holds that StabG
+�
+H(x, ξ)
+�
+⊂ StabG(ξ). In
+addition, If H, H′ are two parallel horospheres based at the same ξ ∈ X(∞) and A ⊂ H is a cocompact
+metric lattice in H, then πH′(A) is a cocompact metric lattice in H′.
+For a point ξ ∈ X(∞) and a flat F with ξ ∈ F(∞), one readily observes that every horoball HB based
+at ξ intersects F in a Euclidean half space. In particular for every ζ ∈ X(∞) with dT (ξ, ζ) < π
+2 , for every
+geodesic ray η with η(∞) = ζ and every horoball HB based at ξ there is some T for which η↾t>T ⊂ HB.
+Parabolic and Horospherical Subgroups.
+The isometries of X are classified into elliptic, hyperbolic,
+and parabolic isometries. Most significant for this paper are the parabolic isometries, i.e. those g ∈ G whose
+displacement function x �→ gx does not attain a minimum in X. Every such isometry fixes (at least) one
+point in X(∞). A group P ≤ Isom(X) is called geometrically parabolic if it is of the form Gξ := StabG(ξ) for
+some ξ ∈ X(∞). Such groups act transitively on X, and in particular act transitively on the set of geodesic
+rays in the equivalence class of ξ. The same holds also for the identity component G◦
+ξ. An element g ∈ Gξ
+acts by permutation on the set of horoballs based at ξ. This permutation is a translation with respect to
+the filtration of the space X by horospheres based at ξ. Put differently, if {Ht}t∈R is a filtration of X by
+horospheres based at ξ, then for every g ∈ Gξ there is l(g) ∈ R such that gHt = Ht+l(g). It is quite clear
+from all of the above that for every horosphere based on ξ, the group GH := StabG(H) acts transitively on
+H, and the same holds for G◦
+H.
+I now present a fundamental structure theorem for geometrically parabolic groups. Denote g := Lie(G),
+and let g = k ⊕ p be a Cartan decomposition defined using the maximal compact subgroup K ≤ G. Recall
+that the Lie exponential map exp : g → G gives rise to a family of 1-parameter subgroups of the form
+exp(tX) for each X ∈ p.
+Proposition 2.6 (Proposition 2.17.3 in [20]). Let x ∈ X(∞), and let X ∈ p be the tangent vector of the
+unit speed geodesic [x0, ξ). Let ht
+ξ be the 1-parameter subgroup defined by t �→ exp(tX). Then an element
+g ∈ G fixes ξ if and only if limt→∞ h−t
+ξ ght
+ξ exists.
+Proposition 2.7 (Langlands Decomposition, Propositions 2.17.5 and 2.17.25 in [20]). Let ξ ∈ X(∞) and
+ht
+ξ as in Proposition 2.6. Let F be a flat containing [x0, ξ) and A ≤ G the maximal abelian subgroup such
+that Ax0 = F. Denote Gξ := StabG(ξ), and define Tξ : Gξ → G by g �→ limn→∞ h���n
+ξ
+ghn
+ξ . Then Tξ is a
+homomorphism, and there are subgroups Nξ, Aξ, Kξ ≤ Gξ such that:
+1. Aξ = exp
+�
+Z(X) ∩ p
+�
+, where Z(X) is the centralizer of X in g. Moreover, every element a ∈ Aξ lies in
+some conjugate Ag = gAg−1 with the property that [x0, ξ) ⊂ F g := Agx0.
+2. Kξ ≤ K = StabG(x0) is the compact subgroup fixing the bi-infinite geodesic determined by [x0, ξ).
+3. KξAξ = AξKξ.
+4. Nξ = Ker(Tξ). It is a connected normal subgroup of Gξ.
+5. Gξ = NξAξKξ, and the indicated decomposition of an element is unique.
+6. G = NξAξK, and the indicated decomposition of an element is unique. In case ξ is a regular point at
+X(∞), this decomposition is the Iwasawa decomposition.
+7. Gξ has finitely many connected components, and G◦
+ξ = (KξAξ)◦Nξ.
+8. Gξ is self normalizing.
+Viewing G as an algebraic group, the geometrically parabolic subgroups are exactly the (algebraically)
+non-trivial parabolic subgroups, i.e. proper subgroups of G that contain a normalizer of a maximal unipotent
+subgroup. Proposition 2.7 is a geometric formulation of the algebraic Langlands decomposition of parabolic
+groups. Recall that a horospherical subgroup is the unipotent radical of a non-trivial parabolic group, or
+equivalently groups of the form Ug := {u ∈ G | limn→∞ g−nugn = idG}. The latter implies that Nξ a
+horospherical subgroup of G.
+8
+
+Limit Set.
+An important set associated to a discrete group ∆ ≤ G acting by isometries on X is the limit
+set L∆. By definition L∆ := ∆ · x ∩ X(∞), i.e. it is the intersection with X(∞) of the closure, in the
+compactification X = X ∪ X(∞), of an orbit ∆ · x. It is clear that L∆ does not depend on the choice of
+x ∈ X. The limit set of any lattice is always the entire X(∞). In fact, much more is true:
+Definition 2.8. Let ∆ ≤ G = Isom(X), and SX the unit tangent bundle. A vector v ∈ SX is ∆-periodic
+if there is δ ∈ ∆ and s > 0 such that δη(t) = η(t + s) for all t ∈ R, where η is the bi-infinite geodesic
+determined by the vector v.
+For a flat F ⊂ X (including geodesics), denote ∆F := {δ | δF = F}. The flat F is called a ∆-periodic
+flat if there exists a compact set C ⊂ F such that ∆F C = F.
+Proposition 2.9 (Propositions 4.7.3., 4.7.5, 4.7.7 in [20], Lemma 8.3′ in [41]). If Γ ≤ G = Isom(X) is a
+lattice, then:
+1. The subset in SX of Γ-periodic vectors is dense.
+2. Let F ⊂ X be any flat, η any bi-infinite geodesic in F, and denote v = ˙η(0) ∈ SX the initial velocity
+vector. There is a sequence vn ∈ SX of regular vectors such that
+(a) limn→∞ vn = v.
+(b) The bi-infinite geodesics ηn determined by vn are all Γ-periodic.
+(c) Denote by Fn the (unique) flat containing ηn. Each Fn is Γ-periodic.
+Put differently, the set of Γ-periodic flats is dense in the set of flats of X.
+Remark 2.10. See Definition 2.25 for the notion of regular tangent vectors.
+Points in the limit set of a group are classified according to how the orbit approaches them.
+Definition 2.11. Let ∆ ≤ G be a discrete subgroup, ξ ∈ L∆.
+1. The point ξ is called conical if for some (hence any) x and some (hence every) geodesic ray η with
+η(∞) = ξ there is a number D = D(x, η) such that for every T ∈ R>0 there is t > T for which
+B
+�
+η(t), D
+�
+∩ ∆ · x ̸= ∅. Since ∆ is discrete, this is equivalent to ∆ · x ∩ ND(η) being infinite.
+2. The point ξ is called horospherical if for every horoball HB based at ξ and every x ∈ X, ∆ · x ∩ HB is
+non-empty. In particular, a conical limit point is horospherical.
+3. The point ξ is non-horospherical if it is not horospherical.
+As a corollary of Proposition 2.9, one has:
+Corollary 2.12. If Γ ≤ Isom(X) is a lattice, then
+1. The set of Γ-conical limit points is dense in X(∞) with the cone topology.
+2. LΓ = X(∞).
+I finish this section with some results on geometrically finite subgroups of isometries in R-rank 1.
+Definition 2.13. Let X be a R-rank 1 symmetric space and ∆ ≤ Isom(X) a discrete subgroup. Denote by
+Hull(∆) the closed convex hull in X = X ∪X(∞) of the limit set L∆, and Hull(∆) = X ∩Hull(∆). By virtue
+of negative curvature, Hull(∆) is the union of all geodesics η such that η(∞), η(−∞) ∈ L∆. The convex core
+of ∆ is defined to be ∆\Hull(∆) ⊂ ∆\X, i.e., the quotient of Hull(∆) by the ∆-action.
+Definition 2.14 (Bowditch [9], see Theorem 1.4 in [30]). Let X be a R-rank 1 symmetric space, i.e. a
+symmetric space of pinched negative curvature. A discrete group ∆ ≤ G = Isom(X) is geometrically finite if
+for some δ > 0, the uniform δ-neighbourhood in ∆\X of the convex core Nδ
+�
+∆\Hull(∆)
+�
+, has finite volume
+and there is a bound on the orders of finite subgroups of ∆. A group is geometrically infinite if it is not
+geometrically finite.
+9
+
+Immediately from the definition of geometrical finiteness, one gets a simple criterion for a subgroup to
+be a lattice:
+Corollary 2.15. Let X be a R-rank 1 symmetric space. If ∆ ≤ Isom(X) is geometrically finite and admits
+L∆ = X(∞), then ∆ is a lattice in Isom(X).
+Sublinear distortion does not effect the limit set, as the following lemma shows.
+Lemma 2.16. Let Γ, Λ ≤ G be discrete subgroups and u : R≥0 → R>0 a sublinear function. If Γ ⊂ Nu(Λ),
+then LΓ ⊂ LΛ, i.e. every Γ-limit point is a Λ-limit point. In particular, if Γ is a lattice then LΛ = X(∞).
+Proof. By definition, one has to show that given a point ξ ∈ X(∞) and a sequence γn ∈ Γ such that γnx0 → ξ
+(in the cone topology on X), there is a corresponding sequence λn ∈ Λ with λnx0 → ξ. Define λn := λγn to
+be the closest point to γn in Λ, and to ease notation denote xn := γnx0, x′
+n = λnx0. Let also ηn := [x0, xn]
+and η′
+n := [x0, x′
+n] be unit speed geodesics. Finally, let Tn denote the time in which ηn terminates, i.e.
+ηn(Tn) = xn.
+Convergence in the cone topology xn → ξ is equivalent to the fact that the geodesics ηn converge to
+η := [x0, ξ) uniformly on compact sets. This means in particular that Tn → ∞. Non-positive curvature
+guarantees that the functions Fn(t) := d
+�
+η(t), ηn(t)
+�
+, F ′
+n(t) := d
+�
+η(t), η′
+n(t)
+�
+and Gn := d
+�
+ηn(t), η′
+n(t)
+�
+are
+convex (F ′
+n is just a notation, completely unrelated to the derivative of Fn). Since Gn(0) = Fn(0) = F ′
+n(0) = 0
+any of these functions is either constant 0 or monotonically increasing, so proving uniform convergence of
+η′
+n to η amounts to proving limn F ′
+n(T ) = 0 for every T ∈ R≥0.
+Triangle inequality gives F ′
+n(T ) ≤ Fn(T ) + Gn(T ), and by assumption limn Fn(T ) = 0. Notice that
+Gn(Tn) = d(xn, x′
+n) ≤ u(|γn|) = u(Tn). Writing T =
+T
+Tn · Tn, convexity of Gn implies
+Gn(T ) ≤ (1 − T
+Tn
+)Gn(0) + T
+Tn
+Gn(Tn) ≤ 0 + T
+Tn
+u(Tn) = T · u(Tn)
+Tn
+As limn Tn = ∞ it follows from sublinearity that limn Gn(T ) = 0. I conclude that η′
+n converge to η
+uniformly on compact sets, therefore ξ lies in the limit set of Λ.
+Remark 2.17. In Section 4.2 I prove that in the setting of Proposition 4.12, the set of Λ-conical limit points
+contains the set of Γ-conical limit points (Corollary 4.27). It holds that the conical limit points of Γ are
+dense in X(∞) (in the cone topology, see Corollary 2.12) and therefore LΛ = LΓ = X(∞). In particular,
+every Γ-limit point is a Λ-limit point. The strength of Lemma 2.16 is that it does not assume anything on Γ
+other than that it is sublinearly covered by Λ. In particular, Lemma 2.16 does not require Γ to be a lattice.
+2.2.2
+Cusps, Compact Core, and the Rational Tits Building
+In this section I present some of the structure theory of non-compact quotients of X. The focus is on the
+structure of ‘cusps’ in noncompact finite volume quotients of symmetric spaces, and the ‘location’ of cusps
+on the visual boundary.
+Cusps and Compact Core.
+Consider V = Γ\X, for Γ ≤ G a non-uniform lattice. This is a locally
+symmetric space of finite volume. The term ‘cusps’ is an informal name given to those areas in a locally
+symmetric space through which one can ‘escape to infinity’. Another description is that cusps are the ends
+of the complement of a large enough compact set in V. In strictly negative curvature, i.e. in R-rank 1 locally
+symmetric spaces, these cusps have a precise description as submanifolds of the form C × R≥0 for a compact
+manifold C, and metrically (C, t) gets narrower as t → ∞. There are finitely many cusps, each corresponding
+to a point at X(∞) called a ‘parabolic point’. See e.g. Introduction in [3] or [19]. A fundamental feature
+of the cusps is that one can ‘chop’ them out of the quotient manifold V and get a compact manifold. This
+could be done in such a way so that:
+1. The lifts of the chopped parts to the universal cover X are disjoint.
+10
+
+2. Each cusp is covered in X by the Γ-orbit of a horoball, that is, the lift of a cusp is the Γ-orbit of a
+horoball. The respective base points are called parabolic points of Γ in X(∞).
+3. Γ acts on X \
+� �
+i∈I HBi
+�
+cocompactly, where {HBi}i∈I is the set of horoballs coming from the lifts
+of cusps.
+Since there are only finitely many cusps and Γ discrete, there are exactly countably many such horoballs.
+See for example Section 12.6 in [17] where this is illustrated in the case of the real hyperbolic spaces Hn.
+Formally, one has;
+Theorem 2.18 (Theorem 3.1 in [19], see also Introduction therein). Assume X is of R-rank 1, and Γ ≤ G a
+non-uniform lattice. The space V = Γ\X has only finitely many (topological) ends and each end is parabolic
+and Riemannian collared. In particular, each cusp is a quotient of a horoball HB based at a parabolic limit
+point ξ such that Γ ∩ Gξ acts cocompactly on H = ∂HB.
+For symmetric spaces of higher rank, a similar construction is available (see [37]). By removing a countable
+family of horoballs from X, one obtains a subspace on which Γ acts cocompactly. There are two main
+differences from the situation in R-rank 1. One is that an orbit map γ �→ γx is a quasi-isometric embedding
+of Γ (with the word metric) into X.
+Theorem 2.19 (Lubozki-Mozes-Raghunathan, Theorem A in [38]). Let G be a semisimple Lie group of
+higher R-rank, dG a left invariant metric induced from some Riemannian metric on G. Let Γ an irreducible
+lattice, dΓ the corresponding word metric on Γ. Then dG↾Γ×Γ and dΓ are Lipschitz equivalent.
+This result plays a significant preliminary role in the proofs of quasi-isometric rigidity for non-uniform
+lattices in higher rank symmetric spaces in both [16] and [22]. The second difference in higher rank spaces
+is that the horoballs could not in general be taken to be disjoint. However, in the special case of Q-rank 1
+lattices the horoballs can be taken to be disjoint. Recall that the Q-structure of (G, Γ) is a Q-structure on
+G = G(R) in which Γ is an arithmetic lattice. The following theorem sums up the relevant properties for
+Q-rank 1 lattices.
+Theorem 2.20 (Theorem 4.2 and Proposition 2.1 in [34], see also Remarks 3 and 4 in [37], Section 13
+in [50], and Proposition 2.1 in [49]). Assume X is of higher rank, and Γ ≤ G an irreducible torsion-free
+non-uniform lattice. On the locally symmetric space V = Γ\X there exists a continuous and piece-wise real
+analytic exhaustion function h : V → [0, ∞) such that, for any s > 0, the sublevel set V(s) := {h < s} is a
+compact submanifold with corners of V. Moreover the boundary of V(s), which is a level set of h, consists of
+projections of subsets of horospheres in X.
+The Γ-action on the above set of horospheres has finitely many orbits, and the following conditions are
+equivalent:
+1. The corresponding horoballs bounded by these horospheres can be taken to be disjoint.
+2. For each such horosphere H the action of Γ ∩ StabG(H) on H is cocompact.
+3. The Q-structure of (G, Γ) is of Q-rank 1.
+4. The lattice Γ is of Q-rank 1.
+Definition 2.21. In the setting of Theorem 2.18 and Theorem 2.20, the horoballs and horospheres that
+appear in the statement are called (global)
+horoballs (horospheres) of Γ. Base points of these are called
+parabolic limit points of Γ.
+The above geometric characterization of Q-rank 1 lattices is all that I use in order to prove the key
+Proposition 4.12. The compact core of Γ is the complement in X of the horoballs of Γ. The group Γ acts on
+it cocompactly. The following corollary describes the orbit of Q-rank 1 lattices in X, and especially some
+finiteness properties which I will use.
+11
+
+Corollary 2.22. Let Γ ≤ G be a Q-rank 1 lattice, x ∈ X, and ξ a parabolic limit point. There is a unique
+horosphere H based at ξ such that both following conditions hold:
+1. Γ · x ∩ H is a cocompact metric lattice in H.
+2. Γ · x ∩ HB = ∅, where HB is the horoball bounded by H.
+Call H an x-horosphere of Γ at ξ, and the corresponding bounded horoball an x-horoball of Γ. Moreover,
+one has:
+1. For every C there exists a bound K = K(x, C) so that B(x, C) intersects at most K x-horospheres of
+Γ.
+2. There is D = D(x) > 0 such that H ⊂ ND(Γ · x ∩ H) for any x-horosphere H of Γ. The constant D is
+called the compactness number of (Γ, x).
+3. There is a number N = N(Γ) such that every point x ∈ X admits exactly N x-horospheres of Γ that
+intersect x. These are called the horospheres of (Γ, x).
+Proof. Let x ∈ X. Since Γ is a Q-rank 1 lattice, the stabilizer in Γ of a parabolic limit point ξ ∈ X(∞) acts
+cocompactly on each horosphere based at ξ, and in particular on Hx := H(x, ξ). Let D(x, H) be such that
+Hx ⊂ ND(Γ · x ∩ Hx). A priori D(x) depends on H, but the fact that Γ acts by isometries implies that for
+every horosphere of the form H′ := γH = H(γx, γξ) for some γ ∈ Γ, one has H′ ⊂ ND(Γ · x ∩ H′). So D(x)
+depends only on the Γ-orbit of H. Since Γ acts on the set of parabolic limit points with finitely many orbits
+(that is to say Γ\X has finitely many cusps) one may take D = D(x) to be the maximum of the respective
+bounds on each orbit. This gives the required compactness number.
+Thinking of horoballs of γ as lifts of ends of the complement of some compact subset of V = Γ\X, one
+sees that there is some horoball HB based at ξ so that Γ · x /∈ HB. Denote H = ∂HB, and y = PHB(x) ∈ H
+be the projection on the closed convex set that is the closure of the horoball HB. Finally, Let η = [x, y]
+and denote l = d(x, y). The existence of the required horosphere is equivalent to the fact that the following
+non-empty set admits a maximum:
+Hx,ξ := {t ∈ [0, l] | Γ · x ∩ H
+�
+η(t), ξ
+�
+̸= ∅}
+Indeed 0 ∈ Hx,ξ and it is a bounded set, so it admits a supremum T . Moreover, this set is discrete.
+If t were an accumulation point, then for any small ε > 0 the geodesic segment η↾(t−ε,t+ε) would intersect
+D-cocompactly infinitely many horospheres of (Γ, x). Orbit points on different horospheres are in particular
+different points, therefore the set B
+�
+η(t), D + ε
+�
+∩ Γ · x would be infinite, contradicting discreteness of Γ. I
+conclude that T is a maximum, and that H
+�
+η(T ), ξ
+�
+is the unique desired horosphere.
+The argument above generally shows that there cannot be an accumulation point in X of x-horospheres
+of Γ. In particular, for every C, the ball B(x, C) intersects only finitely many x-horospheres of Γ, say K(C),
+proving item 1 in the ‘moreover’ statement.
+For the last statement, simply note that the horoballs of Γ are the Γ-translates of finitely many horoballs.
+In the terminology of the statement, infinitely many horospheres of (Γ, x) imply that infinitely many of
+them are in the same Γ-orbit. Suppose these are {Hn}n∈N, with base points ξn that are evidently pairwise
+different. Finally let γn ∈ Γ for which γnH1 = Hn. Since Γ ∩ StabG(Hn) acts cocompactly on Hn, there
+is γ′
+n ∈ Γ ∩ StabG(Hn) that maps γnx to B(x, D). Discreteness of Γ implies that the set γ′
+nγnx is finite,
+hence infinitely many of these points are the same point. However γ′
+nγnξ1 = ξn and therefore γ′
+nγn ̸= IdX,
+contradicting the fact that Γ is torsion-free.
+Remark 2.23. For the most part, I am interested in a fixed base point x0 and the x0-horospheres and
+horoballs. By a slight abuse of terminology I omit x0 and call these objects ‘horospheres of Γ’ and ‘horoballs
+of Γ’, respectively, denoting the associated cocompactness number DΓ.
+Corollary 2.22 allows to upgrade a metric lattice of H to a metric lattice coming from StabG(H) only.
+12
+
+Lemma 2.24. Assume that a torsion-free discrete group ∆ ≤ G = Isom(X) admits the following:
+1. There is a bound N such that at each point x ∈ ∆ · x0 there are at most N horoballs that are tangent
+to x and do not intersect ∆ · x0.
+2. There is a horosphere H ⊂ X such that
+(a) The set ∆ · x0 ∩ H is a cocompact metric lattice in H.
+(b) The horoball HB bounded by H is ∆-free, i.e. ∆ · x0 ∩ HB ⊂ H.
+Then
+�
+∆ ∩ StabG(H)
+�
+· x0 is also a cocompact metric lattice in H.
+Proof. This is the Pigeonhole Principle. Fix x ∈ ∆ · x0 ∩ H, and let {HBi}i∈{1,...,N} be the finite set of
+horoballs tangent to x that do not intersect ∆ · x0. Assume w.l.o.g that H is the bounding horosphere of
+HB1. Fix a ∆-orbit point δ0x0 ∈ ∆ · x0 ∩ H. Up to translating by some element of ∆, I may assume x = x0.
+For any other ∆-orbit point δx0 ∈ ∆ · x0 ∩ H let i(δ) ∈ {1, . . . , N} be the index of the horoball δ−1HB1.
+Notice that from hypothesis 1 it indeed follows that δ−1HB1 ∈ {HBi}i∈{1,...,N}, because the action of ∆ is
+by isometries. Define δi to be the element in ∆ for which:
+1. i(δi) = i, i.e. δ−1
+i
+(HB1) = HBi
+2. d(δix0, x0) is minimal among all such δ ∈ ∆.
+For some i ∈ {1, . . . , N} there is a δ ∈ ∆ with i = i(δ), while for others there might not be. I will only
+care about those i for which there is such δ. Assume w.l.o.g that these are i ∈ {1, . . . M} for M ≤ N, and let
+L := max1≤i≤M{d(x0, δix0)} < ∞. For such an index i0, the ∆-orbit points that share the same i(δ) = i0
+are in the same ∆ ∩ StabG(H) orbit, namely
+{δx0 | i(δ) = i0} ⊂
+�
+∆ ∩ StabG(H)
+�
+· δi0x0
+Indeed, if δ−1HB1 = HBi0 then by definition δδ−1
+i0 HB1 = δHBi0 = HB1, hence δδ−1
+i0
+∈ StabG(H) is
+an element mapping δi0x0 to δx0. Let now δx0 ∈ H. Its distance from the orbit
+�
+∆ ∩ StabG(H)
+�
+· x0 is
+�
+∆ ∩ StabG(H)
+�
+-invariant, therefore
+d
+�
+δx0,
+�
+∆ ∩ StabG(H)
+�
+x0
+�
+= d
+�
+δi(δ)x0,
+�
+∆ ∩ StabG(H)
+�
+x0
+�
+≤ d(δi(δ), x0)
+The right-hand side is uniformly bounded by L, proving that
+(∆ · x0 ∩ H) ⊂ NL
+��
+∆ ∩ StabG(H)
+�
+· x0
+�
+The fact that ∆ · x0 ∩ H is a cocompact metric lattice in H renders (∆ ∩ StabG(H)
+�
+· x0 a cocompact
+metric lattice in h as well, as claimed.
+Real and Rational Tits Buildings.
+The location of the parabolic points in X(∞) also plays an important
+role in the geometry of X. In case Γ is an arithmetic lattice, the natural framework to consider these points
+is the so called rational Tits building.
+This is a building structure on the subset of parabolic points at
+X(∞), sometimes referred to as ‘rational points’ in this case. They are exactly those points in X(∞) whose
+stabilizers are Q-defined (algebraic) parabolic groups of G (see Section 4.4 for more details). I present this
+object, denoted WQ(Γ), together with the more familiar real Tits building structure on X(∞) with the Tits
+metric. The main goal is to present the results of Hattori [25], that give a good description of the rational
+Tits building in terms of conical and horospherical limit points. In case G is of R-rank 1, by WQ(Γ) I mean
+the (countable) set of parabolic limit points of Γ (so that X(∞) \ WQ(Γ) is comprised of conical limit points
+only, see Theorem 2.29).
+13
+
+Definition 2.25. A geodesic η is said to be regular if it is contained in a unique maximal flat F ⊂ X. The
+point η(∞) ∈ X(∞) is called a regular point of X(∞). A point ξ ∈ X(∞) is singular if it is not regular.
+Regularity does not depend on the choice of representative geodesic ray η for ξ.
+A Weyl chamber of X(∞), or an open spherical chamber, is any connected component in the Tits topology
+of X(∞) \ S, where S ⊂ X(∞) is the subset of singular points at X(∞).
+Proposition 2.26 (Propositions 2.2 and 3.2 in [29] and Section 8 in [3]). The Weyl chambers induce a
+simplicial complex structure on X(∞) that is a spherical Tits building. The apartments of the building are
+exactly the sets of the form F(∞) ⊂ X(∞) for all flats F ⊂ X, and the chambers are exactly the Weyl
+chambers at X(∞). Moreover, the Tits metric completely determines the building structure, and vice versa,
+and
+�
+X(∞), dT
+�
+is a metric realization of the Tits building at X(∞).
+None of the rich theory of buildings is used directly in this paper. Given a non-uniform lattice of Γ ≤ G
+the rational Tits building WQ(Γ) is a building structure on the subset of parabolic points. It is not in general
+a sub-building of the real spherical building. Flats of X correspond to real maximal split tori in G. Since
+G is an algebraic group defined over Q, one can consider the maximal Q-split tori. The rational flats of X
+are then the G(Q)-orbits of maximal Q-split tori of G, and the rational boundary are all points ξ ∈ X(∞)
+such that ξ ∈ FQ(∞) for some rational flat FQ(∞). One defines regular rational directions and rational Weyl
+chambers in an analogous way to the real case, this time taking only rational flats into account. For further
+details details see [29], and Section 2 in [25].
+Theorem 2.27 (Theorem A in [25]). Let X = G/K be a symmetric space of non-compact type and of higher
+rank, and let Γ ≤ Isom(X) be an irreducible non-uniform lattice. Then WQ(Γ) does not include horospherical
+limit points. The π
+2 -neighbourhood
+N π
+2
+�
+WQ(Γ)
+�
+:= {ξ ∈ X(∞) | dT
+�
+ξ, WQ(Γ)
+�
+< π
+2 }
+does not include conical limit points.
+In Q-rank 1 , the converse statement also holds:
+Theorem 2.28 (Theorem B in [25]). Let X = G/K be a symmetric space of non-compact type of higher
+rank and Γ ≤ Isom(X) be an irreducible non-uniform lattice. Let
+V = {ξ ∈ X(∞) | dT
+�
+ξ, WQ(Γ)
+�
+≥ π
+2 }
+Suppose that Γ is of Q-rank 1 . Then V consists of conical limit points only.
+In groups of R-rank 1 one has the following well known fact:
+Theorem 2.29 (Proposition 5.4.2 and Theorem 6.1 in [10], see also Theorem 12.29 in [17]). Let X be a
+symmetric space of noncompact type and of R-rank 1, Γ ≤ Isom(X) a lattice. Then every ξ ∈ X(∞) is either
+conical or non-horospherical.
+Corollary 2.30. When Γ is of Q-rank 1 , the following holds:
+1. WQ(Γ) = {ξ ∈ X(∞) | N π
+2 (ξ) does not contain conical limit points}
+2. Any two points ξ, ξ′ ∈ WQ(Γ) are at Tits distance = π.
+Proof. In view of Theorem 2.29, for R-rank 1 both statements hold trivially. In higher rank, both follow
+from the following observation: for any point ξ′ ∈ WQ(Γ) and any point ζ ∈ X(∞) with d(ζ, ξ′) = π
+2 , ζ is
+conical. To see this notice that ζ lies on the boundary of a horosphere based at ξ′: take a flat F ⊂ X with
+ξ′, ζ ∈ F(∞). Any geodesic with limit ζ is contained in (a Euclidean) horosphere based at ξ′. The fact that
+Γ is cocompact on the horospheres based at WQ(Γ) implies that ζ is conical.
+14
+
+The second item of the corollary follows: let ξ, ξ′ ∈ WQ(Γ) and c : [0, α] a Tits geodesic joining them.
+There is a flat F ⊂ X containing both ξ, ξ′ as well as c ⊂ F(∞). Every point ζ that is at Tits distance
+π
+2 from either ξ or ξ′ is conical, and no point inside the π
+2 neighbourhood of either ξ or ξ′ is conical. In
+F(∞) the Tits metric is the same as the Tits metric on the Euclidean space of an equal rank. Therefore
+one may prolong the geodesic c so that c(0) = ξ, c(α) = ξ′ and c(π) = ξ′′. If dT (ξ, ξ′) < π, then there is
+a point along this prolonged geodesic that is at Tits distance exactly π
+2 from ξ (so it is conical by the first
+paragraph), but at Tits distance strictly less than π
+2 from ξ′ (so it cannot be conical by Theorem 2.27).
+Therefore dT (ξ, ξ′) = π.
+For the first item, one containment is just Hattori’s Theorem 2.27. For the other containment, pretty
+much the same argument from above works. Assume for some ξ ∈ X(∞) that N π
+2 (ξ) consists of non-conical
+limit points. In particular ξ itself is not conical, and by Theorem 2.28 it holds that d(ξ, WQ(Γ)) < π
+2 . Let
+ξ′ ∈ WQ(Γ) be a point realizing this distance. As above, this gives rise to a flat F containing ξ, ξ′ and
+another point ζ that is at Tits distance π
+2 from ξ′ but at Tits distance strictly less than π
+2 from ξ. The first
+forces ζ to be conical, and the latter forces it to be non-conical, a contradiction.
+Hattori’s characterization relies on a simple lemma which will also be of use in the sequel. It relates the
+(linear) penetration rate of a geodesic into a horoball to the Tits distance.
+Lemma 2.31 (Lemma 3.4 in [25]). Let X be a symmetric space of higher rank and of noncompact type.
+Let η1, η2 : [0, ∞) → X be two geodesic rays, α := dT
+�
+η1(∞), η2(∞)
+�
+and b2 the Busemann function
+corresponding to η2. Then there exists a positive constant C1, depending only on η1 and η2, such that:
+1. If α > π
+2 , then for all t ≥ 0
+b2
+�
+η1(t)
+�
+≥ −t · cos α − C1
+2. If α = π
+2 , then b2
+�
+η1(t)
+�
+is monotone non-increasing in t and −C1 ≤ b2
+�
+η1(t)
+�
+.
+3. If α < π
+2 , then for all t ≥ 0
+b2
+�
+η1(t)
+�
+≤ −t · cos α − C1
+Remark 2.32. If X is a symmetric spaces of R-rank 1, maximal flats are geodesics. Therefore every two
+points ξ, ζ ∈ X(∞) admit dT (ξ, ζ) = π, and it is clear from the strict negative curvature that Lemma 2.31
+is true also in this case.
+3
+Uniform Lattices
+In this section I prove:
+Theorem 3.1. Let G be a semisimple Lie group without compact factors and with finite centre. Let Γ ≤ G
+be a lattice, Λ ≤ G a discrete subgroup such that Γ ⊂ Nu(Λ) for some sublinear function u. If Γ is uniform,
+then Λ is a uniform lattice.
+The focus of this paper is on sublinear distortion, however for uniform lattices (and also for lattices that
+have property (T), see Section 5), a slightly stronger result holds. I call this ε-linear rigidity.
+Definition 3.2. Let f, g : R≥0 → R>0 be two monotonically increasing functions. Call f asymptotically
+smaller than g if lim sup f
+g ≤ 1. Denote this relation by f ⪯∞ g.
+Theorem 3.3. In the setting of Theorem 3.1, the conclusion holds also under the relaxed assumption that
+u(r) ⪯∞ εr for any 0 < ε < 1.
+Clearly Theorem 3.3 implies Theorem 3.1. From now and until the end of this section, the standing
+assumptions are those of Theorem 3.3.
+15
+
+Lattice Criterion.
+A discrete group is a uniform lattice if and only if it admits a relatively compact
+fundamental domain. The criterion I use is the immediate consequence that if Γ is uniform and u is bounded
+(i.e. Γ ⊂ ND(Λ) for some D > 0), then Λ is a uniform lattice.
+Outline of Proof and Use of ε-Linearity.
+The goal is to show that the ε-linearity of u forces Γ ⊂ ND(Λ)
+for some D > 0, i.e. that Γ actually lies inside a bounded neighbourhood of Λ. The proof is by way of
+contradiction. If there is no such D > 0 then there are arbitrarily large balls that do not intersect Λ. The
+proof goes by finding such large Λ-free balls that are all tangent to some fixed arbitrary point x ∈ X (see
+Figure 1). The ε-linearity then gives rise to concentric Γ-free balls that are arbitrarily large, contradicting
+the fact that Γ is a uniform lattice.
+Remark 3.4. The main difference from the non-uniform case is that for a non-uniform lattice Γ, the space
+X does admit arbitrarily large Γ-free balls. This situation requires different lattice criteria and much extra
+work. Still the proof for the uniform case, though essentially no more than a few lines, lies the foundations
+for and presents the logic of the much more involved case of Q-rank 1 lattices.
+3.1
+Notations and Terminology
+Definition 3.5. Let X be a metric space, Y, Z ⊂ X two closed subsets of X. The closest point projection
+of Y to Z is the set theoretic map pZ : Y → Z defined by pZ(y) := zy, where zy ∈ Z is any point realizing
+the distance d(y, Z) = d(y, z). If X is a proper metric space and Z discrete, then there are at most finitely
+many such points. In any case of multiple points, pZ chooses one arbitrarily.
+The particular case of interest from now on is where the metric space is the pointed symmetric space
+(X, x0), the two subsets are the orbits Γ · x0 and Λ · x0, and the projection is pΛ·x0 : Γ · x0 → Λ · x0. To ease
+notation I often denote this projection by pΛ (there is no risk of ambiguity since the subgroups Λ and Γ are
+always considered in the context of their respective orbits in X and not in G).
+The following definitions will be used repeatedly in both this section and in Section 4. It mainly fixes
+terminology and notation of the geometric situation illustrated in Figure 1.
+Definition 3.6. Let H ≤ G = Isom(X)◦. A set U ⊂ X is called H-free if H · x0 ∩ Int(U) = ∅, where Int(U)
+is the topological interior of U. That is, U is called H-free if its interior does not intersect the H-orbit H ·x0.
+Definition 3.7. Denote PΛ(γx0) = PΛ(γ) = λγx0.
+1. dγ := d(γx0, λγx0).
+2. Bγ := B(γx0, dγ). It is a Λ-free ball centred at γx0 and tangent to λγx0.
+3. x′
+γ := λ−1
+γ γx0. Notice |x′
+γ| = dγ.
+4. B′
+γ := λ−1
+γ Bγ = B(x′
+γ, dγ). It is Λ-free as a Λ-translate of the Λ-free ball Bγ, and is tangent to x0.
+5. For s ∈ R>0 and a ball B = B(x, r), denote sB := B(x, sr), the rescaled ball with same centre and
+radius sr.
+6. For a sequence γn, denote by λn, dn, Bn, B′
+n, x′
+n the respective λγn, dγn, etc.
+3.2
+Proof of Theorem 3.3
+Lemma 3.8. Let x ∈ X. There exists S = S(x, u) ∈ (0, 1) such that for every s ∈ (0, S) there is R = R(s, S)
+such that if r > R and B = B(y, r) is a Λ-free ball tangent to x, then sB is Γ-free.
+In particular, the existence of arbitrarily large Λ-free balls that are all tangent to a fixed point x ∈ X
+implies the existence of arbitrarily large Γ-free balls.
+16
+
+x0
+x′
+γ
+= dγ
+= s · dγ
+Γ − free
+Λ − free
+γx0
+= dγ
+Λ − free
+λγx0
+Lλ−1
+γ
+Figure 1: Basic Setting and Lemma 3.8. A Λ-free ball about γx0 of radius dγ, translated by λ−1
+γ
+to a ball
+tangent to x0. The linear ratio between |x′
+γ| = dγ and the Λ-free radius forces the red ball to be Γ-free.
+There is a slightly stronger version of Lemma 3.8 if u is sublinear:
+Lemma 3.9. Let G, Γ, Λ and u be as in Theorem 3.1 (in particular, u is a sublinear function and Γ ⊂ Nu(Γ)).
+For every x ∈ X and every s ∈ (0, 1) there exists R = R(x, s) > 0 such that for every r > R, if B = B(y, r)
+is a Λ-free ball tangent to x then sB is Γ-free.
+I omit the proof of Lemma 3.9, which is a slightly simpler version of the proof of Lemma 3.8.
+Proof of Lemma 3.8. The proof is more easily read if one assumes x = x0 and u(r) = εr so I begin with this
+case. Assume B = B(y, R) is Λ-free for some y ∈ X. The assumption x = x0 means that |y| = d(y, x0) = r.
+Assume that for a given s ∈ (0, 1), the ball sB intersects Γ · x0. This gives rise to an element γ ∈ Γ such
+that:
+1. |γ| = d(γx0, x0) ≤ d(y, x0) + d(γx0, y) = (1 + s)r (triangle inequality). In particular, d(γx0, Λ · x0) ≤
+ε(1 + s)r
+2. B
+�
+γx0, (1 − s)r
+�
+⊂ B(y, r), so it is Λ-free.
+I conclude that for s for which sB ∩ Γ · x0 ̸= ∅, one has the inequality (1 − s)r ≤ ε(1 + s)r, i.e. 1−s
+1+s ≤ ε.
+The number ε is fixed and smaller than 1, while 1−s
+1+s limit to 1 monotonically from below as s > 0 tend to
+0. I conclude that there is a segment (0, S) ⊂ (0, 1) such that for all s ∈ (0, S), sB is Γ-free.
+Assume now that x ̸= x0 and u(r) ⪯∞ εr. As above, if γx0 ∈ B(y, sr) then it is the centre of a Λ-free
+ball of radius (1 − s)r, and so (1 − s)r ≤ u(|γ|). I wish to use the ε-linear bound on u as I did before,
+only this time u is only asymptotically smaller than εr. To circumvent this I just need to show that |γ|
+is large enough. Indeed since B
+�
+γx0, (1 − s)r
+�
+is Λ-free it does not contain x0 ∈ Λ · x0 and in particular
+(1 − s)r ≤ d(x0, γx0) = |γ|. For some R1(s) = R1(s, u) one therefore has for all r > R1(s)
+(1 − s)r ≤ u(|γ|) ≤ ε|γ|
+On the other hand |y| ≤ d(x, y) + d(x, x0) = r + |x|, and consequently |γ| ≤ (1 + s)r + |x|. For r > R1(s)
+one has
+17
+
+(1 − s)r ≤ u(|γ|) ≤ ε|γ| ≤ ε
+�
+(1 + s)r + |x|
+�
+This means that s for which Γ · x0 ∩ B(y, sr) ̸= ∅ must admit, for all r > R1(s),
+1 − s
+1 + s + |x|
+r
+=
+(1 − s)r
+(1 + s)r + |x| ≤ ε < 1
+(1)
+The rest of the proof is just Calculus 1, and concerns with finding S = S(x, ε, u) ∈ (0, 1) so that for any
+s ∈ (0, S) there is R(s) such that all r > R(s) satisfy
+ε <
+1 − S
+1 + S + |x|
+r
+≤
+1 − s
+1 + s + |x|
+r
+(2)
+The lemma readily follows from inequalities 1,2.
+Explicitly, fix ε′ > ε. As before, monotonic approach of
+1−s
+1+s to 1 allows to fix S ∈ (0, 1) for which
+ε < ε′ < 1−s
+1+s for all s ∈ (0, S). Next note that for any fixed s ∈ (0, S), limr→∞
+1−s
+1+s+ |x|
+r = 1−s
+1+s, and that the
+approach in monotonically increasing with r. Since ε < ε′, this limit implies that for some R2 > R1(S), all
+r > R2 admit ε <
+1−S
+1+S+ |x|
+r . Finally notice that for any fixed r the function
+1−s
+1+s+ |x|
+r
+is again monotonically
+increasing as s tends to 0 from above. Therefore inequality 2 holds for every s ∈ (0, S) and all r > R2(S)
+(capital S is intentional and important).
+To conclude the proof, notice that if moreover r > R1(s) (again lowercase s is intentional and important)
+then inequalities 1,2 both hold. This means that for any s ∈ (0, S) there is R(s) := max{R1(s), R2(S)} such
+that r > R(s) ⇒ B(y, sr) is Γ-free. The constants R1(s), R2(S) have the desired dependencies, hence so
+does R(s), proving the lemma.
+Corollary 3.10 (Theorem 3.3). There is a uniform bound on {dγ}γ∈Γ, i.e., Γ ⊂ ND(Λ) for some D > 0.
+In particular, Λ is a uniform lattice.
+4
+Q-rank 1 Lattices
+In this section I prove:
+Theorem 4.1. Let G be a semisimple Lie group without compact factors and with finite centre, Γ ≤ G an
+irreducible non-uniform Q-rank 1 lattice, Λ ≤ G a discrete irreducible subgroup. If Γ ⊂ Nu(Λ) for some
+sublinear function u, then Λ is a lattice. Moreover, if Γ ̸⊂ ND(Λ) for any D > 0, then Λ is also of Q-rank 1.
+If Γ ⊂ ND(Λ) for some D > 0, then Λ could be a uniform lattice. An obvious obstacle for that is if
+Λ ⊂ Nu′(Γ) for some sublinear function u′. This condition turns out to be sufficient for commensurability.
+Proposition 4.2. Let G be a semisimple Lie group without compact factors and with finite centre, Γ ≤ G an
+irreducible non-uniform Q-rank 1 lattice, Λ ≤ G a discrete subgroup such that Γ ⊂ ND(Λ) for some D > 0,
+and Λ ⊂ Nu(Γ) for some sublinear function u. Then Λ ⊂ ND′(Γ) for some D′.
+As a result of Eskin’s and Schwartz’s arguments (see Section 4.3), I conclude:
+Corollary 4.3. In the setting of Proposition 4.2 and unless G is locally isomorphic to SL2(R), Λ is com-
+mensurable to Γ.
+Theorem 1.8 is a result of Theorem 4.1 and Corollary 4.3
+18
+
+4.1
+Strategy
+Lattice Criteria.
+I use three different lattice criteria, depending on the R-rank of G and on whether or
+not Γ ⊂ ND(Λ) for some D > 0. My proof for Q-rank 1 lattices is motivated by a criterion of Benoist
+and Miquel, resolving a conjecture of Margulis. It can be viewed as an algebraic converse to the geometric
+structure of the compact core described in Theorem 2.20.
+Theorem 4.4 (Theorem 2.16 in [5]). Let G be a semisimple real algebraic Lie group of real rank at least 2
+and U be a non-trivial horospherical subgroup of G. Let ∆ be a discrete Zariski dense subgroup of G that
+contains an indecomposable lattice ∆U of U. Then ∆ is a non-uniform irreducible arithmetic lattice of G.
+Remark 4.5. See Definition 4.47 for the precise meaning of an indecomposable horospherical lattice.
+For R-rank 1 groups, one has the following theorem by Kapovich and Liu, stating that a group is
+geometrically finite so long as ‘most’ of its limit points are conical. Recall L(∆) is the limit set of ∆ ≤
+Isom(X), and Lcon(∆) is the set of its conical limit points.
+Theorem 4.6 (Theorem 1.5 in [30]). Let X be a R-rank 1 symmetric space. A discrete subgroup ∆ ≤
+Isom(X) is geometrically infinite if and only if the set L(∆) \ Lcon(∆) of non-conical limit points has the
+cardinality of the continuum.
+As a direct corollary I obtain the following criterion:
+Corollary 4.7. Let X be a R-rank 1 symmetric space, Γ ≤ G = Isom(X) a non-uniform lattice and Λ ≤ G
+a discrete subgroup. If L(Λ) = X(∞) and Lcon(Γ) ⊂ Lcon(Λ), then Λ is a lattice.
+Proof. Since Γ is a lattice, L(Γ) = X(∞) and it is geometrically finite. Theorem 4.6 implies the cardinality
+of X(∞) \ Lcon(Γ) is strictly smaller than the continuum. The assumption Lcon(Γ) ⊂ Lcon(Λ) implies the
+same holds for Λ, and in particular that Λ is geometrically finite. The assumption that L(Λ) = X(∞) implies
+that Λ is geometrically finite if and only if it is a lattice.
+Corollary 4.8. Let X be a R-rank 1 symmetric space, Γ ≤ G = Isom(X) a non-uniform lattice and Λ ≤ G
+a discrete subgroup. If Γ ⊂ ND(Λ) for some D > 0, then Λ is a lattice.
+Proof. By definition of the limit set and of conical limit points, it is clear that every Γ-limit point is a Λ-limit
+point, and every Γ-conical limit point is also Λ-conical limit point. I conclude from Corollary 4.7 that Λ is
+a lattice.
+Also in higher rank the inclusion Γ ⊂ ND(Λ) implies that Λ is a lattice. This result is due to Eskin.
+Theorem 4.9 (Eskin, see Remark 4.11 below). Let G be a semisimple Lie group without compact factors and
+of higher rank, Γ ≤ G an irreducible non-uniform lattice, Λ ≤ G a discrete subgroup such that Γ ⊂ ND(Λ)
+for some D > 0. Then Λ is a lattice.
+Theorem 4.9 was used in the proof of quasi-isometric rigidity for higher rank non-uniform lattices in [16],
+[22]. In the (earlier) R-rank 1 case, Schwartz [52] used an analogous statement, which requires one extra
+assumption.
+Theorem 4.10 (Schwartz, see Section 10.4 in [52] and Remark 4.11 below). Let G be a real simple Lie group
+of R-rank 1 and with finite centre, Γ ≤ G an irreducible non-uniform lattice, Λ ≤ G a discrete subgroup such
+that both Γ ⊂ ND(Λ) and Λ ⊂ ND(Γ) for some D > 0. Then Λ is a lattice and commensurable to Γ.
+Remark 4.11. Theorem 1.6 should be viewed as a generalization of the bounded case depicted in Theo-
+rems 4.9 and 4.10, which were known to experts in the field in the late 1990’s. Complete proofs for these
+statements were never given in print, and I take the opportunity to include them here. See Section 4.3, where
+I also prove Proposition 4.2. I thank Rich Schwartz and Alex Eskin for supplying me with their arguments
+and allowing me to include them in this paper. I also thank my thesis examiner Emmanuel Breuillard for
+encouraging me to find and make these proofs public.
+19
+
+Outline of Proof and Use of Sublinearity.
+Lattices of Q-rank 1 admit a concrete geometric structure
+(see Section 2.2). This structure is manifested in the geometry of an orbit of such a lattice in the corresponding
+symmetric space X = G/K. One important geometric property is the existence of a set of horoballs which
+the orbit of the lattice intersects only in the bounding horospheres, and in each such horosphere the orbit
+forms a (metric) cocompact lattice.
+Corollary 4.8 and Theorem 4.9 reduce the proof to the case where Γ ̸⊂ ND(Λ) for any D > 0. In that
+case, the essence lies in proving the existence of horospheres in X which a Λ-orbit intersects in a cocompact
+lattice.
+This is proved purely geometrically, using the geometric structure of Q-rank 1 lattices and the
+sublinear constraint. Together with some control on the location of these horospheres, I prove two major
+statements:
+1. Λ · x0 intersects a horosphere H ⊂ X in a cocompact lattice (Proposition 4.12).
+2. Every Γ-conical limit point is also a Λ-conical limit point (Corollary 4.27).
+The R-rank 1 case of Theorem 4.1 follows directly from Corollary 4.7 using the second item above. The
+higher rank case requires a bit more. namely it requires to deduce from the above items that Λ meets
+the hypotheses of the Benoist-Miquel Theorem 4.4. To that end I use a well known geometric criterion
+(Lemma 4.50) in order show that Λ is Zariski dense, and a lemma of Mostow (Lemma 4.36) to show that Λ
+intersects a horospherical subgroup in a cocompact lattice.
+Outline for Section 4.
+Section 4.2 is the core of the original mathematics of this paper. It is devoted
+to proving that Λ · x0 intersects some horospheres in a cocompact lattice. The proof is quite delicate and
+somewhat involved, and I include a few figures and a detailed informal overview of the proof. The figures
+are detailed and may take a few moments to comprehend, but I believe they are worth the effort.
+Section 4.3 deals with the case where Γ ⊂ ND(Λ), and elaborates on Schwartz’s and Eskin’s proofs
+of Theorem 4.9 and Theorem 4.10. Section 4.4 is devoted to the translation of the geometric results of
+Section 4.2 to the algebraic language used in Theorem 4.4.
+Though the work is indeed mainly one of
+translation, some of it is non-trivial. Finally, in Section 4.5 I put everything together for a complete proof
+of Theorem 4.1.
+I highly recommend the reader to have a look at the uniform case in Section 3 before reading this one.
+4.2
+A Λ-Cocompact Horosphere
+Recall that dγ := d(γx0, λγx0). In this section I prove:
+Proposition 4.12. If {dγ}γ∈Γ is unbounded, then there exists a horosphere H based at WQ(Γ) such that
+�
+Λ∩StabG(H)
+�
+·x0 intersects H in a cocompact metric lattice. Moreover, the bounded horoball HB is Λ-free.
+Throughout Section 4.2 the standing assumptions are that {dγ}γ ∈ Γ is unbounded, and Γ is an irreducible
+Q-rank 1 lattice.
+The Argument.
+The proof is by chasing down the geometric implications of unbounded dγ.
+These
+implications are delicate, but similar in spirit to the straight-forward proof for uniform lattices. The proof
+consists of the following steps:
+1. Unbounded dγ results in Λ-free horoballs HBΛ tangent to Λ-orbit points. Each such horoball is based
+at WQ(Γ), giving rise to corresponding horoballs of Γ, denoted HBΓ.
+2. If dγ is large, then γx0 must lie deep inside a unique Λ-free horoball tangent to λγx0. I use:
+(a) A bound on the distance d(HΛ, HΓ).
+(b) A bound on the angle ∠λγx0([λγx0, γx0], [λγx0, ξ)), where ξ ∈ X(∞) is the base point of a suitable
+Λ-free horoball tangent to λγx0.
+20
+
+3. There exist horospheres of Γ, say HΓ, such that if γx0 ∈ HΓ then large dγ implies large Λ-free areas
+along the bounding horosphere of some HBΛ.
+4. If HBΛ is boundedly close to some Λ-orbit point, then HΛ is almost Λ-cocompact, that is HΛ ⊂
+ND(Λ·x0) for some universal D = D(Λ). Together with the previous step, this yields a uniform bound
+on dγ along certain horospheres of Γ.
+5. Finally I elevate the almost cocompactness to actual cocompactness and show HΛ ⊂ ND(Λ · x0 ∩ HΛ)
+for some D > 0. This immediately elevates to HΛ ⊂ (Λ ∩ StabG(HΛ)) · x0, proving the proposition.
+The Properties of Γ.
+The geometric properties of Γ that are used in the proof are:
+1. In higher rank, the characterization of WQ(Γ) using conical / non-horospherical limit points (Corol-
+lary 2.30). In R-rank 1, the dichotomy of limit points being either non-horospherical or conical (The-
+orem 2.29).
+2. Γ-cocompactness along the horospheres of Γ.
+3. For every point x ∈ X and C > 0 there is a bound K(C) on the number of horospheres of Γ that
+intersect B(x, C) (Corollary 2.22).
+4.2.1
+Λ-Free Horoballs
+I retain the notations and objects defined in Section 3.1.
+Lemma 4.13. There exists a Λ-free horoball tangent to x0.
+Proof. Since {dγ}γ∈Γ is unbounded, there are γn ∈ Γ with dn = dγn = d(γn, λn) → ∞ monotonically, where
+λn ∈ Λ is a Λ-orbit point closest to γn. Denote x′
+n = λ−1
+n γnx0, ηn := [x0, x′
+n], and vn ∈ Sx0X the initial
+velocity vectors vn :=
+˙ηn(0). The tangent space Sx0X is compact, so up to a subsequence, vn converge
+monotonically in angle to a direction v ∈ Sx0X. Let η be the unit speed geodesic ray emanating from x0
+with initial velocity ˙η(0) = v. Denote ξ := η(∞) the limit point of η in X(∞).
+I claim that the horoball HB := ∪t>0B
+�
+η(t), t
+�
+, based at ξ and tangent to x0, is Λ-free. Let t > 0 and
+consider η(t). For every ε > 0, there is some angle α = α(t, ε) such that any geodesic η′ with ∠x0(η, η′) < α
+admits d
+�
+η(t), η′(t)
+�
+< ε/2. The convergence vn → v implies d
+�
+η(t), ηn(t)
+�
+< ε/2 for all but finitely many
+n ∈ N. In particular, B
+�
+η(t), t
+�
+⊂ Nε
+�
+B
+�
+ηn(t), t
+��
+for all such n ∈ N.
+For a fixed t ≤ dn, it is clear from the definitions that B
+�
+ηn(t), t
+�
+⊂ B′
+n = B(x′
+n, dn). One has dn → ∞,
+and so for a fixed t > 0 it holds that t < dn for all but finitely many n ∈ N. I conclude that for any fixed
+t > 0 there is n large enough such that
+B
+�
+η(t), t
+�
+⊂ Nε
+�
+B
+�
+ηn(t), t
+��
+⊂ NεB′
+n
+I conclude that for every ε > 0, HB ⊂ �
+n Nε(B′
+n) = Nε
+� �
+n B′
+n
+�
+. This implies that any point in the
+interior of HB is contained in the interior of one of the Λ-free balls B′
+n, proving HB is Λ-free.
+Lemma 4.14. Suppose HB is a Λ-free horoball, based at some point ξ ∈ X(∞). Then ξ ∈ WQ(Γ).
+Proof. For any geodesic η with limit ξ, the size d(x0, γx0) of the Γ-orbit points γx0 that lie boundedly close
+to η grows linearly in the distance to any fixed horosphere based at ξ, and in particular to H = ∂HB. The
+sublinear constraint d(γx0, λγx0) ≤ u(|γ|) together with the fact that HB is Λ-free imply that the size of
+such γ is bounded. In R-rank 1 every limit point is either conical or in WQ(Γ), proving the lemma in this
+case. For higher rank, the above argument actually shows more: it shows that a point ξ′ ∈ N π
+2 (ξ) is not
+conical, because every geodesic with limit ξ′ ∈ N π
+2 (ξ) entres HB at a linear rate (Lemma 2.31). Hattori’s
+characterization of WQ(Γ) (Corollary 2.30) implies ξ ∈ WQ(Γ).
+21
+
+Definition 4.15. Given a Λ-free horoball HBΛ, Lemma 4.14 gives rise to a horoball of Γ based at the same
+point at X(∞). Call this the horoball corresponding to HBΛ, and denote it by HBΓ. The corresponding
+horosphere is denoted HΓ.
+Remark 4.16. In the course of my work I had had a few conversations with Omri Solan regarding the
+penetration of geodesics into Λ-free horoballs. Assuming Λ ⊂ Nu(Γ) implies that Λ preserves WQ(Γ) (see
+Lemma 4.32). This is the case in the motivating setting where Λ is an abstract finitely generated group that
+is SBE to Γ, see Claim 6.4 in Chapter 6. In the case of SL2(R) Omri suggested to use the action of Λ on
+the Bruhat-Tits tree of SL2(Qp) (for all primes p) and the classification of these elements into elliptic and
+hyperbolic elements (separately for each p) in order to deduce that Λ actually lies in SL2(Z). We did not
+pursue that path nor its possible generalization to the SLn case and general Bruhat-Tits buildings.
+4.2.2
+A Γ-orbit point Lying Deep Inside a Λ-Free Horoball
+I established the existence of Λ-free horoballs. It may seem odd that the first step in proving Λ · x0 is
+‘almost everywhere’ is proving the existence of Λ-free regions. But this fits perfectly well with the algebraic
+statement that non-uniform lattices must admit unipotent elements (see Proposition 5.3.1 in [39]). The goal
+of this section is to obtain some control on the location of the Λ-free horoballs, in order to conclude that
+some γx0 lies deep inside HBΛ. This results in yet more Λ-free regions, found on the bounding horosphere.
+I need one property of sublinear functions.
+I thank Panos Papazoglou for noticing a mistake in the
+original formulation.
+Lemma 4.17. Let u be a sublinear function, f, g : R≥0 −→ R>0 two positive monotone functions with
+limx→∞ f(x) + g(x) = ∞. If for all large enough x it holds that f(x) ≤ u
+�
+f(x) + g(x)
+�
+, then for every 1 < s
+and all large enough x it holds that f(x) ≤ u
+�
+s · g(x)
+�
+. In particular f(x) ≤ u′�
+g(x)
+�
+for some sublinear
+function u′.
+Proof. Assume as one may that u is non-decreasing. By definition of sublinearity limx→∞
+u
+�
+f(x)+g(x)
+�
+f(x)+g(x)
+= 0,
+so by hypothesis limx→∞
+f(x)
+f(x)+g(x) = 0. This means that for every ε > 0 one has f(x) ≤ ε · g(x) for all large
+enough x, resulting in
+f(x) ≤ u
+�
+f(x) + g(x)
+�
+≤ u
+�
+(1 + ε) · g(x)
+�
+Notice that for a fixed s > 0, the function u′(x) = u(sx) is sublinear, as
+lim
+x→∞
+u(sx)
+x
+= lim
+x→∞ s · u(sx)
+sx
+= 0
+Lemma 4.18. Let C > 0. There is L = L(C) such that if HBΛ is any Λ-free horoball tangent to a point
+x ∈ B(x0, C) then d(HΛ, HΓ) ≤ L. Moreover, there is a sublinear function u′ such that:
+L(C) ≤
+�
+u′(C)
+if HBΓ ⊂ HBΛ
+C
+if HBΛ ⊂ HBΓ
+Proof. If HBΛ ⊂ HBΓ, then clearly d(HΛ, HΓ) ≤ C, simply because HBΓ is Γ-free and in particular cannot
+contain x0. Therefore HΓ must separate HΛ from x0 and in particular d(HΛ, HΓ) ≤ C.
+Assume that HBΓ ⊂ HBΛ, and denote l = d(HΛ, HΓ). The horoball HBΓ is a horoball of Γ, hence Γ · x0
+is DΓ-cocompact along HΓ and there is an element γ ∈ Γ with |γ| ≤ C + l + DΓ and γx0 ∈ HΓ. Since HBΛ
+is Λ-free one has
+l ≤ d(γx0, λγx0) ≤ u(|γ|) ≤ u(C + l + DΓ)
+C, DΓ are fixed, so this inequality can only occur for boundedly small l, say l < L′(C) (DΓ is a universal
+constant and may be ignored). Consult figure 2 for a geometric visualization of this situation.
+22
+
+Rad = C
+x0
+Λ − free HBΛ
+ξ
+Γ − free HBΓ
+≤ L(C)
+γx0
+PHΓ(x0)
+≤ L(C) + DΓ
+Figure 2: Lemma 4.18. A Λ-free horoball HBΛ intersects a ball of radius C about x0. The associated
+Γ-free horoball HBΓ is boundedly close, essentially due to the uniform cocompactness of Γ · x0 along the Γ
+horospheres.
+It remains to show that L′(C) is indeed sublinear in C. First define L(C) to be the minimal L that
+bounds the distance d(HΛ, HΓ) for all possible HBΛ tangent to a point x ∈ B(x0, C). This is indeed a
+minimum, since by Corollary 2.22 there are only finitely many horoballs of Γ intersecting B
+�
+x0, C + L′(C)
+�
+.
+For every C there is thus a horoball HBΓ
+C and an element γ ∈ Γ such that γx0 ∈ HΓ, d(HΛ
+C, HΓ
+C) = L(C)
+and |γ| ≤ C + L(C) + DΓ. The fact that HBΛ
+C is Λ-free implies
+L(C) = d(HΛ
+C, HΓ
+C) ≤ u(|γ|) ≤ u
+�
+C + L(C) + DΓ
+�
+Lemma 4.17 implies that L(C) ≤ u′(C) for some sublinear function u′.
+The following is an immediate corollary, apparent already in the above proof.
+Corollary 4.19. For every C > 0 there is a bound K = K(C) and a fixed set ξ1, ξ2, . . . ξK ∈ WQ(Γ) ⊂ X(∞)
+so that every Λ-free horoball HBΛ which is tangent to some point x ∈ B(x0, C) is based at ξi for some
+i ∈ {1, . . ., K}. In particular, for any specific point x ∈ NC(Λ · x0) there are at most K Λ-free horoballs
+tangent to x.
+Proof. Let HBΛ be a horoball tangent to a point x ∈ B(x0, C).
+Lemma 4.18 bounds d(HBΛ, HBΓ) by
+L(C), hence HBΓ is tangent to a point x′ ∈ B
+�
+x0, C + L(C)
+�
+. By Corollary 2.22 there are only finitely
+many possibilities for such HBΓ. In particular there are finitely many base-points for these horoballs, say
+ξ1, ξ2, . . . , ξK(C) ∈ WQ(Γ). Finally, recall that a horoball is determined by a base point and a point x ∈ X
+tangent to it, so the last statement of the corollary holds for any x ∈ B(x0, C). But the property in question
+is Λ-invariant so the same holds for any point x ∈ Λ · B(x0, C) = NC(Λ · x0).
+The bound on d(HBΛ, HBΓ) given by Lemma 4.18 further strengthen the relation between HBΛ and HBΓ.
+The ultimate goal is to show that the HBΛ-s play the role of the Γ-horoballs in the geometric structure of
+Q-rank 1 lattices, namely to show that Λ · x0 is cocompact on the HΛ-s. This requires to actually find
+Λ-orbit points somewhere in X, and not just Λ-free regions as was done up to now. As one might suspect,
+these points arise as λγx0 corresponding to points γx0 ∈ HΓ, which exist in abundance since Γ · x0 ∩ HΓ is
+a cocompact lattice in HΓ.
+The hope is that a Λ-free horoball HBΛ tangent to λγx0 would correspond to a horoball of Γ tangent to
+γx0. This would have forced all the λγ to actually lie on the same bounding horosphere, and {λγx0 | γx0 ∈
+HΓ} would then be a cocompact lattice in HΛ. This hope turns out to be more or less true, but it requires
+23
+
+some work. The goal of the rest of this section is to establish a relation between a Λ-free horoball HBΛ
+tangent to λγx0 and γx0. I start with some notations.
+Definition 4.20. In light of Corollary 4.19, there is a finite number N of Λ-free horoballs tangent to x0.
+Denote:
+1. {HBΛ
+i }N
+1=1 are the Λ-free horoballs tangent to x0.
+2. ξi ∈ WQ(Γ) is the base point of HBΛ
+i .
+3. vi ∈ Sx0X is the unit tangent vector in the direction ξi.
+4. ηi := [x0, ξi) is the unit speed geodesic ray emanating from x0 with limit ξi. In particular vi = d
+dtηi(0).
+5. HBΓ
+i is the horoball of Γ that corresponds to HBΛ
+i , based at ξi.
+6. HBΛ
+λ,i, ξi
+λ, ηi
+λ are the respective λ-translates of the objects above. For example, HBΛ
+λ,i := λ · HBΛ
+i .
+7. H decorated by the proper indices denotes the horosphere bounding HB, the horoball with respective
+indices, e.g. HΛ
+i := ∂HBΛ
+i .
+8. For an angle α > 0 and a tangent vector v0 ∈ SxX, define
+(a) The α-sector of v in SxX is the set {v ∈ SxX | v ∈ Nα(v0)}. Recall that the metric on SxX is
+the angular metric.
+(b) The α-sector of v in X are all points y ∈ X for which the tangent vector at 0 of the unit speed
+geodesic [x, y] lies in the α-sector of v in SxX.
+Lemma 4.21. For every angle α ∈ (0, π
+2 ) there exists D = D(α) such that if dγ > D then for some
+i ∈ {1, . . . , N}, γx0 lies inside the α-sector of vi
+λγ at λγx0. Furthermore whenever α is uniformly small
+enough, there is a unique such i = i(γ), independent of α.
+Proof. Translation by the isometry λ−1
+γ
+preserves angles and distances, so it is enough to prove that there
+is an i for which x′
+γ := λ−1γx0 lies inside the α-sector of vi, and that this i is unique if α is uniformly small.
+Assume towards contradiction that there is α ∈ (0, π
+2 ) and a sequence γn ∈ Γ, λn := λγn ∈ Λ with dγn
+unbounded, and x′
+n := λ−1
+n γnx0 not lying in the union of the α-sectors of vi. By perhaps taking smaller α
+I may assume all the α-sectors of the vi in Sx0X are pairwise disjoint. This can be done because there are
+only finitely many vi.
+Compactness of Sx0X allows me to take a converging subsequence v′
+n :=
+˙
+[x0, x′n], with limit direction
+v′. Denote by η′ the geodesic ray emanating from x0 with initial velocity v′. The exact same argument of
+Lemma 4.13 proves that η′(∞) is the base point of a Λ-free horoball tangent to x0. But this means v′ = vi
+for some i ∈ {1, . . . , N}, contradicting the fact that all v′
+n lie outside the α-sectors of the vi. This proves
+that there is a bound D = D(α) such that if dγ > D then x′
+γ lies within the α-sector of some vi.
+The proof clearly shows that whenever α is small enough so that the α-sectors of the vi are disjoint, x′
+γ
+lies in the α-sector of a unique vi as soon as dγ > D(α).
+Remark 4.22. In the proof of Lemma 4.13 I used compactness of Sx0X to induce a converging subsequence
+of directions. Lemma 4.21 actually shows that the fact there are finitely many Λ-free horoballs tangent to
+x0 implies a posteriori that there was not much choice in the process - all directions [x0, x′
+γ] must fall into
+one of the finitely many directions vi.
+Next, I want to control the actual location of certain points with respect to the horoballs of interest, and
+not just the angles. This turns out to be a more difficult of a task than one might suspect, since control on
+angles does not immediately give control on distances.
+Recall that large Λ-free balls near x0 imply large concentric Γ-free balls. The precise quantities and
+bounds are given by Lemma 3.8 (one can use Lemma 3.9 to obtain a slightly cleaner statement).
+24
+
+Proposition 4.23. Let S ∈ (0, 1) be the constant given by Lemma 3.8, and let s ∈ (0, S). There is a bound
+D = D(s) such that dγ > D implies that γx0 lies sdγ deep in HBΛ
+λγ.
+Proof. The proof is a bit delicate but very similar to that of Lemma 3.8. In essence, I use the Γ-free balls
+near x0 to produce a Γ-free cylinder, which would force a certain geodesic not to cross a horosphere of Γ,
+i.e. force it to stay inside a Γ-free horoball.
+As in Lemma 4.21 it is only required to show that x′ = λ−1
+γ γx0 is sdγ deep inside HBΛ
+i(γ). I start with
+proving that x′ ∈ HBΓ
+i(γ). I learned the hard way that even this is not a triviality. Recall the notation
+Bγ = B(γx0, dγ). The ball B′
+γ = λ−1
+γ Bγ is a Λ-free ball of radius dγ about x′ = λ−1
+γ γx0. Denote by x′
+t
+the point at time t along the unit speed geodesic η′ := [x0, x′]. It holds that |x′
+t| = t and, for t ≤ dγ, x′
+t
+is the centre of a Λ-free ball of radius t tangent to x0. The constant s is fixed and by Lemma 3.8 there is
+T ′ = T ′(s) such that if t > T ′, the ball sB
+�
+x′
+t, t
+�
+is Γ-free.
+The next goal is to show that x′
+T ∈ HBΓ for some adequate T . For any time T > 0, let α = α(ε, T ) be the
+angle for which d
+�
+η(T ), ηi(γ)(T )
+�
+< ε for every η in the α-sector of vi(γ). By perhaps taking smaller α I may
+assume that α is uniformly small as stated in Lemma 4.21. Let D(α) be the bound given by Lemma 4.21
+guaranteeing
+D(α) < dγ ⇒ d
+�
+x′
+T , ηi(γ)(T )
+�
+< ε
+For my needs in this lemma ε may as well be chosen to be 1. I now choose a specific time T for which
+I want x′
+T and ηi(γ)(T ) to be close. There are only finitely many Λ-free horoballs {HBΛ
+i }i∈{1,...,N} tangent
+to x0, giving rise to a uniform bound L = maxi∈{1,...,N}{d(HΛ
+i , HΓ
+i )} on the distance d(HΛ
+i(γ), HΓ
+i(γ)). Fix
+T to be any time in the open interval (T ′ + L + ε, dγ). The fact that L + ε < T implies that ηi(γ)(T ) lies
+at least ε-deep inside HBΓ
+i(γ), and therefore η′(T ) ∈ HBΓ
+i(γ). Recall that any point on HΓ is DΓ-close to a
+point γx0 ∈ HΓ. By perhaps enlarging T and shrinking α if necessary, I may assume that DΓ < sT . Thus
+for all T < t ≤ dγ, x′
+t is the centre of a Γ-free ball of radius st > sT > DΓ, hence {x′
+t}T ≤t≤dγ does not cross
+a horosphere of Γ. Since x′
+T ∈ HBΓ
+i(γ), this implies that x′
+t stays in HBΓ
+i(γ) for all T < t ≤ dγ. In particular
+x′
+dγ = x′ ∈ HBΓ
+i(γ).
+To get the result of the proposition, recall that sB′
+γ = B(x′, sdγ) is Γ-free, so x′ must be at distance at
+least sdγ − DΓ from any horosphere of Γ, and in particular from HΓ
+i(γ). In terms of Busemann functions, this
+means that bηi(γ)(x′) ≤ −sdγ + DΓ whenever one can find such T ′ + L + ε < T < dγ. Since HBΛ
+i(γ) is tangent
+to x0, the corresponding horoball HBΓ
+i(γ) lies inside it, and so x′ lies (sdγ − DΓ)-deep inside HBΛ
+i(γ). A close
+look at the argument yields the desired bound D = D(s) such that the above holds whenever dγ > D(s).
+To help the reader take this closer look, I reiterate the choice of constants and their dependencies as they
+appear in the proof:
+1. Fix ε = 1.
+2. Let T ′ = T ′(s) the constant from Lemma 3.8 and L = maxi∈{1,...,N}{d(HBΛ
+i , HBΓ
+i )}.
+3. Fix T > T ′ + L + 1.
+4. Fix α = α(1, T ).
+5. Fix D(s) = max{D(α), T + 1}.
+I remark, for the reader worried about the DΓ which appears in the final bound but not in the statement,
+that (a) DΓ is a fixed universal constant and may as well be ignored, and (b) the discrepancy can be formally
+corrected by taking a slightly larger s < s′ to begin with and as a result perhaps enlarging the bound D for
+dγ). Also note that L = L(Λ) is a universal constant.
+25
+
+4.2.3
+Intersection of Λ-Free Regions and the Existence of a Λ-Cocompact Horosphere
+In this section I find Λ-orbit points that lie close to the bounding horosphere of a Λ-free horoball HBΛ. In
+order to find such points I need to make sure HBΛ is not contained inside a much larger Λ-free horoball. I
+introduce the following definition.
+Definition 4.24. A Λ-free horoball HBΛ is called maximal if it is tangent to a point x = λx0 ∈ Λ · x0. It
+is called ε-almost maximal if d(Λ · x0, HΛ) < ε.
+Remark 4.25. It may happen that a discrete group admits free but not no maximally free horoballs - see
+discussion in section 4 of [55]. In any case it is clear that any Λ-free horoball can be ‘blown-up’ to an ε-almost
+maximal Λ-free horoball, for every ε > 0. Moreover, every two ε-almost maximal horoballs based at the
+same point ξ ∈ WQ(Γ) lie at distance at most ε of one another. For my needs any fixed ε would suffice, and
+I fix ε = 1.
+Lemma 4.26. There is DΛ > 0 such that if HBΛ is 1-almost maximal Λ-free horoball then HΛ ⊂ NDΛ(Λ·x0),
+i.e. d(x, Λ · x0) ≤ DΛ for all x ∈ HΛ.
+Notice that Lemma 4.26 does not state Λ · x0 even intersects HΛ.
+Proof. I start with a short sketch of the proof. Consider a 1-maximal horoball and a point x on its bounding
+horosphere with d(x, Λ · x0) = D. One may translate this situation to x0, which results in a Λ-free horoball
+HBΛ intersecting the (closed) D-ball about x0 at a point w with B(w, D) Λ-free.
+The proof differs depending on whether HBΓ ⊂ HBΛ or the other way round, since I use the bounds
+from 4.18:
+1. If HBΓ ⊂ HBΛ, there is a sublinear bound on d(HBΛ, HBΓ), which readily yields a bound on D.
+2. if HBΛ ⊂ HBΓ there is a bound on d(x0, HBΓ) that is independent of D. So there are only finitely
+many possibilities for HBΓ, independent of D. Hence there are only finitely many possible base points
+for HBΓ. These in turn correspond to possible base points for such HBΛ, and this finiteness yields
+a bound on the distance d(HBΓ, HBΛ) < L that is independent of D. The rest of the proof is quite
+routine.
+Let HBΛ be a 1-almost maximal Λ-free horoball. By definition there is λ ∈ Λ and z ∈ HΛ such that
+d(λx0, z) < 1. Fix D > 0. I show that if there is some z′ ∈ HΛ for which d(z′, Λ · x0) ≥ D, then D must be
+uniformly small. Exactly how small will be set in the course of the proof.
+Fix D > 1 and assume that there is z′ ∈ HΛ with d(z, Λ · x0) ≥ D. Up to sliding z′ along HΛ, the
+continuity of the function x �→ d(x, Λ · x0) together with Intermediate Value Theorem allows to assume that
+d(z′, Λ · x0) = D. Let λ′ ∈ Λ be the element for which d(z′, λ′x0) = D. Translating by λ′−1 yields
+1. A Λ-free horoball HBΛ
+0 := λ′−1HBΛ.
+2. A point w := λ′−1z′ ∈ HΛ
+0 for which |w| = d(w, x0) = d(w, Λ · x0) = D.
+Assume first that HBΓ
+0 ⊂ HBΛ
+0 . By Lemma 4.18 there is a sublinear function u′ such that d(HΓ
+0 , HΛ
+0 ) ≤
+u′(D). This yields a point γx0 ∈ HΓ
+0 for which d(w, γx0) ≤ u′(D) + DΓ. Thus |γx0| ≤ D + u′(D) + DΓ and
+the reverse triangle inequality gives
+D −
+�
+u′(D) + DΓ
+�
+≤ d(w, λγx0) − d(w, γx0) < d(γx0, λγx0)
+Together with the bound d(γx0, λγx0) ≤ u(|γx0|) and rearranging, one obtains
+D ≤ u
+�
+D + u′(D) + DΓ
+�
++ u′(D) + DΓ
+The right hand side is clearly a sublinear function in D, hence this inequality may hold only for boundedly
+small D, say D < D1. I conclude that HBΓ
+0 ⊂ HBΛ
+0 may occur only when D < D1. Notice that D1 depends
+only on u and u′, and not on HBΛ.
+26
+
+w
+x0
+Λ − free
+B(w, D)
+τ(t0)
+τ
+Λ − free HBΛ
+ξ
+Γ − free HBΓ
+γx0
+Figure 3: Lemma 4.26, case HBΛ ⊂ HBΓ. The red horosphere of Γ is trapped between x0 and HΛ, and is
+at distance t0 from x0. A Γ-orbit point on the red horosphere close to x0 allows to use sublinearity to get a
+bound on t0.
+Assume next that HBΛ
+0 ⊂ HBΓ
+0 , and that the containment is strict. Since x0 ∈ Γ · x0, the geodesic
+τ := [x0, w] is of length D and intersects HΓ
+0 . Denote by t0 ∈ [0, D) the time in which τ intersects HΓ
+0 , and
+let w′ := ��(t0) ∈ HΓ
+0 be the intersection point. In particular |w′| = t0. It is clear that B(w′, t0) is Λ-free,
+as a subset of the ball B(w, D). Again there is γx0 ∈ B(w′, DΓ) ∩ HΓ
+0 and so |γx0| ≤ t0 + DΓ. By reverse
+triangle inequality
+t0 − DΓ ≤ d(w′, λγx0) − d(w′, γx0) ≤ d(γx0, λγx0)
+and the sublinear constraint gives t0 − DΓ ≤ u(t0 + DΓ). This can only happen for boundedly small t0,
+say t0 < T . I conclude that if HBΛ
+0 ⊂ HBΓ
+0 , then HBΓ
+0 is a horoball of Γ tangent to some point y ∈ B(x0, T ).
+By Corollary 2.22 there are finitely many horoballs of Γ tangent to points in B(x0, T ). In particular
+there is a finite set {ξ′
+1, . . . , ξ′
+K} ∈ WQ(Γ) of possible base points for HBΓ
+0. This set depends only on T , and
+since the choice of T was completely independent of D, the set of possible base points is independent of D
+as well. Let �
+HBΓ
+i be the horoball of Γ based at ξ′
+i.
+I can now bound the distance d(HBΓ
+0 , HBΛ
+0 ). Let 1 ≤ i ≤ K be an index for which there is a Λ-free
+horoball based at ξ′
+i that is contained in �
+HBΓ
+i . There is thus some 1-almost-maximal Λ-free horoball based at
+ξ′
+i. Fix an arbitrary such 1-almost-maximal Λ-free horoball �
+HBΛ
+i for each such i, and let Li := d(�
+HBΛ
+i , �
+HBΓ
+i ).
+Finally, define L := max{Li} + 1 among such i. As stated in Remark 4.25, d(HBΛ
+0 , �
+HBΛ
+i ) ≤ 1 for some i,
+therefore d(HBΓ
+0, HBΛ
+0 ) ≤ L.
+Recall |w| = D and B(w, D) is Λ-free. It holds that d(w, HΓ
+0 ) ≤ L, and so there is γx0 ∈ HΓ
+0 for which
+d(w, γx0) ≤ L + DΓ. In particular |γx0| ≤ D + L + DΓ (in fact it is clear that |γx0| ≤ T + DΓ, but this
+won’t be necessary). Reverse triangle inequality gives
+D − (L + DΓ) ≤ d(w, λγx0) − d(w, γx0) ≤ d(γx0, λγx0)
+and from the sublinear constraint I conclude D − (L + DΓ) ≤ u(D + L + DΓ). Since L, DΓ are fixed
+constants independent of D, this can only hold for boundedly small D, say D < D2. In particular, one gets
+a uniform bound DΛ := max{D1, D2} such that x ∈ HΛ ⇒ d(x, Λ · x0) < DΛ.
+Corollary 4.27. Every Γ-conical limit point is a Λ-conical limit point.
+27
+
+Proof. Let ξ ∈ X(∞) be a Γ-conical limit point. Let η : R≥0 → X be a geodesic with η(∞) = ξ. By
+definition there is a bound D > 0 and sequences tn → ∞, γn ∈ Γ such that d
+�
+γnx0, η(tn)
+�
+< D. Consider the
+corresponding λn := λγn and λnx0. If dn is uniformly bounded, then ξ is Λ-conical by definition. Otherwise,
+assume dn is monotonically increasing to ∞. For some fixed s ∈ (0, 1) it holds that for all but finitely many
+n ∈ N, γnx0 is sdn deep inside HBΛ
+n := HBΛ
+λn,i(γn). I assume dn is large enough so that sdn > D, and in
+particular η(tn) ∈ HBΛ
+n. Let ξn ∈ WQ(Γ) be the respective base points of HBΛ
+n. The point ξ is Γ-conical,
+and by Theorem 2.28 π
+2 ≤ d(ξ, WQ(Γ)) ≤ d(ξ, ξn).
+The proof differs depending on whether the above inequality is strict or not for any n ∈ N. Assume
+first that for some m ∈ N, d(ξ, ξm) = π
+2 . By item 2 of Lemma 2.31, d
+�
+HΛ
+m, η(t)
+�
+is uniformly bounded, i.e.,
+there is C > 0 such that for every t > 0 there is xt ∈ HΛ
+m for which d
+�
+xt, η(t)
+�
+< C. By Lemma 4.26,
+d(xt, Λ · x0) < DΛ, hence d
+�
+η(tn), xtn
+�
+≤ C + DΛ. This means that ξ is Λ-conical.
+Otherwise, for all n ∈ N it holds that π
+2 < d(ξ, ξn). The fact that η(tn) ∈ HBΛ
+n together with Lemma 2.31
+implies that at some later time the geodesic ray η leaves HBΛ
+n. Thus there is sn > tn for which η(sn) ∈ HΛ
+n.
+Since HBΛ
+n are maximal Λ-free horoballs, Lemma 4.26 gives rise to points λnx0 such that d
+�
+λnx0, η(sn)
+�
+≤
+DΛ. This renders ξ as a Λ-conical limit point, as wanted.
+I now prove Proposition 4.12.
+Proof of Proposition 4.12. The strategy is as follows. For HBΛ = HBΛ
+λγ,i(γ), one uses Proposition 4.23 to
+get that HBΓ ⊂ HBΛ and that the distance d(HΛ, HΓ) is large with dγ. The horosphere HΓ admits a
+Γ-cocompact metric lattice, and so the projections of these metric lattice points onto HΛ form a cocompact
+metric lattice in HΛ. It remains to show that for each γ′x0 ∈ Γ · x0 ∩ HΓ, the corresponding λ′ = λγ′ indeed
+lies on the same HΛ and boundedly close to the projection PHΛ(γ′x0). This is done by putting together all
+the geometric facts obtained up to this point, specifically Lemma 4.26. One delicate fact that will be of use
+is that two maximal Λ-free horoballs that are based at the same point must be equal, because none of them
+can contain a Λ-orbit point while on the other hand both bounding horospheres intersect Λ · x0.
+Fix s > 0 for which Proposition 4.23 yields a corresponding bound D(s), and let γ ∈ Γ such that
+sdγ > 2 ·
+�
+DΛ + D(s)
+�
+.
+Consider the (maximal) Λ-free horoball HBΛ
+λγ,i(γ) based at ξi(γ)
+λ
+.
+I show that
+Λ · x0 ∩ HΛ
+λγ,i(γ) is a cocompact metric lattice in HΛ
+λγ,i(γ). I keep the subscript notation because the proof is
+a game between HBΛ
+λγ,i(γ) and another Λ-free horoball.
+Let HBΓ
+λγ,i(γ) be the Γ-horoball corresponding to HBΛ
+λγ,i(γ). I can conclude that HBΓ
+λγ,i(γ) ⊂ HBΛ
+λγ,i(γ),
+because the choice of dγ > D(s) guarantees γx0 is sdγ deep inside HBΛ
+λγ,i(γ). In particular HBΛ
+λγ,i(γ) is not
+Γ-free. Moreover, it holds that L := d(HΛ
+λγ,i(γ), HΓ
+λγ,i(γ)) ≥ sdγ. Let γ′ ∈ Γ be any element in the cocompact
+metric lattice Γ · x0 ∩ HΓ
+λγ,i(γ), and consider two associated points: (a) λ′x0 = λγ′x0 and (b) the projection
+of γ′x0 on HΛ
+λγ,i(γ), denoted p′
+γ := PHΛ
+λγ ,i(γ)(γ′x0) ∈ HΛ
+λγ,i(γ). The horoball HBΛ
+λγ,i(γ) is a maximal Λ-free
+horoball so it is also 1-almost maximal, hence d(p′
+γ, Λ · x0) ≤ DΛ and the following holds:
+sdγ ≤ L ≤ dγ′ ≤ d(γ′x0, p′
+γ) + d(p′
+γ, Λ · x0) ≤ L + DΛ
+(3)
+Consider ξi(γ′)
+λ′
+, and assume towards contradiction that ξi(γ′)
+λγ′
+̸= ξi(γ)
+λγ . Both points lie in WQ(Γ) and
+therefore must be at Tits distance π of each other. Therefore the fact that γ′x0 lies in HBΛ
+λγ,i(γ) implies that
+the geodesic [γ′x0, ξi(γ′)] leaves HBΛ
+λγ,i(γ) at some point z ∈ HΛ
+λγ,i(γ).
+The fact that D(s) ≤ sdγ ≤ dγ′ implies that γ′x0 lies s2dγ deep inside HBΛ
+λγ′ ,i(γ′). Therefore the point z
+also lies at least s2dγ deep inside HBΛ
+λγ′ ,i(γ′), and therefore z is the centre of a Λ-free horoball of radius at
+least s2dγ.
+By choice of dγ the point z therefore admits a 2DΛ neighbourhood that is Λ-free. But z lies on HΛ
+λγ,i(γ),
+a maximal horosphere of Λ, contradicting Lemma 4.26. I conclude that ξi(γ′)
+λγ′
+= ξi(γ)
+λγ , so HBΛ
+λγ,i(γ) and
+28
+
+γx0
+dγ
+γ′x0
+λγ′x0
+HBΛ
+λγ′ ,i(γ′)
+z′ := πHBΛ
+λγ′ ,i(γ′)(γ′x0)
+D1
+����
+1
+ξi(γ′)
+λγ′
+ξi(γ)
+λγ
+≥ 1
+2dγ
+z
+≥ 1
+2 dγ′
+Λ − free ball
+B
+�
+z, 1
+2dγ
+�
+HBΛ
+λγ,i(γ)
+HBΓ
+λγ,i(γ)
+λγx0
+Figure 4: Proposition 4.12. Assuming towards contradiction that ξi(γ)
+λγ
+̸= ξi(γ′)
+λγ′
+results in a point z ∈ HBΛ
+λγ,i(γ)
+(blue coloured and bold faced in the bottom part of the figure) admitting a large Λ-free neighbourhood,
+contradicting almost cocompactness.
+HBΛ
+λγ′ ,i(γ′) are two Λ-free horoballs that are tangent to a Λ · x0 point and based at the same point at
+∞. This implies HBΛ
+λγ,i(γ) = HBΛ
+λγ′ ,i(γ′), and in particular λγ′x0 ∈ HΛ. Finally, it is clearly seen from
+Inequality 3 that
+d(λ′x0, p′
+γ) ≤ d(λ′x0, γ′x0) + d(γ′x0, p′
+γ) ≤ dγ′ + L ≤ L + DΛ + L
+The element γ′x0 ∈ Γ · x0 ∩ HΓ
+λγ,i(γ) was as arbitrary element, and the above argument shows that the
+corresponding Λ-orbit points satisfy:
+1. λ′x0 all lie on HΛ
+λγ,i(γ).
+2. Each p′
+γ is 2L + DΛ close to the point λ′x0.
+This shows that the cocompact metric lattice {p′
+γ | γ′x0 ∈ HΓ
+λγ,i(γ)} lies in a bounded neighbourhood of
+the set of points Λ · x0 ∩ HΛ
+λγ,i(γ), proving that Λ · x0 ∩ HΛ
+λγ,i(γ) is a cocompact metric lattice in HΛ
+λγ,i(γ).
+Lemma 2.24 elevates this to
+�
+Λ ∩ StabG(HΛ
+λγ,i(γ))
+�
+· x0 ∩ HΛ
+λγ,i(γ)
+being a cocompact metric lattice in HΛ
+λγ,i(γ), completing the proof.
+4.3
+The Bounded Case
+Proposition 4.12 is enough in order to prove Theorem 4.1 in the case Γ ̸⊂ ND(Λ) for any D > 0, i.e. in case
+Γ does not lie in a bounded neighbourhood of Λ. The case where Γ and Λ lie at bounded Hausdorff distance,
+i.e. where Γ ⊂ ND(Λ) and Λ ⊂ ND(Γ), arose naturally in the context of the quasi-isometric classification
+of non-uniform lattices in the works of Schwartz [52] (R-rank 1), Drut¸u [16] and Eskin [22] (higher rank). I
+restate the theorems in the bounded setting.
+29
+
+Theorem 4.28 (Eskin [22], Theorem 4.9 above). Let G be a real centre-free semisimple Lie group without
+compact factors and of higher rank, Γ ≤ G an irreducible non-uniform lattice, Λ ≤ G a discrete subgroup. If
+Γ ⊂ ND(Λ) for some D > 0, then Λ is a lattice, and if moreover Λ ⊂ ND(Γ) then Λ and Γ are commensurable.
+Theorem 4.29 (Schwartz [52], Theorem 4.10 above). Let G be a real simple Lie group of R-rank 1, Γ ≤ G
+an irreducible non-uniform lattice, Λ ≤ G a discrete subgroup. If both Γ ⊂ ND(Λ) and Λ ⊂ ND(Γ) for some
+D > 0, then Λ is a lattice, and if moreover G is not locally isomorphic to SL2(R), then Λ is commensurable
+to Γ.
+The notable difference between the two statements is that for higher rank groups, the inclusion Λ ⊂ ND(Γ)
+is only required to prove commensurability. In view of Corollary 4.8, this allows me to omit that assumption
+from Theorem 1.6. Notice also that for groups with property (T) the result easily follows from the (much
+more recent) result by Leuzinger in Theorem 5.7.
+In the context of commensurability in the sublinear setting, I can only prove a limited result, Namely
+that Λ is commensurable to Γ if Γ is an irreducible Q-rank 1 lattice and both Γ ⊂ ND(Λ) and Λ ⊂ Nu(Γ)
+for some constant D > 0 and a sublinear function u. This is done via a reduction to the bounded case.
+Proposition 4.30. Let G be a real semisimple Lie group without compact factors and with finite centre,
+Γ ≤ G an irreducible lattice of Q-rank 1, Λ ≤ G a discrete subgroup, and u a sublinear function. If Γ ⊂ ND(Λ)
+for some D > 0 and Λ ⊂ Nu(Γ), then actually Λ ⊂ ND′(Γ) for some D′ > 0. Moreover, if G is of R-rank 1,
+the conclusion holds under the relaxed assumption that u(r) ⪯∞ εr for some ε < 1.
+Remark 4.31. While the setting of Proposition 4.2 is indeed rather limited, the situation that both Γ ⊂
+Nu(Λ) and Λ ⊂ Nu(Γ) arises naturally from the motivating example of SBE-rigidity in Theorem 6.9. Notice
+however that Theorem 6.9 is not known for groups G that admit R-rank 1 factors, which is the only setting
+for which I can prove Proposition 4.2.
+4.3.1
+A Reduction
+I start with the proof of Proposition 4.30. The first step is to establish the fact that Λ must preserve WQ(Γ).
+Lemma 4.32. Let G be a real semisimple Lie group without compact factors and with finite centre, Γ ≤ G
+an irreducible non-uniform lattice of Q-rank 1, Λ ≤ G a discrete subgroup. Assume that Γ ⊂ Nu(Λ) and
+that Λ ⊂ Nu′(Γ) for sublinear functions u, u′. Then Λ · WQ(Γ) ⊂ WQ(Γ). Moreover, if G is of R-rank 1, the
+conclusion holds under the relaxed assumption that u′(r) ⪯∞ εr for some ε < 1.
+Proof. The proof is similar to the argument of Lemma 4.14, and uses the linear penetration rate of a geodesic
+into a horoball. Let ξ ∈ WQ(Γ), and let HΓ be a horosphere bounding a Γ-free horoball HBΓ with HΓ ∩Γ·x0
+a metric lattice in HΓ. Assume first that u′ is sublinear. Since HBΓ is Γ-free and Λ ⊂ Nu′(Γ), I can conclude
+that Λ · x0 ∩ HBΓ ⊂ Nu′(HΓ). Recall (Lemma 2.31) that every geodesic ray η with limit point ξ′ ∈ N π
+2 (ξ)
+penetrates HBΓ at linear rate. Therefore for every such geodesic ray η and every sublinear function v there
+is R = R(η, v) > 0 for which Nv(η↾r>R) is Λ-free.
+On the other hand, let λ ∈ Λ, and assume towards contradiction that λξ /∈ WQ(Γ). Then by Proposi-
+tion 2.30 there is a Γ-conical limit point ξ′ ∈ N π
+2 (λξ). The hypothesis that Γ ⊂ Nu(Λ) then implies that for
+every R > 0, Nu(η↾r>R) ∩ Λ · x0 ̸= ∅. Translating by λ−1 yields a contradiction to the previous paragraph.
+I conclude that λξ ∈ WQ(Γ).
+I now modify the argument to include u′(r) ⪯∞ εr when G is of R-rank 1. In this case, the only point
+ξ′ ∈ N π
+2 (λξ) is λξ itself. Therefore by the same argument as above, the assumption that λξ /∈ WQ(Γ)
+implies that the u-sublinear neighbourhood of every geodesic ray with limit point ξ intersects Λ · x0. I.e., for
+every η with limit point ξ and every R > 0 it holds that Nu(η↾r>R) ∩ Λ · x0 ̸= ∅. On the other hand, every
+such geodesic penetrates HBΓ at 1-linear rate. This amounts to the following fact: if v′(r) = εr for some
+ε ∈ (0, 1), then for some R > 0, the set Nu(η↾r>R)∩Nv′(HΓ) = ∅. This is a contradiction to Λ ⊂ Nu′(Γ).
+30
+
+Proof of Proposition 4.30. Assume towards contradiction that there is a sequence λn such that d(λnx0, Γ ·
+x0) > n. Recall that Γ·x0 is a cocompact metric lattice in the compact core of Γ. This implies that there is a
+number D′ > 0 such that any λ ∈ Λ for which λx0 /∈ ND′(Γ · x0) must lie at least 1
+2D′-deep inside a horoball
+of Γ. I can assume that for all n ∈ N there are corresponding horoballs of Γ, which I denote HBΓ
+n, for which
+λn · x0 ∈ HBΓ
+n. The fact that Γ ⊂ ND(Λ) then implies that ND(Λ · x0) covers a cocompact metric lattice in
+HΓ
+n, namely the metric lattice Γ · x0 ∩ HΓ
+n. In the terminology of Section 4.2, HΓ
+n is almost Λ-cocompact, or
+D-almost Λ-cocompact.
+I first prove that every horoball of Γ contains a Λ-free horoball (this is of course immediate if Λ ⊂ NC(Γ)
+for some C > 0). Assume towards contradiction that there is a horoball HBΓ of Γ that does not contain a
+Λ-free horoball. Denote HΓ := ∂HBΓ. In the notations of the previous paragraph, I can assume without loss
+of generality that HBΓ = HBΓ
+n for all n ∈ N. Denote by ξ the base point of HBΓ, fix some arbitrary x ∈ HΓ
+and consider the geodesic ray η := [x, ξ). The constraint that Λ ⊂ Nu(Γ) implies that for every R > 0 there
+is some L > 0 for which the ball B
+�
+η(L + t), R
+�
+is Λ-free, for all t ≥ 0. In particular, for all large enough
+n ∈ N (depending on R), the horosphere H(ξ, λnx0) that is parallel to HΓ and that passes through λnx0
+contains a point that is the centre of Λ-free ball of radius R. This property is Λ-invariant, as well as the fact
+that HBΓ is based at WQ(Γ). In particular, these two properties hold for the horoballs HBn := λ−1
+n
+· HBΓ,
+whose respective base points I denote ξn := λ−1
+n ξ ∈ WQ(Γ).
+Fix R = D +2DΓ (recall that DΓ is such that every horosphere H of Γ admits H ⊂ NDΓ(Γ·x0 ∩H)). Let
+L = L(D+2DΓ) be the corresponding bound from the previous paragraph. For every n > L the horoball HBn
+has bounding horosphere Hn that admits a point zn ∈ Hn for which B(zn, D+2DΓ) is Λ-free. Moreover, the
+same is true for every horosphere that is parallel to Hn which lies inside HBn. Since Γ ⊂ ND(Λ), this means
+that every horosphere that lies inside HBn admits a point that is the centre of a Γ-free ball of radius 2DΓ.
+I conclude that none of those horospheres could be the horosphere of Γ corresponding to the parabolic limit
+point ξn ∈ WQ(Γ). Since x0 ∈ Hn it must therefore be that Hn is a horosphere of Γ. But this contradicts
+the fact that zn ∈ Hn and B(zn, 2DΓ) is Γ-free. This shows that no horoball of Γ contains a sequence of
+Λ-orbit points that lie deeper and deeper in that horoball. Put differently, it shows that every horoball of Γ
+contains a Λ-free horoball.
+I remark that the above argument shows something a bit stronger, which I will not use but which I find
+illuminating. It proves that as soon as d(λx0, Γ · x0) is uniformly large enough, say more than M, then
+λx0 must lie on a (D + 2DΓ)-almost Λ-cocompact horosphere parallel to HΓ, where HΓ is the bounding
+horosphere of any horoball of Γ in which λx0 lies (recall that it must lie in at least one such horoball). On
+the other hand if d(λx0, Γ · x0) < M, then since every point in the Γ-orbit lies on a horosphere of Γ one
+concludes that λx0 lies on a horosphere H based at WQ(Γ) that is (M + D)-almost Λ-cocompact.
+I can now assume that every HBΓ
+n contains a Λ-free horoball. In particular it contains a 1-almost maximal
+Λ-free horoball HBΛ
+n (see Definition 4.24). By definition there is a point λ′
+nx0 that is at distance at most 1
+from HΛ
+n = ∂HBΛ
+n. Up to enlarging d(λnx0, Γ · x0) or decreasing it by at most 1, I can assume λn = λ′
+n to
+begin with. Consider HBn := λ−1
+n
+· HBΛ
+n with Hn = ∂HBn. This is a sequence of horoballs, each of which
+contains a Λ-free horoball at depth at most 1, based at corresponding parabolic limit points ξn ∈ WQ(Γ),
+and tangent to points that are at distance at most 1 from x0, i.e., Hn ∩ B(x0, 1) ̸= ∅.
+Since Γ ⊂ ND(Λ) I conclude that each of the HBn contain a horoball of depth at most D + 1 that
+is Γ-free. Therefore the horoball of Γ that is based at ξn must have its bounding horosphere intersecting
+B(x0, D + 2). By Corollary 2.22 there are only finitely such horoballs. I conclude that there are finitely
+many points ξ′
+1, . . . , ξ′
+K ∈ WQ(Γ) such that for every n ∈ N there is i(n) ∈ {1, . . . , K} with ξn = ξ′
+i(n). From
+the Pigeonhole Principle there is some ξ′ ∈ {ξ′
+1 . . . , ξ′
+K} for which ξn = ξ′ for infinitely many n ∈ N. Passing
+to a subsequence I assume that this is the case for all n ∈ N.
+To begin with the HBΓ
+n are horoballs of Γ, and therefore as in the first case the bounding horospheres
+HΓ
+n are D + 2DΓ-almost Λ-cocompact. This is a Λ-invariant property and therefore the same holds for
+the λ−1
+n
+translate of it. These are the horospheres which are based at ξ′ and lie outside HBn at distance
+d(λnx0, Γ · x0) > n − 1 from Hn. They form a sequence of outer and outer horospheres based at the same
+point at WQ(Γ), all of which are D + 2DΓ-almost Λ-cocompact. This is a contradiction, since the union of
+such horospheres intersect every horoball of X, contradicting the existence of Λ-free horoballs. Formally,
+31
+
+take some ζ ∈ WQ(Γ) different from ξ′. Since both ξ′ and ζ lie in WQ(Γ), they admit dT (ζ, ξ′) = π and
+there is a geodesic η with η(−∞) = ξ′ and η(∞) = ζ. Let HBΓ
+ζ be the horoball of Γ that is based at ζ. By
+the first step of this proof, every such horoball must contain a Λ-free horoball HBΛ
+ζ . Therefore there is some
+T > 0 such that for all t > T the point η(t) lies 2(D + 2DΓ) deep in HBΛ
+ζ . I conclude that for all t > T ,
+B
+�
+η(t), 2D
+�
+is Λ-free. On the other hand, for arbitrarily large t it holds that the horosphere based at ξ′ and
+tangent to η(t) is D+2DΓ-almost Λ-cocompact, and in particular d
+�
+η(t), Λ·x0
+�
+< D+2DΓ, a contradiction.
+I conclude that Λ ⊂ ND′(Γ) for some D′ > 0, as claimed.
+Corollary 4.33. In the setting of Proposition 4.30, Λ is a lattice commensurable to Γ.
+4.3.2
+The Arguments of Schwartz and Eskin
+The R-rank 1 case.
+The statement of Theorem 4.10 is a slight modification of his original formulation.
+His framework leads to a discrete subgroup ∆ ≤ G such that:
+1. Every element of ∆ quasi-preserves the compact core of the lattice Γ. Namely, each element of ∆ is
+an isometry of X that preserves WQ(Γ) and that maps every horosphere of Γ to within the D = D(∆)
+neighbourhood of some other horosphere of Γ.
+2. It holds that Γ ⊂ ND(∆).
+From these two properties Schwartz is able to deduce that ∆ has finite covolume, i.e. that ∆ is a lattice
+in G. Here is a sketch of his argument, which works whenever Γ is a Q-rank 1 lattice.
+Theorem 4.34. In the setting described above, ∆ is a lattice in G.
+Proof sketch. Consider X′
+0 := �
+g∈∆ g · X0, where X0 is the compact core of Γ.
+This space serves as a
+‘compact core’ for ∆: the fact that ∆ quasi-preserves X0 implies that X′
+0 ⊂ ND(X0). It is a ∆-invariant
+space, and therefore one gets an isometric action of ∆ on X′
+0. This action is cocompact: the reason is that
+Γ acts cocompactly on X0, and Γ ⊂ ND(∆). Formally, every point in X′
+0 is D-close to a point in X0. Every
+point in X0 is DΓ-close to a point in Γ · x0. Every point in Γ · x0 is D-close to a point in ∆ · x0. Therefore
+the ball of radius 2D + DΓ contains a fundamental domain for the action of ∆ on X′
+0.
+It remains to see that the action of ∆ on X \ X′
+0 is of finite covolume. As a result of the cocompact
+action of ∆ on X′
+0, there is B := B(x0, R) so that X′
+0 ⊂ ∆·B. X′
+0 is the complement of a union of horoballs,
+which one may call horoballs of ∆, with bounding horospheres of Λ. The fact that Γ is of Q-rank 1 means
+that the horoballs of Γ are disjoint, and therefore those of Λ are almost disjoint: there is some C > 0 such
+that for every horosphere H of Λ and every point x ∈ H, d(x, X′
+0) < C. Up to enlarging the radius of B by
+C, I can assume that H ⊂ ∆ · B for every horosphere H of Λ.
+Each horoball of Λ is based at WQ(Γ), and each lies uniformly boundedly close to the corresponding
+horoballs of Γ. From Corollary 2.22 one therefore sees that there are finitely many horoballs of ∆ that inter-
+sect B. Denote them by HB1, . . . , HBN, their bounding horospheres by Hi = ∂HBi, and their intersection
+with B by Bi := B ∩ HBi. Let also ξi ∈ WQ(Γ) denote the base point of each HBi. Each Bi is pre-compact
+and therefore the projection of each Bi on Hi is pre-compact as well (this is a consequence e.g. of the results
+of Heintze-Im hof recalled in Remark 2.5). Let Di ⊂ Hi be a compact set that contains this projection, i.e.
+PHi(Bi) ⊂ Di ⊂ Hi. In particular B ∩ Hi ⊂ Di.
+Observe now that for every horoball HB of ∆, with bounding horosphere H = ∂HB, the ∆-orbit of every
+point x ∈ H intersects some Di. First notice that for x ∈ H the choice of B implies that the ∆-orbit of x
+must intersect B, say gx ∈ B. In particular gH ∩ B ̸= ∅, and since gHB is a horoball of ∆ then by definition
+gH = Hi for some i ∈ {1, . . ., N}. One concludes that indeed gx ⊂ Di.
+Moreover, let y ∈ X is any point that lies inside a horoball HB of ∆, and x = PH(y) its projection on
+the bounding horosphere H = ∂HB. By the previous paragraph there is some g ∈ ∆ and i ∈ {1, . . . , N} for
+which gx ∈ Di, and therefore it is clear that gy lies on a geodesic emanating from Di to ξi.
+32
+
+Finally, define Cone(Di) to be the set of all geodesic rays that emanate from Di and with limit point ξi.
+The previous paragraph proves that �N
+i=1 Cone(Di) contains a fundamental domain for the action of ∆ on
+X \ X′
+0. Moreover, the fact that Di ⊂ Hi is compact readily implies that each Cone(Di) has finite volume,
+and so this fundamental domain is of finite volume.
+To conclude, B ∪
+� �N
+i=1 Cone(Di)
+�
+is a set of finite volume and it contains a fundamental domain for the
+∆-action on X, as claimed. The proof of commensurability of ∆ and Γ is given in full in [52].
+There is one essential difference between Theorem 4.10 and Theorem 4.34, namely the assumption that
+Λ ⊂ ND(Γ) rather than quasi-preserving the compact core of Γ. In Schwartz’s work, the fact that ∆ ⊂ ND(Γ)
+is not relevant (even though it easily follows from the construction of his embedding of ∆ in G). He only
+uses the two properties described above, namely the quasi-preservation of X0 and Γ ⊂ ND(∆).
+The assumption that Λ quasi-preserves the compact core of Γ does not feel appropriate in the context
+of my thesis, while the metric condition Λ ⊂ ND(Γ) seems much more natural. It is a stronger condition
+as I now show. By Lemma 4.32, Λ · WQ(Γ) ⊂ WQ(Γ). Let HΓ
+1 be a horosphere of Γ, based at ξ ∈ WQ(Γ),
+and let γx0 ∈ HΓ
+1 be some point on the metric lattice of Γ · x0 on HΓ
+1 . There is an element λ ∈ Λ such that
+d(λx0, γx0) < D. Moreover, since Λ ⊂ ND(Γ) one knows that the parallel horoball that lies D-deep inside
+HBΓ
+1 is Λ-free.
+Let λ′ ∈ Λ be an arbitrary element of Λ, and consider λ′ · HΓ
+1 .
+The last statement in the previous
+paragraph is Λ-invariant, and so the horoball that lies D-deep inside λ′ · HBΓ
+1 is Λ-free.
+The fact that
+Γ ⊂ ND(Λ) then implies that the parallel horoball that lies 2D deep inside λ′HBΓ
+1 is Γ-free. Let HΓ
+2 be the
+horosphere of Γ that is based at λ′ξ. The last statement amounts to saying that HΓ
+2 lies at most 2D-deep
+inside λ′HBΓ
+1 . On the other hand, one has d(λ′λx0, λ′HΓ
+1 ) = d(λx0, HΓ
+1 ) ≤ D, so there is a Λ-orbit point that
+lies within D of λ′HΓ
+1 . The parallel horoball that lies D-deep inside HBΓ
+2 must also be Λ-free, so I conclude
+that HΓ
+2 must be contained in the parallel horoball to λ′HBΓ
+1 which contains it and that is at distance D
+from it. I conclude that d(λ′HΓ
+1 , HΓ
+2 ) ≤ 2D, and so that Λ quasi-preserves X0.
+Remark 4.35. It is interesting to note that Schwartz’s arguments are similar in spirit to my arguments
+in Section 4.2. In fact, one could also prove Theorem 4.10 using the same type of arguments that appear
+repeatedly in section 4.2, namely by moving Λ-free horoballs around the space, specifically the proof of
+Proposition 4.30. I do not present it here.
+Higher rank.
+Eskin’s proof is ergodic, and based on results of Mozes [42] and Shah [54]. I produce it here
+without the necessary preliminaries, which are standard.
+Proof of Theorem 4.9. To prove that Λ is a lattice amounts to finding a finite non-zero G-invariant measure
+on Λ\G. By Theorem 2 in [42], if P ≤ G is a parabolic subgroup then every P-invariant measure on Λ\G is
+automatically G-invariant. Fix a minimal parabolic subgroup P ≤ G and let µ0 be some fixed probability
+measure on Λ\G. Since P is amenable it admits a tempered Følner sequence Fn ⊂ P, and one can average
+µ0 along each Fn to get a sequence of probability measures µn. The weak* compactness of the unit ball
+in the space of measures on Λ\G implies that there exists a weak* limit µ of the µn. The measure µ is
+automatically a finite P-invariant measure. It remains to show that µ is not the zero measure. To see this
+it is enough to show that for some compact set CΛ ⊂ Λ\G and some Λg = x ∈ Λ\G, one has
+0 < lim inf
+n
+1
+|Fn|
+�
+Fn
+1CΛ(xp−1)dp
+(4)
+Fix some compact neighbourhood CΓ ⊂ Γ\G of the trivial coset Γe. The hypothesis Γ ⊂ ND(Λ) implies
+that there is a corresponding compact neighbourhood CΛ ⊂ Λ\G of the trivial coset Λe such that for any
+p ∈ P, it holds that Γgp−1 ∈ CΓ ⇒ Λgp−1 ∈ CΛ (simply take CΛ to be the D + 1-blowup of CΓ). The action
+of P on Γ\G is uniquely ergodic, therefore
+0 < µΓ(CΓ) = lim
+n
+1
+|Fn|
+�
+Fn
+1CΓ(Γp−1)dp
+33
+
+where µΓ denotes the natural G-invariant measure on Γ\G. The defining property of CΛ ensures that
+Inequality (4) is satisfied, implying that µ is a non-zero P-invariant probability measure on Λ\G. I conclude
+that µ is also G-invariant, and that Λ is a lattice. If moreover Λ ⊂ ND(Γ), one may use Shah’s Corollary
+[54] to conclude that Λ is commensurable to Γ.
+4.4
+Translating Geometry into Algebra
+The goal of this section is to prove that the results of Section 4.2 imply that Λ satisfies the hypotheses
+of the Benoist-Miquel criterion Theorem 4.4.
+Namely, that Λ is Zariski dense, and that it intersects a
+horospherical subgroup in a cocompact indecomposable lattice.
+These are algebraic properties, and the
+proof that Λ satisfies them is in essence just a translation of the geometric results of Section 4.2 to an
+algebraic language. The geometric data given by Section 4.2 is that for some horosphere H bounding a
+Λ-free horoball, Λ ∩ StabG(H) · x0 intersects H in a cocompact metric lattice (Proposition 4.12), and that
+the set of Λ-conical limit points contains the set of Γ-conical limit points (Corollary 4.27). Note that since
+K is compact the former implies that Λ ∩ StabG(H) is a uniform lattice in StabG(H).
+4.4.1
+A Horospherical Lattice
+I assume that Λ ∩ StabG(H) is a lattice in StabGH, and I want to show that Λ intersects a horospherical
+subgroup U of G in a lattice. This step requires quite a bit of algebraic background, which I give below in full.
+In short, the first goal is to show that StabG(H) admits a subgroup U ≤ StabG(H) that is a horospherical
+subgroup of G. A lemma of Mostow (Lemma 4.36 below) allows to conclude that Λ intersects U in a lattice.
+Lemma 4.36 (Lemma 3.9 in [40]). Let H be a Lie group having no compact connected normal semisimple
+non-trivial Lie subgroups, and let N be the maximal connected nilpotent normal Lie subgroup of H. Let
+Γ ≤ H be a lattice. Then N/N ∩ Γ is compact.
+Remark 4.37. In the original statement Mostow uses the term ‘analytic group’, which I replaced here with
+‘connected Lie subgroup’. This appears to be Mostow’s definition of an analytic group. See e.g. Section
+10, Chapter 1 in [33]. In Chevalley’s Theory of Lie Groups, he defines a Lie group as a locally connected
+topological group whose identity component is an analytic group (Definition 1, Section 8, Chapter 4 in [12]),
+and proves (Theorem 1, Section 4, Chapter 4 therein) a 1-1 correspondence between analytic subgroups of
+an analytic group and Lie subalgebras of the corresponding Lie algebra.
+Lemma 4.36 lays the rationale for the rest of this section. Explicitly, I prove that StabG(H) admits
+a subgroup that is a horospherical subgroup U of G (Corollary 4.39), and that U is maximal connected
+nilpotent normal Lie subgroup of StabG(H) (Corollary 4.45).
+In order to use Lemma 4.36, I show that the horospherical subgroup Nξ is a maximal normal nilpotent
+connected Lie subgroup of StabG(H)◦, and that StabG(H)◦ admits no compact normal factors. This requires
+to establish the structure of StabG(H)◦.
+Definition 4.38. In the notation ht
+ξ = exp(tX) and Aξ = exp
+�
+Z(X) ∩ p
+�
+of Proposition 2.7, define A⊥
+ξ to
+be the codimension-1 submanifold of Aξ that is orthogonal to {hξ(t)}t∈R (with respect to the Killing form
+in the Lie algebra).
+Claim. Every element a ∈ A⊥
+ξ stabilizes H = H(x0, ξ).
+Proof. An element in Aξ is an element that maps x0 to a point on a flat F ⊂ X that contains the geodesic
+ray [x0, ξ). If a ∈ A⊥
+ξ , then the geodesic [x0, ax0] is orthogonal to [x0, ξ), and lies in F. From Euclidean
+geometry and structure of horospheres in Euclidean spaces, it is clear that ax0 ∈ H(x,ξ). Since a ∈ Gξ, this
+means aH = H(ax0, ξ) = H(x, ξ) = H.
+Corollary 4.39. Let H be a horosphere based at ξ. Then StabG(H)◦ = (KξA⊥
+ξ )◦Nξ, and in particular it
+contains a horospherical subgroup of G. Moreover, StabG(H)◦ is normal in StabG(ξ)◦ and acts transitively
+on H.
+34
+
+Proof. Clearly (KξA⊥
+ξ )◦Nξ is a codimension-1 subgroup of StabG(ξ)◦. Since StabG(H) ̸= StabG(ξ) (e.g.
+ht
+ξ /∈ StabG(H) for t ̸= 0), it is enough to show that (KξA⊥
+ξ )◦Nξ ≤ StabG(H). Let kan ∈ (KξA⊥
+ξ )◦Nξ. It
+fixes ξ, so it is enough to show that kanx0 ∈ H. Since k ∈ Kξ and kx0 = x0, it stabilizes H. From Claim 4.4.1
+a ∈ StabG(H). So it remains to check that Nξ stabilizes H, but this is more or less the definition: fixing
+a base point x0, the horospheres based at ξ are parameterized by R.
+Denote them by {Ht}t∈R, where
+H = H0. In this parameterization, any element g ∈ Gξ acts on {Ht}t∈R by translation. I can thus define for
+g ∈ StabG(ξ) the real number l(g) to be that number for which gHt = Ht+l(g). Clearly l
+�
+hξ(t)
+�
+= t. The
+element n fixes ξ, so one has
+h−t
+ξ nht
+ξH0 = h−t
+ξ Ht+l(n) = Ht+l(n)−t = Hl(n)
+The fact that n ∈ Ker(Tξ), i.e. that limt→∞ h−t
+ξ nht
+ξ = eG readily implies that necessarily l(n) = 0. I
+conclude that (KξA⊥
+ξ )◦Nξ = StabG(H)◦, as wanted.
+Next recall that StabG(H)◦ acts transitively on X. Let x, y ∈ H, and consider g ∈ StabG(ξ)◦ with gx = y.
+Writing an element g ∈ Gξ as kata⊥n ∈ Kξht
+ξA⊥
+ξ Nξ, the argument above shows that kht
+ξa⊥nH0 = H0 if
+and only if t = 0, i.e., if and only if g ∈ StabG(H)◦.
+Finally, let g ∈ StabG(ξ) and h ∈ StabG(H). By the discussion above h · Ht = Ht for all t ∈ R. Clearly
+−l(g) = l(g−1), and therefore
+ghg−1H0 = ghH−l(g) = g · H−l(g) = H0
+Therefore StabG(H) is normal in StabGξ, and the same is true for the respective identity components.
+Corollary 4.40. StabG(H)◦ is a connected Lie group with no connected compact normal semisimple non-
+trivial Lie subgroups.
+Proof. Every compact subgroup of G fixes a point. Let H ≤ G be some closed subgroup. It is standard to
+note that a normal N ≤ H that fixes a point x ∈ X must fix every point in the orbit H · x: hnh−1hx = hx.
+Since H = StabG(H)◦ acts transitively on H, it shows that a normal compact subgroup of StabG(H)◦ fixes
+every point in H. An isometry fixing a horosphere pointwise while fixing its base point is clearly the identity,
+proving the claim.
+The following fact is well known but I could not find it in the literature.
+Corollary 4.41. A horosphere in X is not convex.
+Proof. Let H′ be some horosphere in X, with base point ζ ∈ X(∞), and assume towards contradiction that
+it is convex. Fix x ∈ H′ and a′
+t the one parameter subgroup with η′(∞) = a′
+tx, and denote H′
+t = H(a′
+tx, ζ).
+Let eG ̸= n ∈ Nζ (Nζ defined with respect to a′
+t in a corresponding Langlands decomposition), and consider
+the curve η′
+n(t) := a′
+tnx. I claim that this is a geodesic. On the one hand, the fact that H′ is convex implies
+that the geodesic segment [x, nx] is contained in H′. Therefore a′
+t[x, nx] = [a′
+tx, a′
+tnx] ⊂ H′
+t. More generally
+it is clear that because a′
+tH′
+s = H′
+s+t it holds that H′
+t is convex for every t as soon as it is convex for some t.
+On the other hand, for every point y ∈ [x, nx], d(y, H′
+t) = t, and more generally for any y ∈ [a′
+snx, a′
+sx]
+it holds that d(y, H′
+t) = |s − t|. In particular this is true for η′
+n(t) = yt := a′
+tnx. I get that d
+�
+η′
+n(t), η′
+n(s)
+�
+=
+|s − t|. Therefore η′
+n is a geodesic (to be pedantic one has to show that η′
+n is a continuous curve, which is a
+result of the fact that a′
+t is a one parameter subgroup of isometries). Clearly
+d
+�
+η′
+n(t), η′(t)
+�
+= d(a′
+tnx, a′
+tx) = d(nx, x)
+and therefore η′
+n is at uniformly bounded distance to η′. This bounds d(ηn, η′
+n) as bi-infinite geodesics,
+i.e. for all t ∈ R, not just as infinite rays. The Flat Strip Theorem (Theorem 2.13, Chapter 2.2 in [11]), then
+implies that the geodesics ηn, η′
+n bound a flat strip: an isometric copy of R × [0, l] (where l = d(x, nx)).
+Up to now I did not use the fact that n ∈ Nξ, only that the point nx lies on a geodesic that is contained
+in H′ = H′
+0. Therefore the entire bi-infinite geodesic that is determined by [x, nx] lies on a 2-dimensional
+flat F that contains η′. The two elements n, a′
+t therefore admit nx, a′
+tx ∈ F. It is a fact that two such
+elements must commute. I can conclude therefore that [n, a′
+t] = eG, which contradicts the fact that that
+n ∈ Nζ = Ker(Tζ).
+35
+
+Lemma 4.42 (Theorem 11.13 in [51]). Let N be a connected real Lie group. Then Lie(N) is a nilpotent Lie
+algebra if and only if N is a nilpotent Lie group.
+Proposition 4.43 (Proposition 13, Section 4, Chapter 1 in [7]). In the notation of Proposition 2.7, nξ =
+Lie(Nξ) is a maximal nilpotent ideal in gξ = Lie(Gξ).
+Remark 4.44.
+1. The presentation of nξ in [8] is given by means of the root space decomposition of
+StabG(ξ), that appears in Proposition 2.17.13 in [20].
+2. There are two main objects in the literature that are referred to as the nilpotent radical or the nilradical
+of a Lie algebra. These are: (a) the maximal nilpotent ideal of the Lie algebra, and (b) the intersection
+of the kernels of all irreducible finite-dimensional representations.
+Proposition 13 in Section 4 of
+Chapter 9 in [7] shows that in the case of Lie algebras of parabolic Lie groups, these notions coincide.
+Corollary 4.45. Nξ is a maximal connected nilpotent normal Lie subgroup of the identity connected com-
+ponent StabG(H)◦.
+Proof. Lemma 4.42 implies Nξ is nilpotent. Since StabGH ⊳ StabG(ξ), every normal subgroup of
+StabG(H) containing Nξ is in fact a normal subgroup of StabG(ξ), still containing Nξ. It remains to
+prove maximality of Nξ among all connected nilpotent normal Lie subgroups of StabG(ξ). Any such
+subgroup N ′ ⊳ StabG(ξ) gives rise to an ideal n′ of gξ = Lie
+�
+StabG(ξ)
+�
+, and by Lemma 4.42 it is a
+nilpotent ideal. Therefore by Proposition 4.43 it is contained in nξ = Lie(Nξ), implying that N ′ ≤ Nξ.
+Corollary 4.46. A lattice in StabG(H) intersects the horospherical subgroup Nξ in a lattice.
+Proof. Corollaries 4.40 and 4.45 imply that the pair Nξ ⊳ StabG(H) satisfy the hypotheses of Mostow’s
+Lemma 4.36.
+4.4.2
+Indecomposable Horospherical Lattices
+The Benoist and Miquel criterion requires the horospherical lattice to be indecomposable. It is shown in [5]
+that if this lattice is contained in a Zariski dense discrete subgroup, then the indecomposability condition is
+equivalent to irreducibility of the ambient group. The precise definitions and statements are as follows.
+Definition 4.47 (Definition 2.14 in [5]). For a semisimple real algebraic Lie group G and U a horospherical
+subgroup of G, let ∆U be a lattice in U.
+1. ∆U is irreducible if for any proper normal subgroup N of G◦, one has ∆U ∩ N = {e}.
+2. ∆U is indecomposable if one cannot write G◦ as a product G◦ = N ′N ′′ of two proper normal subgroups
+N ′, N ′′ ⊳ G with finite intersection such that the group
+∆′
+U := (∆U ∩ N ′)(∆U ∩ N ′′)
+has finite index in ∆U.
+Definition 4.48 (See Section 2.4.1 in [5]). Let G be a semisimple real algebraic Lie group. A discrete
+subgroup Λ ≤ G is said to be irreducible if, for all proper normal subgroups N ⊳ G, the intersection Λ ∩ N
+is finite.
+Lemma 4.49 (Lemma 4.3 in [5]). Let G be a semisimple real algebraic Lie group, U ⊂ G a non-trivial
+horospherical subgroup, and ∆U ≤ U a lattice of U which is contained in a discrete Zariski dense subgroup
+∆ of G. Then the following are equivalent:
+1. ∆ is irreducible.
+2. ∆U is irreducible.
+3. ∆U is indecomposable.
+36
+
+4.4.3
+Zariski Density
+The last requirement is for Λ to be Zariski dense. I use a geometric criterion which is well known to experts.
+Lemma 4.50 (Proposition 2 in [31]). Let X be a symmetric space of noncompact type, G = Isom(X)◦. A
+subgroup ∆ ≤ G is Zariski dense if and only if:
+1. ∆ does not globally fix a point in X(∞), i.e. ∆ ̸≤ StabG(ζ) for any ζ ∈ X(∞).
+2. The identity component of the Zariski closure of ∆ does not leave invariant any proper totally geodesic
+submanifold in X.
+In the proof I use several facts - mostly algebraic, and two geometric. I warmly thank Elyasheev Leibtag
+for his help and erudition in algebraic groups. The first property I need is very basic.
+Lemma 4.51. Let ∆ ≤ G be a discrete subgroup, and let H ≤ G be the Zariski closure of ∆. Then ∆ ∩ H◦
+is of finite index in ∆.
+Proof. H◦ is normal and of finite index in H.
+The following fact is probably known to experts. It appears in a recent work by Bader and Leibtag[2].
+Lemma 4.52 (Lemma 3.9 in [2]). Let k be a field, G a connected k algebraic group, P ≤ G = G(R) a
+parabolic subgroup. Then the centre of G contains the centre of P.
+Still on the algebraic side, I need a Theorem of Dani, generalizing the Borel Density Theorem.
+Theorem 4.53 (See [15]). Let S be a real solvable algebraic group. If S = S(R) is R-split, then every lattice
+ΓS ≤ S is Zariski dense.
+Remark 4.54. It is a fact (see Theorem 15.4 and Section 18 in [6]) that:
+1. Every unipotent group over R is R-split.
+2. For a field k of characteristic 0, a solvable linear algebraic k-group is k-split if and only if its maximal
+torus is k-split.
+Finally I need two geometric facts. The first is a characterization determining when does a unipotent
+element belongs to Nζ for some ζ ∈ X(∞).
+Proposition 4.55 (Proposition 4.1.8 in [20]). Let X be a symmetric space of noncompact type and of higher
+rank, n ∈ G = Isom(X)◦ a unipotent element, and ζ ∈ X(∞). The following are equivalent:
+1. For Nζ as in Proposition 2.7, n ∈ Nζ.
+2. For some geodesic ray η with η(∞) = ζ it holds that limt→∞ d
+�
+nη(t), η(t)
+�
+= 0.
+3. For every geodesic ray η with η(∞) = ζ it holds that limt→∞ d
+�
+nη(t), η(t)
+�
+= 0.
+The last property I need is a characterization of the displacement function for unipotent elements.
+Proposition 4.56 (See proof of Proposition 3.4 in [4]). Let X be a symmetric space of noncompact type,
+ζ ∈ X(∞) some point and n ∈ Nζ an element of the unipotent radical of StabG(ζ).
+The displacement
+function x �→ d(nx, x) is constant on horospheres based at ζ, and for every ε > 0 there is a horoball HBε
+based at ζ such that d(nx, x) < ε for every x ∈ HBε.
+Corollary 4.57. Assume that:
+1.
+�
+Λ ∩ StabG(H)
+�
+· x0 is a cocompact metric lattice in a horosphere H ⊂ X bounding a Λ-free horoball.
+37
+
+2. Every Γ-conical limit point is a Λ-conical limit point.
+Then Λ is Zariski dense.
+Proof. I show the criteria of Lemma 4.50 are met, starting with Λ ̸≤ StabG(ζ) for any ζ ∈ X(∞). To this
+end, I first prove that Λ · x0 is not contained in any bounded neighbourhood of any horosphere H′. Let
+ξ′ be the base point of H′. By Hattori’s Lemma 2.31 (and Remark 2.32), it is enough to find a Λ-conical
+limit point ζ′ with dT (ξ′, ζ′) ̸= π
+2 . Take some ζ′′ ∈ X(∞) at Tits distance π of ξ′, i.e. take a flat F on
+which ξ′ lies and let ζ′′ be the antipodal point to ξ′ in F.
+Fix ε =
+π
+4 .
+By Proposition 2.3, there are
+neighbourhoods of the cone topology U, V ⊂ X(∞) of ξ′, ζ′′ (respectively) so that every point ζ′ ∈ V admits
+dT (ξ′, ζ′) ≥ dT (ξ′, ζ′′)− π
+4 = 3
+4π. Recall that the set of Γ-conical limit points is dense (in the cone topology),
+so the second hypothesis implies there is indeed a Λ-conical limit point in V and therefore at Tits distance
+different (in this case larger) than π
+2 from ξ′. I conclude that Λ · x0 is not contained in any bounded metric
+neighbourhood of any horosphere of X.
+Assume towards contradiction that Λ ≤ StabG(ζ).
+I show that this forces Λ ∩ Nζ ̸= ∅. By Propo-
+sition 4.55 it is enough to find a unipotent element λ ∈ Λ and a geodesic η with η(∞) = ζ such that
+limt→∞ d
+�
+λη(t), η(t)
+�
+= 0. Let F be a maximal flat with ξ, ζ ∈ F(∞), x ∈ F some point and X, Y ∈ a ≤ p
+two vectors such that exp(tY ) = η(t) for the unit speed geodesic η = [x, ζ), and exp(tX) = η′(t) for the unit
+speed geodesic η′ = [x, ξ) (where a ≤ p a maximal abelian subalgebra in a suitable Cartan decomposition
+g = p ⊕ k). Let StabG(ξ) = KξAξNξ be the decomposition described in Proposition 2.7 with respect to Y
+(notice that Nξ does not depend on choice of Y , see item 3 of Proposition 2.17.7 in [20]).
+The assumption that Λ ≤ StabG(ζ) implies that for any λ ∈ Λ the distance d
+�
+λη(t), η(t)
+�
+either tends
+to 0 as t → ∞ or is uniformly bounded for t ∈ R. In the latter case there is some constant c > 0 for which
+d
+�
+λη(t), η(t)
+�
+= c for all t ∈ R. As in the proof of Corollary 4.41, the Flat Strip Theorem (Theorem 2.13,
+Chapter 2.2 in [11]) implies that λ and at := exp(tY ) commute.
+From the first hypothesis of the statement and Mostow’s result (Corollary 4.46) I know that Λ ∩ Nξ is a
+cocompact lattice in Nξ (attention to subscripts). Therefore Λ ∩ Nξ is Zariski dense in Nξ (Theorem 4.53).
+Moreover, since commuting with an element is an algebraic property, an element g ∈ G that commutes with
+Λ∩Nξ must also commute with its Zariski closure, namely with Nξ. This means that if at commutes with all
+Λ∩Nξ then it commutes with Nξ, i.e. atn = nat for all t ∈ R and all n ∈ Nξ. I know that at ∈ Aξ commutes
+with both Kξ and Aξ (Proposition 2.7) therefore if at also commutes with Nξ then at lies in the centre
+of StabG(ξ). This means that at is central in G (Lemma 4.52). For a group G with compact centre this
+cannot happen, so there is indeed some unipotent element λ ∈ Λ ∩ Nξ for which limt→∞ d
+�
+λη(t), η(t)
+�
+= 0.
+I conclude from Proposition 4.55 that Λ ∩ Nζ ̸= ∅.
+The first paragraph of the proof implies in particular that Λ·x0 does not lie in any bounded neighbourhood
+of a horosphere H′ based at ζ.
+The assumption that Λ ⊂ StabG(ζ) implies that every λ ∈ Λ acts by
+translation on the filtration {H′
+t}t∈R by horospheres based at ζ. Therefore as soon as Λ · x0 ̸⊂ Ht for some
+t ∈ R one concludes that ζ is a horospherical limit point of Λ, i.e. that every horoball based at ζ intersects
+the orbit Λ · x0.
+By Proposition 4.56 it holds that for a unipotent element g ∈ Nζ the displacement function x �→ d(gx, x)
+depends only on the horosphere H′
+t in which x lies and that, for xt ∈ H′
+t it holds that limt→∞ d(gxt, xt) = 0
+(up to reorienting the filtration t ∈ R so that η(t) ∈ H′
+t).
+For a non-trivial element λζ ∈ Λ ∩ Nζ the
+previous paragraph therefore yields a sequence of elements λn ∈ Λ such that limn→∞ d(λζλnx0, λnx0) = 0,
+contradicting the discreteness of Λ. I conclude that Λ ̸≤ StabG(ζ) for every ζ ∈ X(∞).
+Assume that H :=
+�
+Λ
+Z�◦, the identity connected component of the Zariski closure of Λ, stabilizes a totally
+geodesic submanifold Y ⊂ X. By Lemma 4.51, Λ0 := Λ ∩ H is of finite index in Λ, therefore Λ0 ∩ StabG(H)
+is also a cocompact lattice in StabG(H). The fact that
+�
+Λ0 ∩ StabG(H)
+�
+· x0 is a cocompact metric lattice
+in H readily implies that
+�
+Λ0 ∩ StabG(H)
+�
+· y is a cocompact metric lattice in Hy = H(y, ξ). This goes to
+show that there is no loss of generality in assuming x0 ∈ H ∩ Y . It follows that Λ0 ∩ StabG(H) · x0 ⊂ Y ∩ H,
+and therefore H ⊂ ND(Y ) for some D > 0. A horosphere is a codimension-1 submanifold, implying that Y
+is either all of X or of codimension-1. The latter forces Y = H, which is impossible since H is not totally
+geodesic (H is not convex, see Corollary 4.41). I conclude that H does not stabilize any totally geodesic
+38
+
+proper submanifold, and hence that Λ is Zariski dense.
+4.5
+Proof of Theorem 4.1
+I now complete the proof of the main sublinear rigidity theorem for Q-rank 1 lattices.
+Proof of Theorem 4.1. If {dγ}γ∈Γ is bounded, then Λ is a lattice by Corollary 4.7 or Theorem 4.9, depending
+on the R-rank of G.
+If {dγ}γ∈Γ is unbounded, then Proposition 4.12 and Corollary 4.27 both hold. In R-rank 1 the proof
+again follows immediately from Corollary 4.7 using Lemma 2.16 and Corollary 4.27.
+In higher rank, Section 4.4 allows one to conclude that Λ is an irreducible, discrete, Zariski dense subgroup
+that contains a horospherical lattice. By Theorem 4.4, this renders Λ a lattice. It is a Q-rank 1 lattice as a
+result of Theorem 2.20.
+Remark 4.58. The sublinear nature of the hypothesis in Theorem 1.6 induces coarse metric constraints.
+A horospherical lattice on the other hand is a very precise object. It is not clear how to produce unipotent
+elements in Λ, or even general elements that preserve some horosphere. The proof sketched above produces
+a whole lattice of unipotent elements in Λ (this is Corollary 4.46); it is also the only proof that I know which
+produces even a single unipotent (or parabolic) element in Λ.
+5
+Lattices with Property (T)
+Recall that a lattice in a locally compact group G has property (T) if and only if G has property (T). In
+this section I prove:
+Theorem 5.1. Let G be a real centre-free semisimple Lie group without compact factors, Γ ≤ G a lattice,
+Λ ≤ G a discrete subgroup such that Γ ⊂ Nu(Λ) for some sublinear function u. If Γ has property (T), then
+Λ is a lattice.
+As in the case of uniform lattices, lattices with property (T) admit the stronger version of ε-linear rigidity,
+for suitable ε = ε(G):
+Theorem 5.2. Let G be a real centre-free semisimple Lie group without compact factors, Γ ≤ G a lattice
+and Λ ≤ G a discrete subgroup. If Γ has property (T) then there exists ε = ε(G) > 0 depending only on G
+such that if Γ ⊂ Nu(Λ) for some function u(r) ⪯∞ εr, then Λ is a lattice.
+Clearly Theorem 5.2 implies Theorem 5.1. I thank Emmanuel Breuillard for suggesting this generaliza-
+tion. From now and until the end of this section, the standing assumptions are those of Theorem 5.2.
+Lattice Criterion.
+For groups with property (T) I use a criterion by Leuzinger, stating that being a
+lattice is determined by the exponential growth rate. The formulation requires a definition.
+Definition 5.3. Given a pointed metric space (X, dX, x0), denote:
+1. bX(r) = |B(x0, r)|
+2. bu
+X(r) = supx∈X |B(x, r)|
+When a group ∆ acts on a pointed metric space X, the orbit ∆ · x0 together with the metric induced
+from X is a pointed metric space (∆ · x0, dX↾∆·x0, x0). In this setting b∆·x0(r) = |BX(xo, r) ∩ ∆ · x0|. When
+the action is by isometries, i.e. ∆ ≤ Isom(X), it is straightforward to observe that this quantity does not
+depend on the centre of the ball, and so bu
+∆·x0(r) = b∆·x0(r). The pointed metric spaces of interest in this
+section are the Γ and Λ orbits in the symmetric space X = G/K.
+39
+
+Definition 5.4. Let X be a symmetric space and ∆ ≤ G = Isom(X)◦ a subgroup of isometries. The critical
+exponent of ∆ is defined to be
+δ(∆) := lim sup
+r→∞
+log
+�
+b∆·x0(r)
+�
+r
+Remark 5.5. Throughout this section there is no risk of ambiguity, and I allow myself to ease notation and
+let b∆(r) = b∆·x0(r), bu
+∆(r) = bu
+∆·x0(r).
+To a semisimple Lie group G one can associate a quantity ∥ρ∥, where ρ = ρ(G) is the half sum of positive
+roots in the root system of (g, a) (see Section 2 in [36]).
+Theorem 5.6 (Theorem 2 in [36]). Let G be a real centre-free semisimple Lie group without compact factors.
+Let ∆ be a discrete, torsion-free subgroup of G that is not a lattice. If G has Kazhdan’s property (T ), then
+there is a constant c∗(G) (depending on G but not on ∆) such that δ(∆) ≤ 2∥ρ∥ − c∗(G).
+It is known that the critical exponent of a discrete subgroup ∆ ≤ G is bounded above by 2∥ρ∥ (see
+Section 2.2 in [36]). Moreover, every lattice Γ ≤ G admits δ(Γ) = 2∥ρ∥ (Example 2.3.5 in [36], Theorem C
+in [1]). Combining these facts with Theorem 5.6 yield:
+Theorem 5.7 (Theorem B in [36]). Let G be a real centre-free semisimple Lie group without compact
+factors. Let ∆ be a discrete, torsion-free subgroup of G. If G has Kazhdan’s property (T ), then ∆ is a lattice
+iff δ(∆) = 2∥ρ∥.
+Line of Proof and the Use of ε-Linearity.
+The proof of Theorem 5.1 goes by showing that ε-linear
+distortion cannot decrease the exponential growth rate by much. This fact is essentially manifested in a
+proposition by Cornulier [13], stated here in Proposition 5.8. This is the only use I make of ε-linearity, and
+the computations involved are straightforward. Theorem 5.2 is then an immediate consequence of Leuzinger’s
+criterion Theorem 5.6.
+5.1
+Proof of Theorem 5.1
+In his study on SBE maps, Cornulier proves the following growth discrepancy result.
+Proposition 5.8 (Proposition 3.6 in [13]). Let X, Y be two pointed metric spaces. Let u be a non-decreasing
+sublinear function and p : X → Y a map such that for some L, R0 > 0:
+1. |p(x)| ≤ max(|x|, R0), i.e. p(|x|) ≤ |x| for all large enough x ∈ X.
+2. dY
+�
+p(x), p(x′)
+�
+≥ 1
+LdX(x, x′) − u(max{|x|, |x′|})
+Then for all r > R0, bY (r) ≥ bX(r)/bu
+X
+�
+L · u(r)
+�
+.
+I need a slightly modified version of Proposition 5.8:
+Proposition 5.9. Let X, Y be two pointed metric spaces. Let u be a non-decreasing function that admits
+u(r) ⪯∞ εr for some ε < 1, and p : X → Y a map such that for some L, R0 > 0:
+1. |p(x)| ≤ max(|x| + u(|x|), R0), i.e. |p(x)| ≤ |x| + u(|x|) for all large enough x ∈ X.
+2. dY
+�
+p(x), p(x′)
+�
+≥ 1
+LdX(x, x′) − u(max{|x|, |x′|})
+Then for all r > R0,
+bY (r) ≥ bX
+�
+r − u(r)
+�
+/bu
+X
+�
+L · u
+�
+r − u(r)
+��
+Proof. Repeat verbatim the proof for Proposition 3.6. in [13].
+40
+
+Corollary 5.10. Let u(r) ⪯∞ ε · r for some ε < 1
+2. Assume that Γ ⊂ Nu(Λ). Then δ(Λ) ≥ (1 − 4ε) · δ(Γ).
+Remark 5.11. Restricting to ε < 1
+2 stems from a 2 factor that appears in the proof and could possibly be
+dropped using a slightly more sophisticated approach. For my needs this is more than enough, since in any
+case I eventually restrict attention to a small interval around 0.
+Corollary 5.10 can be formulated in a slightly more general fashion.
+Using the notation δ(W) :=
+lim supr→∞
+log
+�
+bW (r)
+�
+r
+for a general subset W in a general metric space X, the following general version
+holds:
+Corollary 5.12. Let (X, x0) be a pointed metric space, Y, Z ⊂ X two subsets, and u(r) ⪯∞ εr for some
+ε < 1
+2. Assume that bu
+Z(r) = bZ(r). If Z ⊂ Nu(Y ), then δ(Y ) ≥ (1 − 4ε) · δ(Z). Moreover, if u is sublinear,
+then δ(Z) = δ(Y ).
+In particular, Corollary 5.10 holds even when the group G does not have property (T ). Corollary 5.10
+follows from Corollary 5.12 because the fact that Γ is a group of isometries implies bu
+Γ(r) = bΓ(r).
+Proof (Corollary 5.12). Observe that the closest point projection pY : Z → Y defined by z �→ zy for some
+point in the closed ball zy ∈ B
+�
+z, u(|z|)
+�
+admits:
+1. |yz| ≤ |z| + u(|z|)
+2. d
+�
+yz, yz′�
+≥ d(z, z′) − 2u(max{|z|, |z′|})
+The first item follows from triangle inequality: |yz| ≤ d(yz, z) + d(z, x0). The second item follows from
+the quadrilateral inequality, i.e., using triangle inequality twice along the quadrilateral [z, z′, yz′, yz].
+The above properties allow me to use Proposition 5.9 with constant L = 1 and function u′ = 2u to get
+bY (r) ≥ bZ
+�
+r − u′(r)
+�
+/bu
+Z
+�
+u′�
+r − u′(r)
+��
+Since I assume bu
+Z = bZ, I can omit the superscript u in the last expression. Recalling the definition
+δ(W) = lim supr→∞
+bW (r)
+r
+, it remains to prove:
+lim sup
+r→∞
+1
+r · log
+�
+bZ
+�
+r − u′(r)
+�
+/bZ
+�
+u′�
+r − u′(r)
+���
+≥ (1 − 4ε) · δ(Z)
+The proof of this inequality involves nothing more than log rules and arithmetic of limits:
+lim sup
+r→∞
+1
+r · log
+�
+bZ
+�
+r − u′(r)
+�
+/bZ
+�
+u′�
+r − u′(r)
+���
+= lim sup
+r→∞
+1
+r ·
+�
+log
+�
+bZ
+�
+r − u′(r)
+��
+− log
+�
+bZ
+�
+u′�
+r − u′(r)
+����
+≥ lim sup
+r→∞
+�
+1
+r · log
+�
+bZ
+�
+r − u′(r)
+��
+− lim sup
+s→∞
+�1
+s log
+�
+bZ
+�
+u′�
+s − u′(s)
+�����
+= lim sup
+r→∞
+1
+r · log
+�
+bZ
+�
+r − u′(r)
+��
+− lim sup
+s→∞
+1
+s log
+�
+bZ
+�
+u′�
+s − u′(s)
+���
+≥ lim sup
+r→∞
+1
+r · log
+�
+bZ
+�
+r − 2εr
+��
+− lim sup
+s→∞
+1
+s log
+�
+bZ
+�
+2εs
+��
+= (1 − 2ε)δ(Z) − 2εδ(Z) = (1 − 4ε)δ(Z)
+(5)
+Below I justify the steps in the above inequalities:
+41
+
+1. First equality is by rules of log.
+2. Second and third inequalities are by arithmetic of limits: let (an)n, (bn)n be two sequences of positive
+numbers, and A = lim supn an, B = lim supn bn. Then lim sup(an − bn) ≥ lim supn(an − B) = A − B.
+3. Fourth inequality: u′(r) < 2ε(r) for all large enough r.
+4. Fifth equality: definition of δ.
+This completes the proof in the general case, which is what is needed for the proof of Theorem 5.2.
+For the more refined statement in the case u is sublinear, one has to show a bit more. From inequality 5
+(specifically from the fourth line of the inequality) it is clearly enough to prove:
+1. lim supr→∞
+1
+r · log
+�
+bZ
+�
+r − u′(r)
+��
+= δ(Z).
+2. lim sups→∞
+1
+s log
+�
+bZ
+�
+u′�
+s − u′(s)
+���
+= 0.
+Starting from the second item, indeed it holds that
+1
+s log
+�
+bZ
+�
+u′�
+s − u′(s)
+���
+= u′�
+s − u′(s)
+�
+s
+·
+log
+�
+bZ
+�
+u′�
+s − u′(s)
+���
+u′�
+s − u′(s)
+�
+Clearly lim sup of the right factor in the above product is bounded by δ(Z), and in particular it is
+uniformly bounded. On the other hand sublinearity of u′ implies that the left factor tends to 0. I conclude
+that this product tends to 0 as s tends to ∞.
+It remains to prove lim supr→∞
+1
+r · log
+�
+bZ
+�
+r − u′(r)
+��
+= δ(Z). In a similar fashion,
+1
+r log
+�
+bZ
+�
+r − u′(r)
+��
+= r − u′(r)
+r
+·
+log
+�
+bZ
+�
+r − u′(r)
+��
+r − u′(r)
+The left factor limits to 1 by sublinearity of u′. The right factor is nearly the expression in the definition
+of δ(Z), and I want to prove that indeed taking lim sup of it equals δ(Z). A priori {r − u′(r)}r∈R>0 is just a
+subset of R>0, so changing variable and writing t := r−u′(r) requires a justification. But there is no harm in
+assuming that u′ is a non-decreasing continuous function, hence R≥R ⊂ {r − u′(r)}r∈R>0 for some R ∈ R>0.
+Therefore for any sequence rn → ∞ there is a sequence r′
+n with rn = r′
+n − u′(r′
+n) for all large enough n (note
+that in particular r′
+n → ∞). In the other direction, for every sequence r′
+n → ∞ there is clearly a sequence
+rn → ∞ for which rn = r′
+n − u′(r′
+n). I conclude
+lim sup
+r→∞
+log
+�
+bZ
+�
+r − u′(r)
+��
+r − u′(r)
+= lim sup
+r→∞
+1
+r · log
+�
+bZ(r)
+�
+= δ(Z)
+This completes the proof.
+Proof of Theorem 5.2. Define ε(G) = c∗(G)
+4·2∥ρ∥, and assume u(r) ⪯∞ ε(G) · r. Notice that ε(G) < 1
+2, and since
+δ(Γ) = 2∥ρ∥ Corollary 5.10 gives
+δ(Λ) ≥
+�
+1 − 4ε(G)
+�
+· 2∥ρ∥ = 2∥ρ∥ − 4ε(G) · 2∥ρ∥ ≥ 2∥ρ∥ − c∗(G)
+By Theorem 5.6, Λ is a lattice.
+42
+
+Remark 5.13. The question of existence of interesting groups that coarsely cover a lattice is a key question
+that arises naturally in the context of this paper. The first question that comes to mind is whether there
+exist groups that are not commensurable to a lattice but that sublinearly, or even ε-linearly, cover one.
+Perhaps the growth rate point of view could be used to rule out groups that cover a lattice ε-linearly but
+not sublinearly.
+6
+SBE Rigidity for Lattices of Higher Q-Rank
+6.1
+Sublinear Distortion and SBE Maps
+Denote a ∨ b := max(a, b) for a, b ∈ R. For a pointed metric space (X, x0, dX) and x, x1, x2 ∈ X, denote
+|x|X := dX(x, x0) and |x1 −x2|X := dX(x1, x2) (or simply |x| and |x1 −x2| when there is no ambiguity about
+the space X).
+Following Cornulier [14], Pallier [44] makes the following definition:
+Definition 6.1. A function u : R≥0 → R is admissible if it satisfies the following conditions:
+• u is non-decreasing
+• u grows sublinearly: lim sup
+r�→∞
+u(r)
+r
+= 0.
+• u is doubling:
+u(tr)
+u(r) is bounded above for all t > 0.
+• u ≥ 1.
+The focus in this paper is condition 2, namely that the function u is strictly sublinear. I moreover require
+it to be subadditive, resulting in the following terminology which I use from now on.
+Definition 6.2. A function u : R≥0 → R is sublinear if it is admissible and subadditive, i.e. u(t + s) ≤
+u(t) + u(s) for all t, s > 0.
+From now on by an SBE I mean an (L, u)-SBE where u is sublinear in the sense of Definition 6.2.
+6.2
+SBE Rigidity
+Two finitely generated groups Γ and Λ are said to be SBE if they are SBE when viewed as metric spaces with
+some word metrics and base points eΓ, eΛ. Observe that every quasi-isometry is an SBE, and in particular
+the word metric is an SBE-invariant. An SBE admits an SBE inverse, defined as quasi-inverse maps are
+defined for quasi-isometries.
+Definition 6.3. A class of groups A is said to be SBE complete if, for every finitely generated group Λ that
+is SBE with some group Γ ∈ A, there is a short exact sequence
+1 → F → Λ → Λ1 → 1
+for a finite group F ≤ Λ and some Λ1 ∈ A.
+In this chapter I prove:
+Theorem 6.4. Let G be a real centre-free semisimple Lie group without compact or R-rank 1 factors.
+1. The class of uniform lattices of G is SBE complete.
+2. The class of non-uniform lattices of G is SBE complete.
+43
+
+Remark 6.5. The proof I present is quite indifferent to whether the lattice Γ is uniform or not. In order
+to have a unified proof and to ease notation, I fix the convention that for both uniform and non-uniform
+lattices, X0 denotes the compact core of the lattice. This just means that X = X0 in case Γ is uniform.
+My works heavily relies on that of Drut¸u in [16], where the theorems are stated for non-uniform lattices.
+Nonetheless one readily sees that her proofs work perfectly well for uniform lattices. Indeed the arguments
+of [16] are only much simpler in the uniform case.
+6.2.1
+The Quasi-Isometry Case
+The outline of the proof I present for Theorem 6.4 is identical to that of quasi-isometric rigidity, which I
+now describe briefly. The main step is that for any quasi-isometry f : X0 → X0 of the compact core of Γ,
+there exists an isometry g : X → X such that f, g are boundedly close, i.e. there is some D > 0 for which
+d
+�
+f(x), g(x)
+�
+< D for all x ∈ X0.
+Let Λ be an abstract group with a quasi-isometry q : Λ → Γ.
+Using Lubotzki-Mozes-Raghunathan
+([38], see Theorem 2.19 above), Γ is quasi-isometrically embedded in X as the compact core X0. One can
+thus extend q to a quasi-isometry q0 : Λ → X0. A conjugation trick allows to associate to each λ ∈ Λ a
+quasi-isometry fλ : X0 → X0 defined by fλ := q ◦ Lλ ◦ q−1 (Lλ : Λ → Λ is the left multiplication by λ in Λ).
+By the first paragraph, there exists gλ ∈ Isom(X) that is boundedly close to fλ. Moreover, the proof also
+shows that the bound D depends only on the quasi-isometry constants of fλ. These could be seen to depend
+only on q and not on any specific λ. From this one concludes that the map λ �→ gλ is a group homomorphism
+Φ : Λ → G. It is then straightforward to show that Φ has finite kernel and that Γ ⊂ ND
+�
+Im(Φ)
+�
+. One then
+uses Theorem 4.3.2 (for higher rank groups, see Section 4.3) to deduce that Im(Φ) is a non-uniform lattice
+in G that is commensurable to Γ.
+6.2.2
+The SBE Case
+Moving to SBE rigidity, one starts with an SBE q : Λ → Γ. The first step is to find an isometry of X that is
+close to an SBE of X0. Drut¸u’s proof is preformed in the asymptotic cone of X, which allows for a smooth
+transition to the SBE setting. Indeed, given an SBE f : X0 → X0, one can find an isometry g : X → X that
+is close to it. The difference is that in the SBE setting, these maps are only sublinearly close.
+Definition 6.6. Let (X, x0) be a pointed metric space, and f, g : X → X be two maps. Maps f, g are said
+to be sublinearly close maps on X if there is a sublinear function u such that d
+�
+f(x), g(x)
+�
+≤ u(|x|).
+Theorem 6.7 (Theorem 6.10). Let f : X0 → X0 be an SBE. Then there exists a unique isometry g ∈
+Isom(X) that is sublinearly close to f (in X0).
+From this point on, one would like to continue as in the quasi-isometry case: define the map Φ : Λ → G
+in a similar fashion and show that Γ ⊂ Nu
+�
+Im(Φ)
+�
+for some sublinear function u. That Im(Φ) is a lattice is
+then a result of Theorem 1.6, proving Theorem 6.4.
+There is however one additional obstacle that is unique to the SBE setting. Namely the SBE constants
+of fλ do depend on λ, and the resulting sublinear bound on d
+�
+fλ(x), gλ(x)
+�
+in Theorem 6.7 is not enough
+to define Φ properly. As far as I can see, one needs to get some uniform control on that bound in terms of
+the SBE constants. The following statement is enough:
+Lemma 6.8 (Lemma 6.11). Let {fr}r∈R>0 be a family of (L′, vr)-SBE maps fr : X0 → X0, where vr(s) =
+L′·v(s)+v(r) for some sublinear function v ∈ O(u) and a constant L′. Let gr be the associated isometry given
+by Theorem 6.10. Then for any x ∈ X0, there is a sublinear function ux ∈ O(u) such that d
+�
+fr(x), gr(x)
+�
+≤
+ux(r).
+This type of uniform control is often needed when working with SBE maps, see e.g. Section I.3 in Pallier’s
+thesis [43]. Using it, I am able to complete the argument as in the quasi-isometry case and prove:
+Theorem 6.9. Let G be as in Theorem 6.4. In the notations described above, the map Φ : Λ → G is a group
+homomorphism with Ker(Φ) finite, and there is a sublinear function u such that for Λ1 := Im(Φ) it holds
+that Γ ⊂ Nu(Λ1) and Λ1 ⊂ Nu(Γ).
+44
+
+Theorem 6.4 is an immediate corollary of Theorem 6.9 and Theorem 1.6.
+6.2.3
+Outline
+This section is divided into two parts that correspond to the steps of the proof. Section 6.3 deals with the
+task of finding an isometry that is sublinearly close to an SBE, and Section 6.4 establishes the properties of
+the map Φ : Λ → G. I keep Section 6.3 slim and concise. The main reason for this choice is that the proof
+of Theorem 6.10 is merely a mimic of Drut¸u’s argument in [16], or an adaptation of it to the SBE setting.
+While these adaptations are somewhat delicate, giving a complete detailed proof would require reproducing
+Drut¸u’s argument more or less in full. I felt that this is not desirable, and instead I only indicate the required
+adaptations. I believe that a reader who is familiar with Drut¸u’s argument and with asymptotic cones could
+easily produce a complete proof using these indications. In particular, there is no preliminary section. I do
+not present buildings or dynamical results that go into Drut¸u’s argument. I only shortly present asymptotic
+cones and some ideas from Drut¸u’s proof of the quasi-isometry version of Theorem 6.10. Section 6.4 is
+elementary.
+6.3
+SBE Maps are Close to Isometries
+In this section I indicate how to adapt Drut¸u’s arguments in [16] in order to prove:
+Theorem 6.10. There is a sublinear function v = v(L, u) such that for every
+�
+L, u
+�
+-SBE f : X0 → X0,
+there exists a unique isometry g = g(f) ∈ Isom(X) such that d
+�
+f(x), g(x)
+�
+≤ v(|x|).
+The proof of Theorem 6.4 requires some control on the sublinear distance between f and g, in terms of
+the sublinear constants of f. This is the meaning of the following lemma.
+Lemma 6.11. Let {fr}r∈R>0 be a family of (L′, vr)-SBE maps fr : X0 → X0, where vr = L′ · v + v(r) for
+some sublinear function v ∈ O(u) and a constant L′. Let gr be the associated isometry given by Theorem 6.10.
+Then for any x ∈ X0, there is a sublinear function ux ∈ O(u) such that d
+�
+fr(x), gr(x)
+�
+≤ ux(r).
+Remark 6.12. Combined with Theorem 6.10, a different way to phrase the above statement is to say that
+the function D : Λ× X0 → R≥0 defined by D(λ, x) = d
+�
+fλ(x), gλ(x)
+�
+is sublinear in each variable. I.e., there
+is a function u : R≥0 × R≥0 → R≥1 such that u is sublinear in each variable and D(λ, x) ≤ u(|λ|, |x|).
+Outline.
+I begin with a short presentation of asymptotic cones. I then give an account of the original
+proof of Theorem 6.10 when f : X0 → X0 is a quasi-isometry. I present the routines required to modify the
+proof for the SBE setting. I exemplify the modification procedure in a specific representative example, and
+finish with a road map for proving Theorem 6.10 and Lemma 6.11 using the aforementioned routines.
+6.3.1
+Asymptotic Cones
+Definition 6.13. Let (X, d) be a metric space. Fix an ultrafilter ω, a sequence of points xn ∈ X and a
+sequence of scaling factors ın −→
+ω 0. The asymptotic cone of X w.r.t. xn, ın, denoted C(X), is the metric
+ω-ultralimit of the sequence of pointed metric spaces (X, 1
+ın · d, xn). The metric on C(X) is denoted dω.
+See Section 2.4 in [16] for an elaborate account, including the definitions of ultrafilters and ultralimits.
+The strength of SBE maps is that they induce bi-Lipschitz maps between the respective asymptotic cones.
+Lemma 6.14 (See e.g. Cornulier [13]). Let f : X → Y be an (L, u)-SBE. Then f induces an L-bi-Lipschitz
+map C(f) : C(X) → C(Y ) between the corresponding asymptotic cones with the same scaling factors C(X) =
+(X, 1
+ın dX, x0
+n) and C(Y ) = (Y, 1
+ın dY , y0
+n).
+45
+
+6.3.2
+The Argument
+A High-Level Description.
+The core of the argument lies in elevating an SBE f0 : X0 → X0 to an
+isometry g0 ∈ G = Isom(X). There are two gaps to fill: first, Γ is non-uniform and so f0 is not even defined
+on the whole space X. And obviously, f0 is just an SBE.
+Assume for a moment that Γ is uniform and that f is defined on the whole space X. Elevating f : X → X
+to an isometry is done by considering the map C(f) : C(X) → C(X) that f induces on an asymptotic cone
+C(X). This map is bi-Lipschitz, and the work of Kleiner and Leeb [32] allows one to conclude that C(f)
+is, up to a scalar, an isometry. In turn, this isometry induces an isometry ∂g on the spherical building
+structure of X(∞). This is done by the relation between the Euclidean building structure of C(X) and the
+spherical building structure of ∂∞X. A theorem of Tits [56] associates to ∂q a unique isometry g ∈ Isom(X)
+that induces ∂f as its boundary map. By construction, it is then not too difficult to see that g and f are
+‘close’. In case Γ is non-uniform, an SBE f : Γ → Γ does not readily yield a cone map on C(X), but only on
+C(X0). Overcoming this difficulty requires substantial work and is the heart of Drut¸u’s proof. In short, she
+uses dynamical results stating that the vast majority of flats in X are close enough to X0. As mentioned
+in Section 2.2, the building structure on X(∞) is determined by the boundaries of flats F ⊂ X. The same
+holds for the (Euclidean) building structure of C(X). Therefore the fact that the majority of flats in X are
+‘close enough’ to X0 results in the fact that C(X0) composes the majority of C(X). This is a very rough
+sketch of the logic behind Drut¸u’s argument.
+The procedure described above results in an isometry g ∈ Isom(X) associated to f0. To complete the
+argument one needs to verify that the map f0 �→ g is a group homomorphism between SBE(Γ) = SBE(X0)
+and Isom(X).
+Composing this map with a representation of Λ into SBE(Γ) yields a map Λ → G by
+λ �→ fλ �→ gλ := gfλ. A computation then shows that this map has finite kernel and that Γ lies in a sublinear
+neighbourhood of the image.
+Flat Rigidity.
+The adaptations that are required for the SBE setting lie mainly in the part of Drut¸u’s
+work that concerns flat rigidity. That is, the proof that the quasi-isometry q0 maps a flat F ⊂ X0 to within
+a uniformly bounded neighbourhood of another flat F ′ ⊂ X0. This is proved by passing to the cone map,
+using an analogous result for bi-Lipschitz maps between Euclidean buildings, which translates back down to
+the space X0.
+Remark 6.15. Drut¸u’s argument requires many geometric and combinatorial definitions - some classical
+and widely known (e.g., Weyl chambers of a symmetric space X) some less known (e.g. an asymptotic cone
+with respect to an ultrafilter ω) and some new (e.g., the horizon of a set A ⊂ X). I use her definitions,
+terminology and notations freely without giving the proper preliminaries or even the definitions. I assume
+most readers who are interested in the question of SBE rigidity are familiar to some extent with most of these
+objects. For the new definitions, I try to say as little as needed to allow the reader to follow the argument.
+The proof consists of 6 steps:
+1. The horizon of an image of a Weyl chamber is contained in the horizon of a finite union of Weyl
+chambers, and the number of chambers in this union depend only on the Lipschitz constant. (Lemmas
+3.3.5, 3.3.6 in [16], consult Remark 6.16 below for a sketchy definition of horizon).
+2. The horizon of an image of a flat coincides with the horizon of a finite union of Weyl chambers, and
+the number of chambers depends only on the Lipschitz constant of the quasi-isometry.
+3. The union of Weyl chambers in the previous step limits to an apartment in the Tits building at X(∞).
+Such a union is called a fan over an apartment.
+4. For each Weyl chamber W there corresponds a unique chamber W ′ such that q0(W) and W ′ have the
+same horizon. This amounts to an induced map on the Weyl chambers of the Tits building at X(∞)
+(Lemma 4.2.1 in [16]).
+46
+
+5. Given a flat F through a point x, the unique flat F ′ asymptotic to the union of Weyl chambers obtained
+in step 3 is at uniform bounded distance from f0(x). The bound depends only on the quasi-isometry
+constants. The flat F ′ is called the flat associated to F.
+6. If F1 and F2 are two flats through x which intersect along a hyperplane H, then the boundaries at
+X(∞) of the associated flats F ′
+1 and F ′
+2 intersect along a hyperplane of the same codimension as H.
+Remark 6.16. For a precise definition of horizon see [16], section 3. For now, it suffices to say the following.
+The horizon of a set A ⊂ X is contained in the horizon of a set B ⊂ X if, looking far away at A from some
+point x ∈ X, A appears to be contained in an ε-neighbourhood of B. This intuition is made precise by
+considering the angle at x that a point a ∈ A makes with the set B. Two sets have the same horizon if each
+set’s horizon is contained in the other. In the case A and B have the same horizon, an important aspect is
+the distance R starting from which A and B seem to be ε-contained in one another. Call this distance the
+horizon radius. It depends on x and ε.
+The proofs for most of these steps have similar flavour: in any asymptotic cone C(X), f0 induces a
+bi-Lipschitz map. Kleiner and Leeb [32] proved many results about such maps between cones of higher rank
+symmetric spaces. One assumes towards contradiction that some assertion fails (say, in step 5, assume that
+there is no bound on the distance between f0(xn) and the associated flat F ′
+n). This gives an unbounded
+sequence of scalars (say, d
+�
+f0(xn), F ′
+n
+�
+= ın → ∞). These scalars are used to define a cone in which one
+obtains a contradiction to some fact about bi-Lipschitz maps (say, that the point [q0(xn)]ω is at dω-distance
+1 from [F ′
+n]ω, while it should lie in [F ′
+n]ω).
+Typically, the bounds obtained this way depend on the quasi-isometry constants. A priori, they also
+depend on the specific point x ∈ X or flat F ⊂ X in which you work (e.g. the horizon radius for the chambers
+in step 3 or the bound on d
+�
+q(x), F ′�
+in step 5). However, it is easy to see that in the quasi-isometry setting,
+the bounds are actually independent of the choice of point/flat/chamber. This independence stems from
+the fact that one can pre-compose f0 with an isometry translating any given point/flat/chamber to a fixed
+point/flat/chamber (resp.), without changing the quasi-isometry constants (see e.g. Remark 3.3.11 in [16]).
+The fact that these bounds depend only on the quasi-isometry constants is essential for the proof that the
+map Λ → Isom(X) has the desired properties.
+Moving to the SBE setting, the essential difference is exactly that the bounds one obtains depend on
+the specific point, Weyl chamber or flat. Indeed it is clear that these bounds should depend on the size |x|,
+as they depend on the additive constant in the quasi-isometry case. It is sensible to guess though that the
+bounds only grow sublinearly in |x|, which is enough in order to push the argument forward. In the next
+section I show how to elevate a typical cone argument from the quasi-isometry setting to the SBE setting. I
+focus on showing that the bound one obtains depend only on the SBE constants (L, u) and sublinearly |x|.
+6.3.3
+Generalization to SBE: Adapting Cone Arguments
+To adapt for the SBE setting, split each step into three sub-steps: the first two amount to proving Theo-
+rem 6.10, and the third step amounts to proving Lemma 6.11.
+Sub-Step 1.
+Repeat the argument of the quasi-isometry setting verbatim, to obtain a bound c = c(x)
+which depend on the point x.
+Sub-Step 2.
+Assume towards contradiction that there is a sequence of points xn for which limω
+c(xn)
+|xn| ̸= 0.
+This means limω
+|xn|
+c(xn) ̸= ∞, and so the point (xn)n lies in the cone C(X) = Cone
+�
+X, x0, c(xn)
+�
+, and one
+may proceed as in the corresponding quasi-isometry setting to obtain a contradiction.
+Sub-step 3.
+Fix x ∈ X and a sequence of SBE maps as in Lemma 6.11, i.e. {fn}n∈N with the same Lipschitz
+constant and with sublinear constants vn(s) = v(s) + u(n), for some sublinear functions u, v. Denote by
+cn(x) the constants that were achieved in the previous steps for x and the SBE map fn, and assume towards
+47
+
+contradiction that |cn(x)| is not bounded above by any function sublinear in n. This means in particular
+that limω
+u(n)
+cn(x) = 0. One concludes that the cone map C(frn) is bi-Lipschitz, and gets a contradiction in the
+same manner as in the first step.
+Example 6.17. In order to give the reader a sense of what is actually required, I now demonstrate this
+procedure in full in a specific claim. I chose to do this for proposition 4.2.7 of [16], which is complicated
+enough to require some attention to details, but not too much. The statement is as follows:
+Proposition 6.18 (SBE version of Proposition 4.2.7 in [16]). Let f : X → X be an (L, u)-SBE, and F ⊂ X
+a flat through x to which f associates a fan over an apartment, ∪p
+i=0Wi. If F ′ is the maximal flat asymptotic
+to the fan, then d
+�
+f(x), F ′�
+≤ c(x) where c(x) = c(|x|) is sublinear.
+Moreover, let fn : X → X be a sequence of (L, vn) SBE maps for vn = v + u′(n) for v, u′ some sublinear
+functions. The constant cn(x) associated to x and the SBE fn achieved in the first part of the proposition
+admits cn(x) ≤ ux(n) for some sublinear function ux.
+F ′ is then said to be the associated flat to F by f.
+Proof. Proceed in two (sub-)steps.
+Step 1.
+I show that for a given x, there exists such a constant c(x) independent of the flat F. This is done
+exactly as in [16], but I repeat the proof here because it contains the terminology and necessary preparation
+for the second step.
+Fix x ∈ X and assume towards contradiction that there exists a sequence Fn of flats through x and a
+sequence fn : X → X of (L, u)-SBE maps such that cn := d
+�
+fn(x), F ′
+n
+�
+→ ∞. In Cone(X, x, c−1
+n ) one can
+show that [∪p
+i=0W n
+i ], the union of Weyl chambers associated to fn(Fn), is a maximal flat (see Proposition
+4.2.6 in [16]). Denote Fω := [∪p
+i=0W n
+i ]. Furthermore, since the bi-Lipschitz flat [fn(Fn)] ⊂ Cone(X, x, c−1
+n )
+is contained in it, it coincides with it: [fn(Fn)] = Fω. On the other hand, since the Hausdorff distance
+between ∪p
+i=0W n
+i and F ′
+n is by assumption cn = d(x, F ′
+n), in the cone the maximal flats Fω and F ′
+ω := [F ′
+n]
+are at Hausdorff distance 1.
+This implies that Fω = F ′
+ω (see Corollary 4.6.4 in [32]). But since d
+�
+q(x), F ′
+n
+�
+= cn the limit point
+yω := Q(xω) = [q(x)] , which is contained in Fω, is at distance 1 from F ′
+ω - a contradiction.
+Step 2.
+Assume c(x) is taken to be the smallest possible for each x, and then modify the function c
+so that c(x) = maxy:|y|=|x| c(y).
+The function c : X → R now only depends on |x|.
+I wish to show
+that c(|x|) = O(u(|x|).
+Assume towards contradiction that there exists a sequence xn with |xn| → ∞
+such that limω
+c(xn)
+u(|xn|) = ∞. Denote cn = c(xn) and consider the cone Cone(X, xn, c−1
+n ). The assumption
+limω
+c(xn)
+u(|xn|) = ∞ implies (x0)ω = (xn)ω hence Cone(X, xn, cn) = Cone(X, x0, cn). By the definition of c(x)
+this means that there is a sequence of flats Fn through xn such that d
+�
+q(xn), F ′
+n
+�
+= cn, so one may proceed
+as in step 1 for a cone with a fixed base point (x0)ω. In this cone the flat F ′
+ω := [F ′
+n] is at distance 1
+from the point [q(xn)], which lies on the maximal flat [∪p
+i=0W n
+i ]. The latter flat is, on the one hand, at
+Hausdorff distance 1 from F ′
+ω (by the definition of the scaling factors cn), so they actually coincide. On
+the other hand, [∪p
+i=0W n
+i ] coincides with Fω := Q(Fω) = [q(Fn)], so Fω = F ′
+ω, contradicting the fact that
+dω([q(xn)], F ′
+ω) = 1. Thus c(|x|) = O(u(|x|)), as wanted.
+Step 3.
+For the moreover part (uniform control on the growth of c(x) as a function of the sublinear
+constants), the proof is identical to Step 1. This time, consider a sequence fn as in the statement, and
+denote by cn = cn(x) the constant obtained in step 1 w.r.t. the SBE constants (L, vn). Assume towards
+contradiction that limω
+u(n)
+cn(x) = 0. The proof goes exactly as in step 1, with the sole difference that now one
+might need convincing in the fact that in C(X), the cone with
+1
+cn as scaling factors, the cone map C(fn) is
+bi-Lipschitz. But indeed for any two cone points (xn), (yn) ∈ C(X) it holds:
+dω
+�
+C(fn)(xn), C(fn)(yn)
+�
+= lim
+ω
+1
+cn
+d
+�
+fn(xn), fn(yn)
+�
+≤ lim
+ω
+1
+cn
+L · d(xn, yn) + vn(|xn| ∨ |yn|)
+48
+
+By definition of the cone metric, limω
+1
+cn L · d(xn, yn) = L · dω
+�
+(xn), (yn)
+�
+.
+It thus remains to show
+limω
+1
+cn vn(|xn| ∨ |yn|) = 0. By definition of vn it amounts to proving limω
+1
+cn v(|xn|) = 0 = limω
+1
+cn v(|yn|)
+and limω
+1
+cn u(n) = 0. The former follows from the fact that (xn), (yn) ∈ C(X) and therefore both limω
+1
+cn |xn|
+and limω
+1
+cn |xn| are finite. The latter follows from the assumption on the cn. One obtains a contradiction
+identical to the one in Step 1.
+Following the claims of Sections 3, 4, 5 in [16] carefully, and making the SBE adaptations as depicted
+in the above example, one obtains flat rigidity in the SBE setting, that is Theorem 6.10 together with the
+uniform control described in Lemma 6.11. Here is the complete list of claims involving cone arguments in
+[16] that should be modified.
+• Section 3. All claims starting from Lemma 3.3.5 through Corollary 3.3.10. All statements should
+consider, instead of a quasi-isometry, a general (L, u)-SBE f and, when relevant, a general point x ∈ X
+with f(x) = y (i.e., f(x) does not necessarily equal x). Also when relevant one should consider a family
+fn of (L, vn) SBE maps as depicted in Lemma 6.11 above.
+• Section 4. Propositions 4.2.6, 4.2.7, 4.2.9. When relevant, statements should be modified so that the
+distance between f(x) and an associated flat of it should be uniformly sublinear in |x|. Also when
+relevant one should consider a family fn of (L, vn) SBE maps as above.
+• Section 5. Lemma 5.4.1 (D = D(x) should be uniformly linear in c = c(x)). Proposition 5.4.2 (the
+constant D = D(x) should be replaced be a sublinear function D(|x|)). When relevant one should
+consider a family fn of (L, vn) SBE maps as above.
+This yields a proof for uniform flat rigidity in the SBE setting. The other major part of Drut¸u’s argument
+concerns the fact that f0 is defined only on X0 and not on X.
+The considerations for this aspect are
+intertwined in the proof, but they all involve only Γ and the quasi-isometry between Γ and X0. For this
+reason, the fact that f0 is an SBE to begin with does not effect any of these arguments. Moreover, this
+argument is indifferent to whether or not Γ is uniform or not. If Γ is uniform all that changes is that that
+part of Drut¸u’s argument dealing with extending the cone map from C(X0) to C(X) is not necessary since
+X = X0. Her proof still works perfectly well also for uniform lattices. Therefore the argument above proves
+Theorem 6.10 and Lemma 6.11.
+6.3.4
+Some Remarks On R-rank 1 Factors
+Quasi-isometric rigidity holds for groups of R-rank 1. It is worth mentioning that Schwartz’s proof also
+relies on ‘flat rigidity’ - but in this case the flats are the horospheres of Γ. While these are not isometrically
+embedded flats, the induced metric on horospheres is flat and Schwartz uses that in order to construct the
+boundary map and find the associated isometry.
+The same phenomena occurs in the SBE setting. Considering the compact core X0 ⊂ X of Γ, one can
+use Proposition 5.6 in Drut¸u-Sapir [18], in order to show that horospheres are mapped boundedly close to
+horospheres. In general, this work characterizes and explores a certain class of spaces they call asymptotically
+tree graded, a class that is very suitable for the setting of the compact core of a non-uniform lattice in R-
+rank 1.
+A key ingredient in the proof is the fact that the boundary map ∂q induced by the quasi-isometry is
+quasi-conformal. This in particular implies that it is almost everywhere differentiable. I spent some time
+trying to generalize the proof of Schwartz to the SBE setting. One obstacle is that it is not clear that the
+boundary map is going to be differentiable almost everywhere. Gabriel Pallier found that there are SBE
+maps of the hyperbolic space whose boundary maps are not quasi-conformal (see Appendix A in [45]). For
+this reason Pallier develops the notion of quasi-conformality [46]. While these maps may be differentiable,
+he told me of examples he constructed where the differential is almost everywhere 0 - a property which also
+nullifies Schwartz’s argument.
+In the context of SBE rigidity, the maps I consider seem to indeed have ‘flat rigidity’, i.e. to map a
+horosphere to within bounded distance of a unique horosphere. As in the higher rank flat rigidity, this
+49
+
+bound is not uniform but rather grows sublinearly with the distance of the horosphere to a fixed base point.
+These are very specific maps, that coarsely preserve the compact core of that lattice X0 ⊂ X and basically
+map horospheres to horospheres. This means there might still be hope for these specific maps to induce
+boundary maps that admit the required analytic properties.
+In a subsequent paper [53], Schwartz proves quasi-isometric rigidity for lattices in products of R-rank 1
+groups, i.e. in Hilbert modular groups. His proof there is different, but it also makes use of the fact that
+horospheres are mapped to within uniformly bounded distance of horospheres. The fact that in the SBE
+setting this bound is not uniform seems like a real obstruction to any attempt of generalizing his proof in
+that case.
+6.4
+From SBE to Sublinearly Close Groups
+In this section I prove Theorem 6.9. I restate it here for convenience
+Theorem 6.19. Let G be a real centre-free semisimple Lie group without compact or R-rank 1 factors. Let
+Γ ≤ G be an irreducible lattice, and Λ an abstract finitely generated group that is SBE to Γ. Then there is
+a group homomorphism Φ : Λ → G with finite kernel such that Γ ⊂ Nu
+�
+Φ(Λ)
+�
+and Φ(Λ) ⊂ Nu(Γ) for a
+sublinear function u.
+The proof is a sublinear adaptation of the classical arguments by Schwartz. The only difference is that
+some calculations are in order, but there is no essential difference from Section 10.4 in [52].
+Before I start, I need one well known preliminary fact, namely that sublinearly close isometries are equal.
+Lemma 6.20. Let X be a symmetric space of noncompact type and with no R-rank 1 factors. Let Γ ≤
+Isom(X) be a non-uniform lattice, X0 its compact core with respect to x0 ∈ X. Let g, h ∈ G = Isom(X) and
+u a sublinear function such that for every x ∈ X0, d
+�
+g(x), h(x)
+�
+≤ u(|x|). Then g = h.
+Proof. The proof is essentially just the fact that a sublinearly bounded convex function is uniformly bounded.
+Up to multiplying by h−1, one may assume h = idX. First I show that the continuous map ∂g : X(∞) →
+X(∞) is the identity map. Recall that the space X(∞) can be represented by all geodesics emanating from
+the fixed point x0. Let η : [x0, ξ) be a Γ-periodic geodesic. By definition, there is some T > 0 and a sequence
+γn ∈ G for which η(nT ) = γnx0. In particular, xn := γnx0 ∈ X0 hence d
+�
+g(xn), xn
+�
+≤ u(|xn|) = u(nT ).
+On the other hand, the distance function d
+�
+η(t), g·η(t)
+�
+is convex. A convex sublinear function is bounded,
+and so by definition in X(∞) one has [η] = [g · η], for all Γ-periodic geodesics η. The manifold X is of non-
+positive curvature, hence ∂g is a homeomorphism of X(∞), and the density of Γ-periodic geodesics implies
+∂g = idX(∞). This implies that g = idX (see Section 3.10 in [20] for a proof of this last implication).
+Remark 6.21. The proof for the fact that ∂g = idX(∞) implies g = idX appears in [20] (section 3.10) as
+part of the proof of the following important theorem of Tits:
+Theorem 6.22 ([56], see Theorem 3.10.1 in [20]). Let X, X′ be symmetric spaces of noncompact type and
+of higher R-rank. Assume X has no R-rank 1 factors, and let φ : X(∞) → X′(∞) be a bijection that is a
+homeomorphism with respect to the cone topology and an isometry with respect to the Tits metric. Then,
+after multiplying the metric of X by positive constants on de Rham factors, there exists a unique isometry
+g : X → X′ such that φ = ∂g.
+This theorem is actually a key ingredient in Drut¸u’s argument. Much of her work is directed towards
+showing that the cone map C(q) will correspond to a map on X(∞) satisfying the above hypothesis. The
+restriction to X with no R-rank 1 factors in Theorem 6.4 comes from this restriction in Tits’ Theorem 6.22
+Remark 6.23. The proof that ∂g = idX(∞) ⇒ g = idX only uses the fact that X has no Euclidean de
+Rham factors (see pg. 251 in [20]). Since I only use it in the setting of no R-rank 1 factors, I added that
+assumption to Lemma 6.20.
+50
+
+The Map Φ : Λ → G.
+The orbit map q0 : Γ → X0 defined by γ �→ γx0 is a quasi-isometric embedding:
+this is ˘Svarc-Milnor in case Γ is uniform and X0 = X, and Lubotzki-Mozes-Raghunathan (Theorem 2.19
+above) if Γ is non-uniform. An SBE f : Λ → Γ thus gives rise to an SBE Λ → X0, which I also denote by f.
+For each λ ∈ Λ let fλ := f ◦ Lλ ◦ f −1 : X0 → X0, where Lλ is the left multiplication by λ. The left
+translation Lλ is an isometry, hence fλ is a self SBE of X0. By Theorem 6.10, there exists a unique isometry
+gλ ∈ Isom(X) that is sublinearly close to fλ. Define the map Φ : Λ → G by λ �→ gλ. The goal in this section
+is to prove Φ is a homomorphism with finite kernel, and that Γ and Φ(Λ) are each contained in a sublinear
+neighbourhood of the other.
+I begin by controlling the SBE constants of the fλ.
+Lemma 6.24. For each λ ∈ Λ, fλ is an (L2, vλ)-SBE, for
+vλ(|x|) := (L + 1)u(|x|) + u(|λ|)
+In particular vλ ∈ O(u).
+Before the proof I state a corollary which follows immediately by combining Lemma 6.24 with Lemma 6.11.
+Corollary 6.25. Assume Lemma 6.11 holds. Then for any x ∈ X there is a sublinear function ux such that
+d
+�
+fλ(x), gλ(x)
+�
+≤ ux(|λ|)
+.
+Proof of Lemma 6.24. The proof is a straightforward computation. Up to an additive constant I may assume
+f −1 is an (L, u)-SBE with f −1(eΓ) = eΛ. Let x1, x2 ∈ X0, and assume w.l.o.g |x2| ≤ |x1|. By the properties
+of an SBE, this also means that for i ∈ {1, 2}:
+|f −1(xi)| ≤ L|xi − x0| + u(|xi|) ≤ L|x1| + u(|x1|)
+(6)
+Notice that fλ(x) = f
+�
+λ · f −1(x)
+�
+, and f is an (L, u)-SBE. The following inequalities, justified below,
+give the required upper bounds:
+��fλ(x1) − fλ(x2)
+�� ≤ L ·
+��λf −1(x1) − λf −1(x2)
+�� + u
+�
+|λf −1(x1)| ∨ |λf −1(x2)|
+�
+≤ L2|x1 − x2| + Lu
+�
+|x1|
+�
++ u
+�
+|λ|) + L|x1| + u(|x1|)
+�
+≤ L2|x1 − x2| + (L + 1)u
+�
+|x1|
+�
++ u(|λ|).
+(7)
+From the first line to the second line I used:
+1. For the first term: left multiplication in Λ is an isometry, and f −1 is an (L, u)-SBE.
+2. For the second term: triangle inequality, left multiplication in Λ is an isometry, and Equation 6.
+From the second to the third line I used the properties of u as an admissible function, namely that it is
+sub-additive and doubling, so u
+�
+(L + 1)|x1|
+�
+≤ (L + 1)u(|x1|) for all large enough x1.
+Remark 6.26. I remark that the proof of Lemma 6.24 is the only place where I use the properties of an
+admissible function and not just the sublinearity of u.
+Claim. Φ : Λ → G is a group homomorphism.
+Proof. Let λ1, λ2 ∈ Λ. I begin with some notations:
+1. f1 = fλ1, f2 = fλ2, f12 = fλ1λ2. By Lemma 6.24, these are all O(u) SBE maps with the same Lipschitz
+constant L′ := L2 and sublinear constants v1, v2, v12 ∈ O(u).
+51
+
+2. g1 = Φ(λ1), g2 = Φ(λ2), g12 = Φ(λ1λ2)
+3. u1, u2, u12 the sublinear functions that bound the respective distances between any g and f, e.g.
+|g1(x) − f1(x)| ≤ u1(|x|).
+One has to prove that gλ2 ◦ gλ1 = gλ1λ2. In view of Lemma 6.20, it is enough to find a sublinear function
+v such that for all x ∈ X0 |g1g2(x) − g12(x)| ≤ v(|x|). By triangle inequality and the above definitions and
+notation, it is enough to show that each of the following four terms are bounded by a function sublinear in
+x:
+1. |g1g2(x) − g1f2(x)| = |g2(x) − f2(x)| ≤ u2(|x|) (g1 is an isometry).
+2. |g1f2(x) − f1f2(x)| ≤ u1
+�
+|f2(x)|
+�
+≤ u1
+�
+L2|x| + v2(|x|)
+�
+3. |f1f2(x) − f12(x)|
+4. |f12(x) − g12(x)| ≤ u12(|x|)
+Clearly items 1, 2, 4 are bounded by a sublinear function in |x|. It remains to bound |f1f2(x) − f12(x)|.
+The map Λ → Aut(Λ) given by λ �→ Lλ is a group homomorphism, i.e. Lλ1λ2 = Lλ1Lλ2, so it remains to
+bound:
+|f1f2(x) − f12(x)| = |fLλ1f −1fLλ2f −1(x) − fLλ1Lλ2f −1(x)|
+f ◦ Lλ is a composition of an isometry with an SBE, so it is still an SBE. Denote the SBE constants of
+fLλ1 by L′, v (clearly one can take L′ = L and v ∈ O(u), but this is not needed). Writing y := Lλ2f −1(x),
+this shows
+|fLλ1f −1f(y) − fLλ1(y)| ≤ L|f −1fy − y| + v(|f −1fy| ∨ |y|)
+By definition of an SBE inverse it holds that |f −1f(y) − y| ≤ u(|y|) and in particular also |f −1f(y)| ≤
+|y| + u(|y|). I conclude that
+|f1f2(x) − f12(x)| ≤ L · u(|y|) + v
+�
+|y| + u(|y|)
+�
+The right-hand side is a sublinear function in |y|, hence it only remains to show that |y| is bounded by a
+linear function in x. Indeed
+|y| = |Lλ2f −1(x)| ≤ |λ2| + |f −1(x)| ≤ |λ2| + L|x| + u(|x|)
+This completes the proof, rendering Φ a group homomorphism.
+Claim. Φ has discrete image and finite kernel.
+Proof. I show that for any radius R > 0, there are finitely many λ ∈ Λ for which gλx0 ∈ B(x0, R). I.e., that
+there is a finite number of Φ(Λ)-orbit points, with multiplicities, inside an R ball in X. In particular the
+set {λ ∈ Λ | gλx0 = x0} is finite, and clearly contains Ker(Φ). In addition, the actual number of Φ(Λ)-orbit
+points inside that R ball is finite, so Φ(Λ) is discrete.
+Let R > 0, and λ ∈ Λ. By the defining property of gλ and the definition of fλ, reverse triangle inequality
+gives
+d
+�
+x0, gλ(x0)
+�
+≥ d
+�
+x0, fλ(x0)
+�
+− d
+�
+gλ(x0), fλ(x0)
+�
+≥ |f(λ)| − uλ(|x0|)
+Corollary 6.25 gives d
+�
+gλ(x0), fλ(x0)
+�
+≤ ux0(|λ|) for some sublinear function ux0 ∈ O(u). On the other
+hand f is an SBE, and so |f(λ)| grows close to linearly in λ. Formally,
+|f(λ)| = d
+�
+f(λ), x0
+�
+= d
+�
+f(λ), f(eΛ)
+�
+≥ 1
+Ld(λ, x0) − u(|λ| ∨ |eΛ|) ≥ 1
+L|λ| − u(|λ|)
+52
+
+To conclude, one has
+d
+�
+x0, gλ(x0)
+�
+≥ 1
+L|λ| − u(|λ|) − ux0(|λ|)
+and both u, ux0 are sublinear in |λ|. Therefore there is a bound S ∈ R>0 such that |λ| > S ⇒ 1
+L|λ| −
+u(|λ|) − ux0(|λ|) > R. The group Λ is finitely generated and so only finitely many λ ∈ Λ admit |λ| ≤ S,
+hence gλ(x0) ∈ B(x0, R) only for finitely many λ ∈ Λ.
+Claim. There exists a sublinear function u′ : R≥0 → R≥1 such that
+Γ · x0 ⊂ Nu′�
+Φ(Λ) · x0
+�
+Proof. I claim that there is a sublinear function u0, depending only on f and q0, such that for all γ ∈ G,
+d
+�
+gλ(x0), γ(x0)
+�
+≤ u0(|γ|).
+As before, I only have control on gλ via fλ, and so I use triangle inequality to get:
+d
+�
+gλ(x0), γ(x0)
+�
+≤ d
+�
+gλ(x0), fλ(x0)
+�
++ d
+�
+fλ(x0), γ(x0)
+�
+By Corollary 6.25, d
+�
+gλ(x0), fλ(x0)
+�
+≤ ux
+0(|λ|) for a sublinear function ux0.
+It is beneficial to distinguish between the SBE fΓ : Λ → Γ and the same SBE composed with the orbit
+quasi-isometry q0 : Γ → X0. From now on I keep the notation fΓ : Λ → Γ for the SBE of the groups and f0
+for the same SBE composed with the orbit quasi-isometry so f0 = q0 ◦ fΓ.
+Define λγ := f −1
+Γ (γ). I show that d
+�
+fλγ(x0), γ(x0)
+�
+is bounded by a function sublinear in γ. Indeed,
+recall that I assumed without loss of generality q0 ◦f −1
+Γ (x0) = eΛ. Moreover, Γ is assumed to be torsion-free,
+and so there is no ambiguity or trouble in defining the restriction of the map q−1 to the orbit Γ · x0 to be of
+the form q−1(γx0) = γ. All together, this gives
+d
+�
+fλγ(x0), γ(x0)
+�
+= d
+�
+f0(λγ), γ(x0)
+�
+= d
+�
+q0 ◦ fΓf −1
+Γ (γ), q0(γ)
+�
+Since fΓ is an SBE d
+�
+fΓf −1
+Γ (γ), γ
+�
+≤ u(γ). The fact that q0 is an (L′, C)-quasi-isometry implies that
+d
+�
+q0 ◦ fΓf −1
+Γ (γ), q(γ)
+�
+≤ Ld
+�
+fΓf −1
+Γ (γ), γ
+�
++ C ≤ L′u(|γ|) + C
+Combining everything, one has
+d
+�
+gλγ(x0), γ(x0)
+�
+≤ ux0(|λγ|) + L′u(|γ|) + C
+As before, |λγ| = |f −1(γ)| ≤ |γ| + u(|γ|) ≤ 2|γ|, where the last inequality holds for all large enough γ.
+What matters is that |λγ| is linear in |γ|. I conclude that indeed Γ · x0 ⊂ Nu′�
+Φ(Λ) · x0
+�
+for the sublinear
+function u′ = ux0 + L′u + C ∈ O(u), as wanted. (To be pedantic, u′ = ux0 ◦ 2 + L′u + C ∈ O(u) where 2 is
+the ‘multiplication by 2’ function, r �→ 2r).
+Claim. There exists a sublinear function u′ : R≥0 → R≥1 such that
+Φ(Λ) · x0 ⊂ Nu′(Γ · x0)
+Proof. Let λ ∈ Λ.
+Let γ = f(λ) and consider the distance d(gλx0, γx0).
+From triangle inequality and
+Corollary 6.25 one has
+d(gλx0, γx0) ≤ d(gλx0, fλx0) + d(fλx0, γx0) ≤ u′(|λ|) + d(fλx0, γx0)
+By definition of fλ it holds that fλ(x0) = f(λ) · x0 = γ · x0 and so d
+�
+fλ(x0), γx0
+�
+= 0.
+Claims 6.4, 6.4, 6.4 and 6.4 result in Theorem 6.9. Theorem 6.4 then follows from Theorem 1.6.
+53
+
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