Rémi Coulon & Denis Osin A non-residually finite group acting uniformly properly on a hyperbolic space Tome 6 (2019), p. 19-30. <http://jep.centre-mersenne.org/item/JEP_2019__6__19_0> © Les auteurs, 2019. Certains droits réservés. Cet article est mis à disposition selon les termes de la licence LICENCE INTERNATIONALE D’ATTRIBUTION CREATIVE COMMONS BY 4.0. https://creativecommons.org/licenses/by/4.0/ L’accès aux articles de la revue « Journal de l’École polytechnique — Mathématiques » (http://jep.centre-mersenne.org/), implique l’accord avec les conditions générales d’utilisation (http://jep.centre-mersenne.org/legal/). Publié avec le soutien du Centre National de la Recherche Scientifique Publication membre du Centre Mersenne pour l’édition scientifique ouverte www.centre-mersenne.org Tome 6, 2019, p. 19–30 DOI: 10.5802/jep.86 A NON-RESIDUALLY FINITE GROUP ACTING UNIFORMLY PROPERLY ON A HYPERBOLIC SPACE by Rémi Coulon & Denis Osin Abstract. — In this article we produce an example of a non-residually finite group which admits a uniformly proper action on a Gromov hyperbolic space. Résumé (Un exemple de groupe non résiduellement fini muni d’une action uniformément propre sur un espace hyperbolique) Dans cet article nous construisons un exemple de groupe qui n’est pas résiduellement fini et qui est muni d’une action uniformément propre sur un espace hyperbolique au sens de Gromov. Contents 1. Introduction.................................................................. 19 2. A short review of hyperbolic geometry........................................ 21 3. The class P0 ................................................................. 22 4. Stability of the class P0 ...................................................... 24 5. Proof of the main theorem.................................................... 28 References....................................................................... 29 1. Introduction By default, all actions of groups on metric spaces considered in this paper are by isometries. Recall that a group is hyperbolic if and only if it acts properly and cocompactly on a hyperbolic metric space. It is natural to ask what kind of groups we get if we remove the requirement of cocompactness from this definition. However, it turns out that every countable group admits a proper action on a hyperbolic space, 2010 Mathematics Subject Classification.— 20F65, 20F67, 20E26, 20F06. Keywords.— Hyperbolic spaces, residually finite group, small cancellation theory, uniformly proper action, bounded geometry. The first author acknowledges the support of the ANR grant DAGGER ANR-16-CE40-0006-01. He is also grateful to the Centre Henri Lebesgue ANR-11-LABX-0020-01 for creating an attractive mathematical environment. The work of the second author has been supported by the NSF grant DMS-1612473. e-ISSN: 2270-518X http://jep.centre-mersenne.org/ 20 R. Coulon & D. Osin namely the parabolic action on a combinatorial horoball [13]. Thus to obtain an interesting class of groups we have to strengthen our properness assumptions. In this paper we propose to study the class of groups that admit a uniformly proper action on a hyperbolic length space. We denote this class of groups by P. Recall that an action of a group G on a metric space X is uniformly proper if for every r > 0, there exists N 2 N, such that for all x 2 X, jfg 2 G j dX (x; gx) 6 rgj 6 N: Having a uniformly proper action on a hyperbolic space is a rather restrictive con- dition. For instance, [16, Th. 1.2] implies that every group G 2 P (as well as each of its subgroups) is either virtually cyclic or acylindrically hyperbolic, which imposes strong restrictions on the algebraic structure of G. In this article we actually focus on a smaller class P0 ⊂ P which is easier to manipulate. It consists of all groups G having an action on a hyperbolic graph with bounded valence, whose restriction to the vertex set is free. Hyperbolic groups and their subgroups obviously belong to P0. Indeed if H is a subgroup of a hyperbolic group G, then the action of H on a Cayley graph of G satisfies the above properties. In general, groups from the class P0 have many prop- erties similar to those of hyperbolic groups. In fact, we do not know the answer to the following question: Does P0 coincide with the class of all subgroups of hyperbolic groups? Although the affirmative answer seems unlikely, we are not aware of any counterexamples. This paper is inspired by the well-known open problem of whether every hyperbolic group is residually finite. Our main result shows that the answer to this question is negative if one replaces the class of hyperbolic groups with the class P0. Theorem 1.1. — There exists a finitely generated non-trivial group G with an action on a hyperbolic graph of bounded valence whose restriction to the vertex set is free such that every amenable quotient of G is trivial. In particular, G 2 P0 and G is not residually finite. In the process of constructing such a group G, we show that P0 is closed under taking certain small cancellation quotients (see Section4). This result seems to be of independent interest and can potentially be used to construct other interesting examples of groups from the class P0. The proof of the second claim of Theorem 1.1 can be illustrated as follows. We first use a variant of the Rips construction given in [1] to construct a subgroup N of a torsion-free hyperbolic group H and two elements a; b 2 N which are “sufficiently independent” in N (more precisely, non-commensurable – see Section2 for the defini- tion) but are conjugate in every finite quotient of N. The fact that these elements are “sufficiently independent” together with the result about small cancellation quotients p q mentioned above imply that the quotient group G = N=hha ; b ii belongs to P0 for some (in fact, all sufficiently large) primes p and q. If p 6= q, the images of a and b are clearly trivial in every finite quotient of G. In particular, G is not residually finite. J.É.P.— M., 2019, tome 6 A non-residually finite group acting uniformly properly on a hyperbolic space 21 A slightly more elaborated version of this idea involving Kazhdan’s property (T) leads to the proof of the first claim of the theorem. Acknowledgments.— We are grateful to Ashot Minasyan for useful comments and suggestions, which allowed us to simplify the original proof of Theorem 5.2. We also thank the referees for their helpful remarks. 2. A short review of hyperbolic geometry In this section we recall a few notations and definitions regarding hyperbolic spaces in the sense of Gromov. For more details, refer the reader to Gromov’s original article [11] or [4, 10]. The four point inequality.— Let (X; d) be a length space. Recall that the Gromov product of three points x; y; z 2 X is defined by 1 hx; yi = fd(x; z) + d(y; z) − d(x; y)g : z 2 In the remainder of this section, we assume that X is δ-hyperbolic, i.e., for every x; y; z; t 2 X, (1) hx; zit > min fhx; yit ; hy; zitg − δ: We denote by @X the boundary at infinity of X, see [4, Chap. 2]. Quasi-convex subsets.— Let Y be a subset of X. Recall that Y is α-quasi-convex if 0 0 for every x 2 X, for every y; y 2 Y , we have d(x; Y ) 6 hy; y ix + α. If Y is path- connected, we denote by dY the length pseudo-metric on Y induced by the restriction of dX on Y . The set Y is strongly quasi-convex if Y is 2δ-quasi-convex and for every y; y0 2 Y we have 0 0 0 dX (y; y ) 6 dY (y; y ) 6 dX (y; y ) + 8δ: We denote by Y +α, the α-neighborhood of Y , i.e., the set of points x 2 X such that d(x; Y ) 6 α. Isometries of a hyperbolic space.— Let G be a group acting uniformly properly on X. An element g 2 G is either elliptic (it has bounded orbits, hence finite order) or loxodromic (it has exactly two accumulation points in @X)[2, Lem. 2.2]. A subgroup of G is either elementary (it is virtually cyclic) or contains a copy of the free group F2 [11, §8.2]. In order to measure the action of g on X, we use the translation length defined as follows kgk = inf d(gx; x) : X x2X If there is no ambiguity, we omit the space X in the notation. A loxodromic element g 2 G fixes exactly two points g− and g+ in @X. We denote by E(g) the stabilizer of fg−; g+g. It is the maximal elementary subgroup containing g. Moreover hgi has finite index in E(g) [8, Lem. 6.5]. J.É.P.— M., 2019, tome 6 22 R. Coulon & D. Osin Given a loxodromic element g 2 G, there exists a g-invariant strongly quasi-convex subset Yg of X which is quasi-isometric to a line; its stabilizer is E(g) and the quo- tient Yg=E(g) is bounded [7, Def. 3.12 and Lemma 3.13]. We call this set Yg the cylinder of g. We say that two elements g; h 2 G are commensurable, if there exist n; m 2 Zrf0g and u 2 G such that gn = uhmu−1. Every loxodromic element is contained in a unique maximal elementary subgroup [7, Lem. 3.28]. Hence two loxodromic elements g and h are commensurable if and only if there exists u 2 G such that g and uhu−1 generate an elementary subgroup.