DOCSLIB.ORG

A Non-Residually Finite Group Acting Uniformly Properly on a Hyperbolic Space

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
A Non-Residually Finite Group Acting Uniformly Properly on a Hyperbolic Space 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.
Load more

Recommended publications
  • Abstract Quotients of Profinite Groups, After Nikolov and Segal
    ABSTRACT QUOTIENTS OF PROFINITE GROUPS, AFTER NIKOLOV AND SEGAL BENJAMIN KLOPSCH Abstract. In this expanded account of a talk given at the Oberwolfach Ar- beitsgemeinschaft “Totally Disconnected Groups”, October 2014, we discuss results of Nikolay Nikolov and Dan Segal on abstract quotients of compact Hausdorff topological groups, paying special attention to the class of finitely generated profinite groups. Our primary source is [17]. Sidestepping all difficult and technical proofs, we present a selection of accessible arguments to illuminate key ideas in the subject. 1. Introduction §1.1. Many concepts and techniques in the theory of finite groups depend in- trinsically on the assumption that the groups considered are a priori finite. The theoretical framework based on such methods has led to marvellous achievements, including – as a particular highlight – the classification of all finite simple groups. Notwithstanding, the same methods are only of limited use in the study of infinite groups: it remains mysterious how one could possibly pin down the structure of a general infinite group in a systematic way. Significantly more can be said if such a group comes equipped with additional information, such as a structure-preserving action on a notable geometric object. A coherent approach to studying restricted classes of infinite groups is found by imposing suitable ‘finiteness conditions’, i.e., conditions that generalise the notion of being finite but are significantly more flexible, such as the group being finitely generated or compact with respect to a natural topology. One rather fruitful theme, fusing methods from finite and infinite group theory, consists in studying the interplay between an infinite group Γ and the collection of all its finite quotients.
    [Show full text]
  • Residual Properties of Free Products
    Residual properties of free products Federico Berlai∗ Abstract Let C be a class of groups. We give sufficient conditions ensuring that a free product of residually C groups is again residually C, and analogous conditions are given for LE-C groups. As a corollary, we obtain that the class of residually amenable groups and the one of LEA groups are closed under taking free products. Moreover, we consider the pro-C topology and we characterize special HNN extensions and amalgamated free products that are residually C, where C is a suitable class of groups. In this way, we describe special HNN extensions and amalgamated free products that are residually amenable. Key words: root property, residually amenable group, sofic group, free product, pro-C topology, proamenable topology. 2010 AMS Subject Classification: 20E26, 43A07, 20E06, 20E22, 20E18. 1 Introduction Amenable groups were defined in 1929 by von Neumann in his work on the Banach-Tarski paradox and, since then, the study of amenability has been remarkably fruitful in a variety of branches of mathematics. Recently, weak variants of amenability have attracted considerable interest. Examples are the Haagerup property, also known as Gromov’s a-T-menability, and coarse amenability, also termed as Guoliang Yu’s Property A. These notions appeared in the study of analytic properties of infinite groups. The main focus of this paper are other, more algebraic, weak forms of amenability: the notions of residual amenability and of LEA (Locally Embeddable into Amenable) groups. Residual amenability is a common generalization of amenability and of residual finiteness. While the stability, or permanence, properties of both the classes of amenable groups and of residually finite groups are well known, the stability properties of the class of residually amenable groups have not yet been investigated.
    [Show full text]
  • On a Question of Bumagin and Wise
    ON A QUESTION OF BUMAGIN AND WISE ALAN D. LOGAN Abstract. Motivated by a question of Bumagin and Wise, we con- struct a continuum of finitely generated, residually finite groups whose outer automorphism groups are pairwise non-isomorphic finitely gener- ated, non-recursively-presentable groups. These are the first examples of such residually finite groups. 1. Introduction In this paper we construct the first examples of finitely generated, resid- ually finite groups G whose outer automorphism groups are finitely gener- ated and not recursively presentable. Indeed, we construct a continuum, so 2ℵ0 , of such groups G with pairwise non-isomorphic outer automorphism groups. Our construction is motivated by a question of Bumagin and Wise, who asked if every countable group Q could be realised as the outer auto- morphism group of a finitely generated, residually finite group GQ. In this paper we solve a finite-index version of this question for Q finitely generated and residually finite, and the aforementioned result then follows. Bumagin and Wise solved the question for Q finitely presented [BW05]. In previous work, the author gave a partial solution for Q finitely generated and recur- sively presentable [Log15b, Theorem A], and a complete solutions for these groups assuming that there the exists a “malnormal version” of Higman’s embedding theorem [Log15b, Theorem B]. arXiv:1509.01848v2 [math.GR] 9 Sep 2015 Residually finite groups. A group G is residually finite if for all g ∈ G \ {1} there exists a homomorphism image φg : G → Fg where Fg is finite and where φg(g) =6 1. Residual finiteness is a strong finiteness prop- erty.
    [Show full text]
  • Profinite Properties of Discrete Groups
    PROFINITE PROPERTIES OF DISCRETE GROUPS ALAN W. REID 1. Introduction This paper is based on a series of 4 lectures delivered at Groups St Andrews 2013. The main theme of the lectures was distinguishing finitely generated residually finite groups by their finite quotients. The purpose of this paper is to expand and develop the lectures. The paper is organized as follows. In x2 we collect some questions that motivated the lectures and this article, and in x3 discuss some examples related to these questions. In x4 we recall profinite groups, profinite completions and the formulation of the questions in the language of the profinite completion. In x5, we recall a particular case of the question of when groups have the same profinite completion, namely Grothendeick's question. In x6 we discuss how the methods of L2-cohomology can be brought to bear on the questions in x2, and in x7, we give a similar discussion using the methods of the cohomology of profinite groups. In x8 we discuss the questions in x2 in the context of groups arising naturally in low-dimensional topology and geometry, and in x9 discuss parafree groups. Finally in x10 we collect a list of open problems that may be of interest. Acknoweldgement: The material in this paper is based largely on joint work with M. R. Bridson, and with M. R. Bridson and M. Conder and I would like to thank them for their collaborations. I would also like to thank the organizers of Groups St Andrews 2013 for their invitation to deliver the lectures, for their hopsitality at the conference, and for their patience whilst this article was completed.
    [Show full text]
  • Profinite Groups, by J. S. Wilson, Clarendon Press, Oxford, 1998, Ix+284 Pp., $115.00, ISBN 0-19-850082-3 Profinite Groups, by L
    BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 38, Number 4, Pages 475{479 S 0273-0979(01)00914-4 Article electronically published on June 12, 2001 Profinite groups, by J. S. Wilson, Clarendon Press, Oxford, 1998, ix+284 pp., $115.00, ISBN 0-19-850082-3 Profinite groups, by L. Ribes and P. Zalesskii, Springer, New York, 2000, xiv+ 435 pp., $109.00, ISBN 3-540-66986-8 Analytic pro-p groups, by J. D. Dixon, M. P. F. du Sautoy, A. Mann, and D. Segal, 2nd edition, Cambridge University Press, Cambridge, UK, 1999, xviii+ 368 pp., $59.95, ISBN 0-521-65011-9 New horizons in pro-p groups, by M. du Sautoy, D. Segal, and A. Shalen (eds.), Birkh¨auser, Boston, 2000, xiii+423 pp., $79.00, ISBN 0-8176-4171-8 The theory of profinite groups is flourishing! This is the first immediate obser- vation from looking at the four books on the subject which have come out in the last two years. The subject, which only two decades ago was somewhat remote, has made its way to mainstream mathematics in several different ways. What is a profinite group? A profinite group G is a topological group which is Hausdorff, compact and totally disconnected. A more meaningful equivalent definition is that G is the inverse limit of finite groups. As such, much of the theory of finite groups can be carried over to profinite groups. The origin of the subject lies in Galois theory: If E is an infinite algebraic Galois extension of a field F and G =Gal(E=F) is the group of the automorphisms of E fixing F , then the classical Galois theory does not hold anymore.
    [Show full text]
  • ON the RESIDUAL FINITENESS of GENERALISED FREE PRODUCTS of NILPOTENT GROUPS by GILBERT BAUMSLAG(I)
    ON THE RESIDUAL FINITENESS OF GENERALISED FREE PRODUCTS OF NILPOTENT GROUPS BY GILBERT BAUMSLAG(i) 1. Introduction. 1.1. Notation. Let c€ be a class of groups. Then tfë denotes the class of those groups which are residually in eS(2), i.e., GeRtë if, and only if, for each x e G (x # 1) there is an epimorph of G in ^ such that the element corresponding to x is not the identity. If 2 is another class of groups, then we denote by %>•Qt the class of those groups G which possess a normal subgroup N in # such that G/NeSi(f). For convenience we call a group G a Schreier product if G is a generalised free product with one amalgamated subgroup. Let us denote by , . _> o(A,B) the class of all Schreier products of A and B. It is useful to single out certain subclasses of g (A, B) by specifying that the amalgamated subgroup satisfies some condition T, say; we denote this subclass by °(A,B; V). Thus o(A,B; V) consists of all those Schreier products of A and B in which the amalgamated subgroup satisfies the condition T. We shall use the letters J5", Jf, and Í» to denote, respectively, the class of finite groups, the class of finitely generated nilpotent groups without elements of finite order, and the class of free groups. 1.2. A negative theorem. The simplest residually finite(4) groups are the finitely generated nilpotent groups (K. A. Hirsch [2]). In particular, then, (1.21) Jf c k3F.
    [Show full text]
  • Engulfing in Word-Hyperbolic Groups 1 Introduction
    ISSN 1472-2739 (on-line) 1472-2747 (printed) 743 Algebraic & Geometric Topology Volume 2 (2002) 743–755 ATG Published: 9 September 2002 Engulfing in word-hyperbolic groups G. A. Niblo B. T. Williams Abstract We examine residual properties of word-hyperbolic groups, ad- apting a method introduced by Darren Long to study the residual properties of Kleinian groups. AMS Classification 20E26; 20F67, 20F65 Keywords Word hyperbolic groups, residual finiteness, engulfing 1 Introduction A group is said to be residually finite if the intersection of its finite index sub- groups is trivial. Equivalently it is residually finite if the trivial subgroup is closed in the profinite topology. It is an open question whether or not word hyperbolic groups are residually finite. Evidence that they may be comes from the observation that many familiar groups in this class are linear and there- fore residually finite by an application of Selberg’s lemma. Furthermore there are geometric methods for establishing the residual finiteness of free groups [5], surface groups [11] and some reflection groups [13] that may generalise. Nonetheless the general question seems hard to settle, hindered by the appar- ent difficulty of establishing that a given group contains any proper finite index subgroups at all. In [8] Long hypothesised this difficulty away by assuming that the groups he studied satisfied an engulfing property: Definition 1.1 A subgroup H in a group G is said to be engulfed if H is contained in a proper finite index subgroup of G. The group G has the engulfing property with respect to a class H of subgroups of G if every subgroup in the class H is engulfed in G.
    [Show full text]
  • Separability of Solvable Subgroups in Linear Groups
    Separability of Solvable Subgroups in Linear Groups Roger Alperin and Benson Farb∗ August 11, 2004 Abstract Let Γ be a finitely generated linear group over a field of charac- teristic 0. Suppose that every solvable subgroup of Γ is polycyclic. Then any solvable subgroup of Γ is separable. This conclusion is false without the hypothesis that every solvable subgroup of Γ is polycyclic. 1 Introduction A subgroup S of a finitely generated group Γ is called separable (in Γ) if S is the intersection of all the subgroups of finite index containing S. This can be restated: for every x 6∈ S, there is a finite quotient π :Γ → G of Γ with π(x) ∈/ π(S); that is, x and S are separated in some finite quotient G. When S is the trivial subgroup, separability of S in Γ is the same as Γ being residually finite. It is a basic problem of infinite group theory to determine which sub- groups S of a given Γ are separable; see, e.g., [ALR, Be, Gi, H, HW, MR] for some recent progress on this problem. Apart from applications in geometric topology (see, e.g. [AH, Mc] and the references therein, and Corollary 1.2 below), separability also arises when studying the profinite topology on Γ: a subgroup S is closed in this topology precisely when it is separable. In this paper we will consider finitely generated linear groups Γ over a field K of characteristic 0, that is subgroups Γ < GL(n, K). Malcev proved that every finitely generated linear group Γ is residually finite.
    [Show full text]
  • Profinite Properties of 3-Manifold Groups
    Profinite Properties of 3-Manifold Groups Gareth Wilkes St John's College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Trinity 2018 Dedicated to my family. Acknowledgements Firstly I give great thanks to Marc Lackenby for his excellent supervision and advice throughout my DPhil and for being a font of knowledge from whom I have learnt a great deal. I would also like to thank him for so carefully reading all my mathematics and rooting out any misconceptions. I would also like to acknowledge several academics and professors who have invited me for visits over the course of my DPhil and with whom I have had very enlightening discussions. In particular I thank Henry Wilton at Cambridge, Stefan Freidl at Regensburg and Michel Boileau at Marseille. I also thank Peter Neumann for his advice on the giving of seminars and on proper mathematical style, as well as for illuminating Kinderseminar with his great knowledge. Huge thanks go to my Mum, my Dad, my sister Lizzie and all my family and friends for all their support over the last few years and for picking me up when I needed it. Among my friends I would particularly like to mention Joseph Mason, In´esDawson, Alexander Roberts, Federico Vigolo, Robert Kropholler, TomaˇsZeman, Josephine French, David Hume, Ric Wade and Guy-Andr´eYerro. Federico gets additional thanks for helping me with some of the diagrams. Finally I thank the plant Camellia sinensis for providing the lifeblood of my mathematics. Abstract In this thesis we study the finite quotients of 3-manifold groups, con- cerning both residual properties of the groups and the properties of the 3-manifolds that can be detected using finite quotients of the fundamental group.
    [Show full text]
  • Grothendieck's Problems Concerning Profinite Completions And
    Annals of Mathematics, 160 (2004), 359–373 Grothendieck’s problems concerning profinite completions and representations of groups By Martin R. Bridson and Fritz J. Grunewald Abstract In 1970 Alexander Grothendieck [6] posed the following problem: let Γ1 and Γ2 be finitely presented, residually finite groups, and let u :Γ1 → Γ2 be a homomorphism such that the induced map of profinite completionsu ˆ : Γˆ1 → Γˆ2 is an isomorphism; does it follow that u is an isomorphism? In this paper we settle this problem by exhibiting pairs of groups u : P→ Γ such that Γ is a direct product of two residually finite, hyper- bolic groups, P is a finitely presented subgroup of infinite index, P is not abstractly isomorphic to Γ, butu ˆ : Pˆ → Γˆ is an isomorphism. The same construction allows us to settle a second problem of Grothendieck by exhibiting finitely presented, residually finite groups P that have infinite index in their Tannaka duality groups clA(P ) for every commu- tative ring A =0. 1. Introduction The profinite completion of a group Γ is the inverse limit of the di- rected system of finite quotients of Γ; it is denoted by Γ.ˆ If Γ is residually finite then the natural map Γ → Γˆ is injective. In [6] Grothendieck discov- ered a remarkably close connection between the representation theory of a finitely generated group and its profinite completion: if A = 0 is a commu- tative ring and u :Γ1 → Γ2 is a homomorphism of finitely generated groups, thenu ˆ : Γˆ1 → Γˆ2 is an isomorphism if and only if the restriction functor ∗ → uA : RepA(Γ2) RepA(Γ1) is an equivalence of categories, where RepA(Γ) is the category of finitely presented A-modules with a Γ-action.
    [Show full text]
  • Virtual Retraction Properties in Groups
    VIRTUAL RETRACTION PROPERTIES IN GROUPS ASHOT MINASYAN Abstract. If G is a group, a virtual retract of G is a subgroup which is a retract of a finite index subgroup. Most of the paper focuses on two group properties: property (LR), that all finitely generated subgroups are virtual retracts, and property (VRC), that all cyclic subgroups are virtual retracts. We study the permanence of these properties under commensurability, amalgams over retracts, graph products and wreath products. In particular, we show that (VRC) is stable under passing to finite index overgroups, while (LR) is not. The question whether all finitely generated virtually free groups satisfy (LR) motivates the remaining part of the paper, studying virtual free factors of such groups. We give a simple criterion characterizing when a finitely generated subgroup of a virtually free group is a free factor of a finite index subgroup. We apply this criterion to settle a conjecture of Brunner and Burns. 1. Introduction Recall that a subgroup H of a group K is called a retract if there is a homomorphism ρ : K ! H which restricts to the identity map on H. This is equivalent to K splitting as a semidirect product N o H, where N = ker ρ. In this case the map ρ is called a retraction of K onto H. Definition 1.1. Let G be a group and let H be a subgroup of G. We will say that H is a virtual retract of G, denoted H 6vr G, if there exists a subgroup K 6 G such that jG : Kj < 1, H ⊆ K and H is a retract of K.
    [Show full text]
  • Residual Properties of Automorphism Groups of (Relatively) Hyperbolic Groups Gilbert Levitt Ashot Minasyan
    Residual properties of automorphism groups of (relatively) hyperbolic groups Gilbert Levitt Ashot Minasyan We show that Out(G) is residually finite if G is one-ended and hyperbolic relative to virtually polycyclic subgroups. More generally, if G is one-ended and hyperbolic relative to proper residually finite subgroups, the group of outer automorphisms preserving the peripheral structure is residually finite. We also show that Out(G) is virtually residually p-finite for every prime p if G is one-ended and toral relatively hyperbolic, or infinitely- ended and virtually residually p-finite. 20F67, 20F28; 20E26 1 Introduction A group G is residually finite if, given any g = 1, there exists a homomorphism ϕ from G to a finite group such that ϕ(g) = 1. Residual finiteness is an important property of groups. It is equivalent to the statement that G embeds into its profinite completion. Well known theorems of Mal’cev claim that finitely generated residually finite groups are Hopfian, and finitely presented residually finite groups have solvable word problem. Many groups are known to be residually finite (in particular, finitely generated linear groups), but it is a big open question whether all (Gromov) hyperbolic groups are residually finite. It is a standard and classical fact [5] that the automorphism group Aut(G) is residually finite if G is finitely generated and residually finite, but this is not true for the outer automorphism group Out(G). Indeed, any finitely presented group may be represented as Out(G) with G finitely generated and residually finite [8]. A special case of our main theorem is: Corollary 1.1 If G is one-ended, and hyperbolic relative to a family P = {P1,..., Pk} of virtually polycyclic groups, then Out(G) is residually finite.
    [Show full text]