Acylindrical group actions on quasi-trees

Abstract

A group G is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. We prove that every acylindrically hyperbolic group G has a generating set X such that the corresponding Cayley graph Γ is a (non-elementary) quasi-tree and the action of G on Γ is acylindrical. Our proof utilizes the notions of hyperbolically embedded subgroups and projection complexes. As an application, we obtain some new results about hyperbolically embedded subgroups and quasi-convex subgroups of acylindrically hyperbolic groups. IntroductionRecall that an isometric action of a group G on a metric space (S, d) is acylindrical if for every ε > 0 there exist R, N > 0 such that for every two points x, y with d(x, y) ≥ R, there are at most N elements g ∈ G satisfying d(x, gx) ≤ ε and d(y, gy) ≤ ε.Obvious examples are provided by geometric (i.e., proper and cobounded) actions; note, however, that acylindricity is a much weaker condition.A group G is called acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. Over the last few years, the class of acylindrically hyperbolic groups has received considerable attention. It is broad enough to include many examples of interest, e.g., non-elementary hyperbolic and relatively hyperbolic groups, all but finitely many mapping class groups of punctured closed surfaces, Out(F n ) for n ≥ 2, most 3-manifold groups, and finitely presented groups of deficiency at least 2. On the other hand, the existence of a non-elementary acylindrical action on a hyperbolic space is a rather strong assumption, which allows one to prove non-trivial results. In particular, acylindrically hyperbolic groups share many interesting properties with non-elementary hyperbolic and relatively hyperbolic groups. For details we refer to [5,10,11,12] and references therein.The main goal of this paper is to answer the following.Question 1.1. Which groups admit non-elementary cobounded acylindrical actions on quasi-trees?In this paper, by a quasi-tree we mean a connected graph which is quasi-isometric to a tree. Quasi-trees form a very particular subclass of the class of all hyperbolic spaces. From the asymptotic point of view, quasi-trees are exactly "1-dimensional hyperbolic spaces".

Random walks, WPD actions, and the Cremona group

“…This result is implicit in the constructions of projection complexes in [1, 3, 5, 15, 51] and thus is likely well known. As we are unable to find a reference in the literature, in the next two subsections we provide a proof using published results of [5].…”

“…Bestvina, Bromberg and Fujiwara [3] showed that is a quasi‐tree for all sufficiently large. Osin [51] defined a slightly different space on which the action of is acylindrically hyperbolic, and Balasubramanya [1] showed that this construction could be modified to guarantee that the space is a quasi‐tree. In fact, we shall use the version from Bestvina, Bromberg, Fujiwara and Sisto [5], which also construct a projection complex which is a quasi‐tree and on which acts acylindrically.…”

Groups acting on hyperbolic spaces with a locally finite orbit

“…Moreover (Corollary 7.4) if further G is finitely generated and X is a quasitree then G is virtually free. This contrasts vastly with [2] which showed that any acylindrical hyperbolic group acts acylindrically on some graph which is a quasitree (but not locally finite).…”

Groups Acting Acylindrically on Hyperbolic Spaces

“…Part (c) of this theorem is especially useful for studying properties of acylindrically hyperbolic groups since it allows to pass from a (possibly non-cobounded) action of G on a general hyperbolic space to the more familiar action on the Cayley graph. In addition, one can ensure that Γ(G, X) is quasi-isometric to a tree [6].…”