Largest acylindrical actions and stability in hierarchically hyperbolic groups

Abstract

We consider two manifestations of non-positive curvature: acylindrical actions (on hyperbolic spaces) and quasigeodesic stability. We study these properties for the class of hierarchically hyperbolic groups, which is a general framework for simultaneously studying many important families of groups, including mapping class groups, right-angled Coxeter groups, most 3-manifold groups, right-angled Artin groups, and many others.A group that admits an acylindrical action on a hyperbolic space may admit many such actions on different hyperbolic spaces. It is natural to try to develop an understanding of all such actions and to search for a "best" one. The set of all cobounded acylindrical actions on hyperbolic spaces admits a natural poset structure, and in this paper we prove that all hierarchically hyperbolic groups admit a unique action which is the largest in this poset. The action we construct is also universal in the sense that every element which acts loxodromically in some acylindrical action on a hyperbolic space does so in this one. Special cases of this result are themselves new and interesting. For instance, this is the first proof that right-angled Coxeter groups admit universal acylindrical actions.The notion of quasigeodesic stability of subgroups provides a natural analogue of quasiconvexity which can be considered outside the context of hyperbolic groups. In this paper, we provide a complete classification of stable subgroups of hierarchically hyperbolic groups, generalizing and extending results that are known in the context of mapping class groups and right-angled Artin groups. Along the way, we provide a characterization of contracting quasigeodesics; interestingly, in this generality the proof is much simpler than in the special cases where it was already known.

Random walks and quasi-convexity in acylindrically hyperbolic groups

“…We refer to [12,13,60] for definitions and background on hierarchically hyperbolic groups. What is relevant for us is that such groups share many properties with mapping class groups; in particular, a hierarchically hyperbolic group G admits an acylindrical action on a hyperbolic metric space X such that a subgroup H ≤ G is stable in G if and only if it is quasi-isometrically embedded in X under the orbit map [2]. We conjecture that the natural analogue of Theorem 1.7 also holds in this setting.…”

“…In many cases when G is a group with an non-elementary partially WPD action on a hyperbolic metric space X, the subgroups of G which are quasi-isometrically embedded in X are precisely the stable subgroups of G in the sense of Durham-Taylor [2,7,24,40]. We prove an analogue of [49, Proposition 1] in the setting of a stable subgroup H of an acylindrically hyperbolic group G (Theorem 5.1).…”

Acylindrical hyperbolicity of groups acting on quasi-median graphs and equations in graph products

“…After the first version of this preprint was made available, it has been brought to the author's attention that most of the results stated in Corollary C follow from the results in [Gen16,Gen18,GM18]. Moreover, a special case of Corollary D (when the vertex groups G v are hierarchically hyperbolic) follows from the results in [ABD17,BR18]. See Remarks 5.4 and 5.5 for details.…”

“…Moreover, the action of G on X induces an action of G on (a space quasi-isometric to) U = x∈X π S (x) ⊆ ĈS, and in [BHS17, Theorem 14.3] it is shown that this action is acylindrical. In [ABD17], this construction is modified so that the action G U represents the largest element of AH(G). If Γ is connected, non-trivial, and the groups G v are infinite and hierarchically hyperbolic (with the intersection property and clean containers), then the proof of Corollary D gives this action ΓG U explicitly.…”

Acylindrical actions for two-dimensional Artin groups of hyperbolic type

“…An acylindrical action of a group G on a hyperbolic metric space X is called universal [Osi16] if the elements of G that are loxodromic for this action are exactly the elements of G that are loxodromic for some acylindrical action of G on a hyperbolic space (such elements are called generalised loxodromic). Universal actions often offer much deeper insight into the structure of the underlying group, and are known to exist for instance for right-angled Artin groups [ABD17]. The goal of this article is to show the existence of such a universal acylindrical action for a large class of Artin groups, and to use the dynamics of the action to understand the structure of certain of its subgroups.…”