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. In the appendix, it is verified that any space satisfying the a priori weaker property of being an "almost hierarchically hyperbolic space" is actually a hierarchically hyperbolic space. The results of the appendix are used to streamline the proofs in the main text.

The local-to-global property for Morse quasi-geodesics

“…We call the function M the Morse gauge of γ . 1 The study of Morse quasi-geodesics arose from trying to understand the "hyperbolic directions" in a non-hyperbolic space [20] and has since received immense interest in the literature (see [1,2,23,34,40] for a sampling). Numerous results from hyperbolic spaces have fruitful generalizations to spaces containing infinite Morse quasi-geodesics, particularly with respect to the study of stable subgroups [5,7,16] and the quasi-isometric classification of spaces [15,17].…”

“…For the mapping class group, Teichmüller space, and graph products of hyperbolic groups, we prove a more general result for the class of hierarchically hyperbolic spaces satisfying a minor technical condition that encompasses these examples. The proof for hierarchically hyperbolic spaces rests upon a characterization of Morse quasi-geodesics in these space due to Abbott, Behrstock, and Durham [1]. Virtually solvable groups and groups with infinite order central elements are examples of spaces where the Morse geodesics have uniformly bounded length.…”

“…Stable subgroups are a strong generalization of quasiconvex subgroups that requires every pair of elements to be joined by a uniform Morse geodesic that stays uniformly close to the subgroup. Stable subgroups were introduced by Durham and Taylor to study the convex cocompact subgroups of the mapping class group [26], but have sense been studied in a variety of non-hyperbolic groups [1,34,45]. Theorem G is particularly interesting in the mapping class group, where it produces combination theorems for the convex cocompact subgroups by the work of Durham and Taylor (see Sect.…”

Random walks and quasi‐convexity in acylindrically hyperbolic groups

“…We refer to [14,15,64] 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 a 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 [2,9,29,45]. We prove an analogue of [54,Proposition 1] in the setting of a stable subgroup H of an acylindrically hyperbolic group G (Theorem 5.1).…”