# 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 groups, right-angled Artin groups, and many others. –manifold

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.

## 1. Introduction

Hierarchically hyperbolic groups were recently introduced by Behrstock, Hagen, and Sisto Reference BHS17b to provide a uniform framework in which to study many important families of groups, including mapping class groups of finite type surfaces, right-angled Coxeter groups, most groups, right-angled Artin groups and many others. A –manifold*hierarchically hyperbolic space* (HHS) consists of: a quasigeodesic space, a set of ;*domains*, which index a collection of , spaces to which –hyperbolic projects; and, some additional information about these projections, including, for instance, a partial order on the domains and a unique largest element in that order, which we denote by (i.e., is comparable to and larger than every other domain in ).

### Largest acylindrical actions

The study of acylindrical actions on hyperbolic spaces, as initiated in its current form by Osin Reference Osi16 building on earlier work of Sela Reference Sel97 and Bowditch Reference Bow08, has proven to be a powerful tool for studying groups with some aspects of non-positive curvature. As established in Reference BHS17b, non-virtually cyclic hierarchically hyperbolic groups admit non-elementary acylindrical actions when the space associated to the maximal element in –hyperbolic has infinite diameter, a property which holds in all the above examples except for those that are direct products.

Any given group with an acylindrical action may actually admit many acylindrical actions on many different spaces. A natural question is to try and find a “best” acylindrical action. There are different ways that one might try to optimize the acylindrical action. For instance, the notion of a *universal acylindrical action*, for a given group is an acylindrical action on a hyperbolic space , such that every element of which acts loxodromically in some acylindrical action on some hyperbolic space, must act loxodromically in its action on As established by Abbott, there exist finitely generated groups which admit acylindrical actions, but no universal acylindrical action .Reference Abb16; we also note that universal actions need not be unique Reference ABO19.

In Reference ABO19, Abbott, Balasubramanya, and Osin introduce a partial order on cobounded acylindrical actions which, in a certain sense, encodes how much information the action provides about the group. When there exists an element in this partial ordering which is comparable to and larger than all other elements it is called a *largest* action. By construction, any largest action is necessarily a universal acylindrical action and unique.

In this paper we construct a largest action for every hierarchically hyperbolic group. Special cases of this theorem recover some recent results of Reference ABO19, as well as a number of new cases. For instance, in the case of right-angled Coxeter groups (and more generally for special cubulated groups), even the existence of a universal acylindrical action was unknown. Further, outside of the relatively hyperbolic setting, our result provides a single construction that simultaneously covers these new cases as well as all previously known largest and universal acylindrical actions of finitely presented groups. The following summarizes the main results of Section 5 (where there are also further details on the background and comparison with known results).

We use this construction of a largest action to characterize stable subgroups (Theorem B) and contracting elements (Corollary 5.5) of hierarchically hyperbolic groups, and to describe random subgroups of hierarchically hyperbolic groups (Theorem E).

### Stability in hierarchically hyperbolic groups

One of the key features of a Gromov hyperbolic space is that every geodesic is uniformly *Morse*, a property also known as *(quasigeodesic) stability*; that is, any uniform quasigeodesic beginning and ending on a geodesic must lie uniformly close to it. In fact, any geodesic metric space in which each geodesic is uniformly Morse is hyperbolic.

In the context of geodesic metric spaces, the presence of Morse geodesics has important structural consequences for the space; for instance, any asymptotic cone of such a space has global cut points Reference DMS10. Moreover, quasigeodesic stability in groups is quite prevalent, since any finitely generated acylindrically hyperbolic group contains Morse geodesics Reference Osi16Reference Sis16.

There has been much interest in developing alternative characterizations Reference DMS10Reference CS15Reference ACGH17Reference ADT17 and understanding this phenomenon in various important contexts Reference Min96Reference Beh06Reference DMS10Reference DT15Reference ADT17. This includes the theory of Morse boundaries, which encode all Morse geodesics of a group Reference CS15Reference Cor17Reference CH17Reference CD19Reference CM19. In Reference DT15, Durham and Taylor generalized the notion of stability to subspaces and subgroups.

In this paper, we obtain a complete characterization of stability in hierarchically hyperbolic groups.

Let be an HHS. We say that a subset has * projections –bounded* when for all non-maximal when the constant does not matter, we simply say the subset has ;*uniformly bounded projections*.

Theorem B generalizes some previously known results. In the case of mapping class groups: Reference Beh06 proved that 2 implies 1 for cyclic subgroups; Reference DT15 proved equivalence of 1 and 3; equivalence of 2 and 3 follows from the distance formula; moreover, Reference KL08Reference Ham yield that these conditions are also equivalent to convex cocompactness in the sense of Reference FM02. The case of right-angled Artin groups was studied in Reference KMT17, where they prove equivalence of 1 and 3.

Section 6 contains a more general version of Theorem B, as well as further applications, including Theorem 6.6, which concerns the Morse boundary of hierarchically hyperbolic groups and proves that all hierarchically hyperbolic groups have finite stable asymptotic dimension.

### On purely loxodromic subgroups

In the mapping class group setting Reference BBKL20 proved that the conditions in Theorem B are also equivalent to being undistorted and purely pseudo-Anosov. Similarly, in the right-angled Artin group setting, it was proven in Reference KMT17 that 1 and 3 are each equivalent to being purely loxodromic.

Subgroups of right-angled Coxeter groups all of whose elements act loxodromically on the contact graph were studied in the recent preprint Reference Tra, Theorem 1.4, which proved that property is equivalent to 3. Since there often exist Morse elements in a right-angled Coxeter group which do not act loxodromically on the contact graph (which plays the role of in the standard HHG structure on the group), his condition is not equivalent to 1. It is the ability to change the hierarchically hyperbolic structure as we do in Theorem 3.7, discussed below, which allows us to prove our more general result which characterizes *all* stable subgroups, not just the ones acting loxodromically on the contact graph.

Mapping class groups and right-angled Artin groups have the property that in their standard hierarchically hyperbolic structure they admit a universal acylindrical action on where , is the hyperbolic space associated to the domain –maximal On the other hand, right-angled Coxeter groups often don’t admit universal acylindrical actions on . in their standard structure. Accordingly, we believe the following questions are interesting. The first item would generalize the situation in the mapping class group as established in Reference BBKL20, and the second item for right-angled Artin groups would generalize results proven in Reference KMT17, and for right-angled Coxeter groups would generalize results in Reference Tra. If the second item is true for the mapping class group, this would resolve a question of Farb–Mosher Reference FM02. See also Reference ADT17, Question 1.

Note that in the context of Question C, an element acts loxodromically on if and only if it has positive translation length. This holds since the action is acylindrical and thus each element either acts elliptically or loxodromically.

In an early version of this paper, we asked if the second part of Question C held for all hierarchically hyperbolic groups. In the general hierarchically hyperbolic setting, however, the undistorted hypothesis is necessary, as pointed out to us by Anthony Genevois with the following example. The necessity is shown by Brady’s example of a torsion-free hyperbolic group with a finitely presented subgroup which is not hyperbolic Reference Bra99. This subgroup is torsion-free and thus purely loxodromic. But, a subgroup of a hyperbolic group is stable if and only if it is quasiconvex. Thus, since this subgroup is not quasiconvex, we see that being purely loxodromic is strictly weaker than the conditions of Theorem B.

### New hierarchically hyperbolic structures

In order to establish the above results, we provide some new structural theorems about hierarchically hyperbolic spaces.

One of the key technical innovations in this paper is provided in Section 3. There we prove Theorem 3.7 which allows us to modify a given hierarchically hyperbolic structure by removing for some and, in their place, enlarging the space For instance, this is how we construct the space on which a hierarchically hyperbolic group has its largest acylindrical action. .

Another important tool is Theorem 4.4 which provides a simple characterization of contracting geodesics in a hierarchically hyperbolic space.

The following is a restatement of that result in the case of groups (see Theorem 4.4 for the precise statement):

Since the presence of a contracting geodesic implies the group has at least quadratic divergence, an immediate consequence of Theorem D is that any hierarchically hyperbolic group has quadratic divergence whenever projects to an infinite diameter subset of .

As a sample application of Theorem D and using work of Taylor–Tiozzo Reference TT16, we prove the following in Section 6.4 as Theorem 6.8.

We note that one immediate consequence of this result is a new proof of a theorem of Maher–Sisto: any random subgroup of a hierarchically hyperbolic group which is not the direct product of two infinite groups is stable Reference MS19. The mapping class group and right-angled Artin group cases of this result were first established in Reference TT16.

Finally, at the end of the paper we discuss a technical condition on hierarchically hyperbolic structures, called having *clean containers*. While in Proposition 7.2 this hypothesis is shown to hold for many groups, it does not hold in all cases. This condition was used in earlier versions of this paper in which it was assumed for the proof of Theorem 3.7, and then the general result was bootstrapped from there. In light of Theorem A.1 in the Appendix, this property is no longer required for this paper. We keep the contents of this section in the paper nonetheless, since they have found independent interest and already been used elsewhere, e.g., Reference BRReference HS16Reference Rus20, as well as in several papers in progress.

## 2. Background

We begin by recalling some preliminary notions about metric spaces, maps between them, and group actions. Given metric spaces we use , to denote the distance functions in respectively. A map , is * –Lipschitz* if there exists a constant such that for every , it is ;* Lipschitz –coarsely* if The map is a * embedding –quasi-isometric* if there exist constants and such that for all ,

If, in addition, is contained in the of –neighborhood then , is a * –quasi-isometry*. For any interval the image of an isometric embedding , is a *geodesic* and the image of a embedding –quasi-isometric is a * –quasigeodesic*.

If any two points in can be connected by a then we say –quasigeodesic, is a * space –quasigeodesic*. If we may simply say that , is a * space –quasigeodesic*. A subspace is * –quasi-convex* if there exists a constant such that any geodesic in connecting points in is contained in the of –neighborhood For all of the above notions, if the particular constants . are not important, we may drop them and simply say, for example, that a map is a quasi-isometry.

Throughout this paper, we will assume that all group actions are by isometries. The action of a group on a metric space which we denote by , is ,*proper* if for every bounded subset the set , is finite. The action is *cobounded* (respectively, *cocompact*) if there exists a bounded (respectively, compact) subset such that If a group . acts on metric spaces and we say a map , is * –equivariant* if for every and every we have A .*quasi-action* of on associates to each a quasi-isometry of with uniform quasi-isometry constants, such that is within uniformly bounded distance of .

### 2.1. Hierarchically hyperbolic spaces

In this section we recall the basic definitions and properties of hierarchically hyperbolic spaces as introduced in Reference BHS17bReference BHS19.

An important consequence of being a hierarchically hyperbolic space is the following distance formula, which relates distances in to distances in the hyperbolic spaces for The notation . means include in the sum if and only if .

We now recall an important construction of subspaces in a hierarchically hyperbolic space called *standard product regions* introduced in Reference BHS17b, Section 13 and studied further in Reference BHS19. First we define a *consistent tuple*, which will be used to define the two factors in the product space.

We often abuse notation slightly and use the notation and , to refer to the image in of the associated set. In Reference BHS19, Construction 5.10 it is proven that these standard product regions have the property that they are “hierarchically quasiconvex subsets” of We leave out the definition of hierarchical quasiconvexity, because its only use here is that product regions have “gate maps,” as given by the following in .Reference BHS19, Lemma 5.5:

We also need the notion of a hierarchy path, whose existence was proven in Reference BHS19, Theorem 4.4 (although we use the word *path*, since they are quasi-geodesics, typically we consider them as discrete sequences of points):

We call a domain *relevant* to a pair of points, if the projections to that domain are larger than some fixed (although possibly unspecified) constant depending only on the hierarchically hyperbolic structure. We say a domain is *relevant* for a particular quasi-geodesic if it is relevant for the endpoints of that quasi-geodesic.