# Largest acylindrical actions and stability in hierarchically hyperbolic groups

## TL;DR

This paper studies hierarchically hyperbolic groups, establishing the existence of a largest universal acylindrical action on hyperbolic spaces and classifying stable subgroups, thus advancing understanding of non-positive curvature properties.

## Contribution

It proves the existence of a unique largest acylindrical action for hierarchically hyperbolic groups and classifies their stable subgroups, extending known results to a broader class of groups.

## Key findings

- Existence of a largest universal acylindrical action for hierarchically hyperbolic groups
- Complete classification of stable subgroups in these groups
- Simplified characterization of contracting quasigeodesics

## 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 studying many important families of groups, including mapping class groups, right-angled Coxeter and Artin groups, most 3-manifold groups, and many others. A group that admits an acylindrical action on a hyperbolic space may admit many such actions on different hyperbolic spaces, so it is natural to search for a "best" one. The set of all cobounded acylindrical actions on hyperbolic spaces admits a natural poset structure; 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 outside the context of hyperbolic groups. We provide a complete classification of stable subgroups of hierarchically hyperbolic groups, generalizing and extending results that are known for mapping class groups and right-angled Artin groups. We also 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.