Hierarchically hyperbolic groups and uniform exponential growth

We give several sufficient conditions for uniform exponential growth in the setting of virtually torsion-free hierarchically hyperbolic groups. For example, any hierarchically hyperbolic group that is also acylindrically hyperbolic has uniform exponential growth. In addition, we provide a quasi-isometric characterizations of hierarchically hyperbolic groups without uniform exponential growth. To achieve this, we gain new insights on the structure of certain classes of hierarchically hyperbolic groups. Our methods give a new unified proof of uniform exponential growth for several examples of groups with notions of non-positive curvature. In particular, we obtain the first proof of uniform exponential growth for certain groups that act geometrically on CAT(0) cubical groups of dimension 3 or more. Under additional hypotheses, we show that a quantitative Tits alternative holds for hierarchically hyperbolic groups.

Actions of small cancellation groups on hyperbolic spaces

We generalize Gruber--Sistos construction of the coned--off graph of a small cancellation group to build a partially ordered set $mathcal{TC}$ of cobounded actions of a given small cancellation group whose smallest element is the action on the Gruber--Sisto coned--off graph. In almost all cases $mathcal{TC}$ is incredibly rich: it has a largest element if and only if it has exactly 1 element, and given any two distinct comparable actions $[Gcurvearrowright X] preceq [Gcurvearrowright Y]$ in this poset, there is an embeddeding $iota:P(omega)tomathcal{TC}$ such that $iota(emptyset)=[Gcurvearrowright X]$ and $iota(mathbb N)=[Gcurvearrowright Y]$. We use this poset to prove that there are uncountably many quasi--isometry classes of finitely generated group which admit two cobounded acylindrical actions on hyperbolic spaces such that there is no action on a hyperbolic space which is larger than both.

Hyperbolic structures on groups

For every group $G$, we introduce the set of hyperbolic structures on $G$, denoted $mathcal{H}(G)$, which consists of equivalence classes of (possibly infinite) generating sets of $G$ such that the corresponding Cayley graph is hyperbolic; two generating sets of $G$ are equivalent if the corresponding word metrics on $G$ are bi-Lipschitz equivalent. Alternatively, one can define hyperbolic structures in terms of cobounded $G$-actions on hyperbolic spaces. We are especially interested in the subset $mathcal{AH}(G)subseteq mathcal{H}(G)$ of acylindrically hyperbolic structures on $G$, i.e., hyperbolic structures corresponding to acylindrical actions. Elements of $mathcal{H}(G)$ can be ordered in a natural way according to the amount of information they provide about the group $G$. The main goal of this paper is to initiate the study of the posets $mathcal{H}(G)$ and $mathcal{AH}(G)$ for various groups $G$. We discuss basic properties of these posets such as cardinality and existence of extremal elements, obtain several results about hyperbolic structures induced from hyperbolically embedded subgroups of $G$, and study to what extent a hyperbolic structure is determined by the set of loxodromic elements and their translation lengths.

Largest acylindrical actions and stability in hierarchically hyperbolic groups

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.