Hyperbolic structures on groups

Abstract

For every group $G$, we define the set of hyperbolic structures on $G$, denoted by $\mathcal{\mathscr{H}}\left(G\right)$, 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 $\mathcal{A}\mathcal{\mathscr{H}}\left(G\right)\subseteq \mathcal{\mathscr{H}}\left(G\right)$ of acylindrically hyperbolic structures on $G$, ie hyperbolic structures corresponding to acylindrical actions. Elements of $\mathcal{\mathscr{H}}\left(G\right)$ 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 $\mathcal{\mathscr{H}}\left(G\right)$ and $\mathcal{A}\mathcal{\mathscr{H}}\left(G\right)$ 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.