Hyperbolic structures on groups

Abstract

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.