#### Volume 19, issue 4 (2019)

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Hyperbolic structures on groups

### Carolyn Abbott, Sahana H Balasubramanya and Denis Osin

Algebraic & Geometric Topology 19 (2019) 1747–1835
##### Abstract

For every group $G\phantom{\rule{0.3em}{0ex}}$, we define the set of hyperbolic structures on $G\phantom{\rule{0.3em}{0ex}}$, denoted by $\mathsc{ℋ}\left(G\right)$, 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 $\mathsc{A}\mathsc{ℋ}\left(G\right)\subseteq \mathsc{ℋ}\left(G\right)$ of acylindrically hyperbolic structures on $G\phantom{\rule{0.3em}{0ex}}$, ie hyperbolic structures corresponding to acylindrical actions. Elements of $\mathsc{ℋ}\left(G\right)$ can be ordered in a natural way according to the amount of information they provide about the group $G\phantom{\rule{0.3em}{0ex}}$. The main goal of this paper is to initiate the study of the posets $\mathsc{ℋ}\left(G\right)$ and $\mathsc{A}\mathsc{ℋ}\left(G\right)$ for various groups $G\phantom{\rule{0.3em}{0ex}}$. 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 $G\phantom{\rule{0.3em}{0ex}}$, and study to what extent a hyperbolic structure is determined by the set of loxodromic elements and their translation lengths.

##### Keywords
hyperbolic structures, group actions on hyperbolic spaces, acylindrical actions
##### Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 20E08, 20F67