Hyperbolic structures on groups

Abbott, Carolyn; Balasubramanya, Sahana; Osin, Denis

doi:10.2140/agt.2019.19.1747

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Hyperbolic structures on groups

Carolyn Abbott, Sahana H Balasubramanya and Denis Osin

Algebraic & Geometric Topology 19 (2019) 1747–1835

DOI: 10.2140/agt.2019.19.1747

Abstract

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

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Keywords

hyperbolic structures, group actions on hyperbolic spaces, acylindrical actions

Mathematical Subject Classification 2010

Primary: 20F65

Secondary: 20E08, 20F67

References

Publication

Received: 14 November 2017

Revised: 17 November 2018

Accepted: 22 December 2018

Published: 16 August 2019

Authors

Carolyn Abbott
University of California, Berkeley Berkeley, CA United States

Sahana H Balasubramanya
Department of Mathematics & Statistics University of North Carolina at Greensboro Greensboro, NC United States

Denis Osin
Department of Mathematics Vanderbilt University Nashville, TN United States

Volume 19, issue 4 (2019)