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Hyperbolic structures on groups

Carolyn Abbott, Sahana H Balasubramanya and Denis Osin

Algebraic & Geometric Topology 19 (2019) 1747–1835

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

hyperbolic structures, group actions on hyperbolic spaces, acylindrical actions
Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 20E08, 20F67
Received: 14 November 2017
Revised: 17 November 2018
Accepted: 22 December 2018
Published: 16 August 2019
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