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2019 Hyperbolic structures on groups
Carolyn Abbott, Sahana H Balasubramanya, Denis Osin
Algebr. Geom. Topol. 19(4): 1747-1835 (2019). DOI: 10.2140/agt.2019.19.1747


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.


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Carolyn Abbott. Sahana H Balasubramanya. Denis Osin. "Hyperbolic structures on groups." Algebr. Geom. Topol. 19 (4) 1747 - 1835, 2019.


Received: 14 November 2017; Revised: 17 November 2018; Accepted: 22 December 2018; Published: 2019
First available in Project Euclid: 22 August 2019

zbMATH: 07121514
MathSciNet: MR3995018
Digital Object Identifier: 10.2140/agt.2019.19.1747

Primary: 20F65
Secondary: 20E08, 20F67

Rights: Copyright © 2019 Mathematical Sciences Publishers


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Vol.19 • No. 4 • 2019
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