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Hyperbolic structures
on groups
Carolyn Abbott, Sahana H Balasubramanya and Denis Osin |
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Algebraic & Geometric Topology 19 (2019) 1747–1835
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Abstract |
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For every group , we define the set of hyperbolic structures on , denoted by , which consists of equivalence classes of (possibly infinite) generating sets of such that the corresponding Cayley graph is hyperbolic; two generating sets of are equivalent if the corresponding word metrics on are bi-Lipschitz equivalent. Alternatively, one can define hyperbolic structures in terms of cobounded –actions on hyperbolic spaces. We are especially interested in the subset of acylindrically hyperbolic structures on , ie hyperbolic structures corresponding to acylindrical actions. Elements of can be ordered in a natural way according to the amount of information they provide about the group . The main goal of this paper is to initiate the study of the posets and for various groups . 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 , 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
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Mathematical Subject Classification 2010
Primary: 20F65
Secondary: 20E08, 20F67
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References |
Publication
Received: 14 November 2017
Revised: 17 November 2018
Accepted: 22 December 2018
Published: 16 August 2019
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Authors |
| Carolyn Abbott | |
| University of California,
Berkeley Berkeley, CA United States |
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| Sahana H Balasubramanya | |
| Department of Mathematics &
Statistics University of North Carolina at Greensboro Greensboro, NC United States |
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| Denis Osin | |
| Department of Mathematics Vanderbilt University Nashville, TN United States |
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