Volume 17, issue 4 (2017)

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Acylindrical group actions on quasi-trees

Sahana H Balasubramanya

Algebraic & Geometric Topology 17 (2017) 2145–2176
Abstract

A group G is acylindrically hyperbolic if it admits a non-elementary acylindrical action on a hyperbolic space. We prove that every acylindrically hyperbolic group G has a generating set X such that the corresponding Cayley graph Γ is a (non-elementary) quasi-tree and the action of G on Γ is acylindrical. Our proof utilizes the notions of hyperbolically embedded subgroups and projection complexes. As an application, we obtain some new results about hyperbolically embedded subgroups and quasi-convex subgroups of acylindrically hyperbolic groups.

Keywords
acylindrically hyperbolic groups, acylindrical actions, projection complex, quasi-trees, hyperbolically embedded subgroups
Mathematical Subject Classification 2010
Primary: 20F67
Secondary: 20F65, 20E08
References
Publication
Received: 4 March 2016
Revised: 19 October 2016
Accepted: 30 October 2016
Published: 3 August 2017
Authors
Sahana H Balasubramanya
Department of Mathematics
Vanderbilt University
1326 Stevenson Center Ln
Nashville, TN
37240
United States