#### 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 $\Gamma$ is a (non-elementary) quasi-tree and the action of $G$ on $\Gamma$ 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
##### 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