Acylindrical group actions on quasi-trees

Balasubramanya, Sahana

doi:10.2140/agt.2017.17.2145

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Acylindrical group actions on quasi-trees

Sahana H Balasubramanya

Algebraic & Geometric Topology 17 (2017) 2145–2176

DOI: 10.2140/agt.2017.17.2145

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

Volume 17, issue 4 (2017)