Acylindrical group actions on quasi-trees

TL;DR

This paper proves that every acylindrically hyperbolic group can be generated so that its Cayley graph is a quasi-tree with an acylindrical action, using hyperbolically embedded subgroups and projection complexes.

Contribution

It establishes that all acylindrically hyperbolic groups admit generating sets with Cayley graphs that are quasi-trees, advancing understanding of their geometric structure.

Findings

Existence of generating sets with Cayley graphs as quasi-trees

New results on hyperbolically embedded subgroups

Insights into quasi-convex subgroups of acylindrically hyperbolic groups

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 is a (non-elementary) quasi-tree and the action of G on the Cayley graph is acylindrical. Our proof utilizes the notions of hyperbolically embedded subgroups and projection complexes. As a by-product, we obtain some new results about hyperbolically embedded subgroups and quasi-convex subgroups of acylindrically hyperbolic groups.