Characterising acylindrical hyperbolicity via permutation actions

TL;DR

This paper provides a new characterization of acylindrical hyperbolicity of groups through properties of their actions on sets, including actions on the group itself and on the Furstenberg-Poisson boundary, without additional structure.

Contribution

It introduces a novel characterization of acylindrical hyperbolicity based solely on group actions on sets, broadening understanding of this property.

Findings

Characterizes acylindrical hyperbolicity via group actions on sets

Applies to actions on the group itself and on the Furstenberg-Poisson boundary

Provides criteria for acylindrical hyperbolicity without extra structure

Abstract

We characterise acylindrical hyperbolicity of a group in terms of properties of an action of the group on a set (without any extra structure). In particular, this applies to the action of the group on itself by left multiplication, as well as the action on a (full measure subset of the) Furstenberg-Poisson boundary.