Faithful invariant random subgroups in acylindrically hyperbolic groups

TL;DR

This paper demonstrates that acylindrically hyperbolic and non-elementary hyperbolic groups possess faithful, weakly mixing invariant random subgroups, expanding understanding of their subgroup structures and dynamics.

Contribution

It establishes the existence of faithful, weakly mixing invariant random subgroups in acylindrically hyperbolic and hyperbolic groups, a novel result in the study of group actions.

Findings

Existence of faithful, weakly mixing IRS in acylindrically hyperbolic groups

Existence of characteristic random subgroups with similar properties in hyperbolic groups

Extension of previous work by Sun and Kechris-Quorning

Abstract

Building on work from Sun and Kechris-Quorning, we prove that every acylindrically hyperbolic group $G$ admits a weakly mixing probability measure preserving action $G \curvearrowright (X,\mathcal{B},\mu)$ which is faithful but not essentially free. In other words, $G$ admits a weakly mixing nontrivial faithful IRS. We also prove that every non-elementary hyperbolic group admits a characteristic random subgroup with the same properties.