A characterization of $C^{\ast }$-simplicity of countable groups via Poisson boundaries

Abstract

We characterize $C^{\ast }$-simplicity for countable groups by means of the following dichotomy. If a group is $C^{\ast }$-simple, then the action on the Poisson boundary is essentially free for a generic measure on the group. If a group is not $C^{\ast }$-simple, then the action on the Poisson boundary is not essentially free for a generic measure on the group.

This preprint superseds and expans upon my previous preprint arXiv:2409.02013