Non-C*-simple groups admit non-free actions on their Poisson boundaries

TL;DR

This paper demonstrates that non-C*-simple groups can have non-free actions on their Poisson boundaries with non-trivial stabilizers, expanding understanding of group actions beyond previously known cases.

Contribution

It shows that all countable non-C*-simple groups admit symmetric measures with non-trivial stabilizers on their Poisson boundaries, including those with trivial amenable radical.

Findings

Existence of symmetric measures with full support and non-trivial stabilizers for all non-C*-simple groups.

Examples of non-C*-simple groups with trivial amenable radical have non-normal stabilizers.

Non-C*-simple groups can have non-free actions on their Poisson boundaries.

Abstract

It is a classical result of Kaimanovich and Vershik and independently of Rosenblatt that a non-amenable group admits a non-degenerate symmetric measure such that the Poisson boundary is trivial. Most if not all examples to date of non-free actions of countable groups on their Poisson boundaries had the stabilizers sitting inside the amenable radical. We show that every countable non-C*-simple group admits a symmetric measure of full support with non-trivial stabilizers. For a class of non-C*-simple groups with trivial amenable radical, which is non-empty as was shown by le Boudec, this gives a wealth of examples with non-normal stabilizers.