A characterization of $C^*$-simplicity of countable groups via Poisson boundaries

TL;DR

The paper connects $C^*$-simplicity of countable groups to the dynamics of their Poisson boundaries under random walks driven by generic measures. It develops a non-freeness criterion for stabilizers, a versatile decomposition lemma for convolution powers, and shows that non-$C^*$-simple groups exhibit generic non-free boundary actions while $C^*$-simple groups have generic essentially free boundary actions via HK measures. The main outcome is a precise characterization: $G$ is $C^*$-simple iff for a generic (symmetric) measure $\\nu$, the action of $G$ on the Poisson boundary $\\partial(G,\\nu)$ is essentially free. The results deepen the bridge between operator-algebraic simplicity and probabilistic boundary theory, extending previous work on Furstenberg boundaries and amenable radicals, and offering a probabilistic lens on $C^*$-simplicity and related rigidity phenomena.

Abstract

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

Theorems & Definitions (39)

• Theorem A
• Definition 1
• Theorem 1
• Corollary 1
• proof
• Corollary 2
• Theorem B
• Theorem C
• Lemma 1
• proof