Stationary boundaries on the space of amenable subgroups and C*-simplicity

TL;DR

The paper establishes a new criterion for the existence of non-trivial stationary boundaries on the space of amenable subgroups, contrasting with known characterizations of C*-simplicity, and applies it to specific groups including Thompson's group F.

Contribution

It provides a sufficient condition for non-trivial stationary boundaries on amenable subgroups, extending understanding of C*-simplicity and boundary actions for certain groups.

Findings

The space of amenable subgroups is not uniquely stationary under certain measures.

The criterion applies to wreath products and Thompson's group F.

Non-trivial boundaries are supported on amenable normalish subgroups.

Abstract

We give a sufficient condition for a countable group $G$ to possess a probability measure $\mu$ that admits a non-trivial $\mu$-boundary modeled in the space $\mathrm{Sub}_{\mathrm{am}}(G)$ of amenable subgroups of $G$. In particular, for such $\mu$ the space $\mathrm{Sub}_{\mathrm{am}}(G)$ is not uniquely $\mu$-stationary. This contrasts with a theorem of Hartman-Kalantar, which states that a countable group $G$ is C*-simple if and only if there exists $\mu\in \mathrm{Prob}(G)$ such that $\mathrm{Sub}_{\mathrm{am}}(G)$ is uniquely $\mu$-stationary. Our criterion applies to (permutational) wreath products, which include groups that are C*-simple, and to Thompson's group $F$, whose C*-simplicity is equivalent to its non-amenability and therefore remains an open problem. We also show that any non-trivial $\mu$-boundary modeled on $\mathrm{Sub}_{\mathrm{am}}(G)$ is supported on amenable normalish subgroups, in the sense of Breuillard-Kalantar-Kennedy-Ozawa. As a consequence, we conclude that a countable group with no finite normal subgroups and no amenable normalish subgroups acts essentially freely on all its Poisson boundaries.