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

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.