The Poisson boundary of Thompson's group $T$ is not the circle

Abstract

Let $\mu$ be a nondegenerate probability measure with finite entropy on a countable group $G\le {\mathrm{Homeo}}_{+}(S^1)$ of orientation-preserving homeomorphisms of the circle acting proximally, minimally and topologically nonfreely on $S^1$. We prove that the circle $S^1$ endowed with its unique $\mu$-stationary probability measure is not the Poisson boundary of $(G,\mu )$. When $G$ is Thompson's group $T$ and $\mu$ is finitely supported, this answers a question posed by B. Deroin [Ergodic Theory Dynam. Systems, 2013] and A. Navas [Proceedings of the International Congress of Mathematicians, 2018].