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arXiv · 2504.01283

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

Martín Gilabert Vio·Cosmas Kravaris·Eduardo Silva
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Abstract

Let $\mu$ be a nondegenerate probability measure with finite entropy on a countable group $G \leq \mathrm{Homeo}_+(S^1)$ 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].

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Martín Gilabert Vio, Cosmas Kravaris, Eduardo Silva. 2025-04-02. The Poisson boundary of Thompson's group $T$ is not the circle. https://arxiv.org/abs/2504.01283

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