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

2025/04/02 by Vio, Martín Gilabert, Kravaris, Cosmas, Silva, Eduardo · 1 citation
#Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Probability (math.PR)

paper · doi:10.48550/arxiv.2504.01283

Abstract

Let μ be a nondegenerate probability measure with finite entropy on a countable group G ≤ Homeo₊(S¹) of orientation-preserving homeomorphisms of the circle acting proximally, minimally and topologically nonfreely on S¹. We prove that the circle S¹ endowed with its unique μ-stationary probability measure is not the Poisson boundary of (G,μ). When G is Thompson's group T and μ 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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• Non-Realizability of the Poisson Boundary

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• Thompson's group F is not Liouville
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