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

Martín Gilabert Vio, Cosmas Kravaris, Eduardo Silva

TL;DR

The paper proves that for groups G acting on the circle with a minimal, proximal, and topologically nonfree action, any nondegenerate μ with finite entropy yields a μ-boundary (S^1,ν) that is strictly smaller than the Poisson boundary; namely, (S^1,ν) cannot be the Poisson boundary of (G,μ). The authors develop an entropy-based argument using Kaimanovich’s conditional entropy criterion and an Erschler-style machinery, augmented by a dynamical input showing a linear number of dominating intervals along a random walk. In parallel, they construct a breakpoint boundary (R^{Br}, ν̃) from the derivative breakpoints, proving it is a μ-boundary not obtainable from (S^1,ν), and in the Thompson group T they even produce explicit μ-harmonic functions not arising via the circle boundary. These results yield concrete obstructions to circle-boundary identification in this setting and connect to broader questions about amenability and boundary rigidity for groups acting on the circle.

Abstract

Let $μ$ 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 $μ$-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].

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

TL;DR

The paper proves that for groups G acting on the circle with a minimal, proximal, and topologically nonfree action, any nondegenerate μ with finite entropy yields a μ-boundary (S^1,ν) that is strictly smaller than the Poisson boundary; namely, (S^1,ν) cannot be the Poisson boundary of (G,μ). The authors develop an entropy-based argument using Kaimanovich’s conditional entropy criterion and an Erschler-style machinery, augmented by a dynamical input showing a linear number of dominating intervals along a random walk. In parallel, they construct a breakpoint boundary (R^{Br}, ν̃) from the derivative breakpoints, proving it is a μ-boundary not obtainable from (S^1,ν), and in the Thompson group T they even produce explicit μ-harmonic functions not arising via the circle boundary. These results yield concrete obstructions to circle-boundary identification in this setting and connect to broader questions about amenability and boundary rigidity for groups acting on the circle.

Abstract

Let be a nondegenerate probability measure with finite entropy on a countable group of orientation-preserving homeomorphisms of the circle acting proximally, minimally and topologically nonfreely on . We prove that the circle endowed with its unique -stationary probability measure is not the Poisson boundary of . When is Thompson's group and 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].

Paper Structure

This paper contains 21 sections, 27 theorems, 86 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Main results
  3. Further background on identification of Poisson boundaries
  4. Structure of the proofs
  5. Organization
  6. Acknowledgements
  7. Preliminaries
  8. Random walks on groups and Poisson boundaries
  9. Conditional probabilities with respect to a $\mu$-boundary
  10. Entropy
  11. Groups acting on the circle
  12. Lower bounds on entropy
  13. Estimating asymptotic entropy
  14. Estimating conditional asymptotic entropy
  15. Good collections
  16. ...and 6 more sections

Key Result

Theorem 1

Let $G \leq \mathrm{Homeo}_+(S^1)$ be a countable group whose action on $S^1$ is minimal, proximal and topologically nonfree, and let $\mu$ be a finite entropy nondegenerate probability measure on $G$. Then $(S^1,\nu)$ is not the Poisson boundary of $(G, \mu)$.

Figures (3)

  • Figure 1: All red intervals $y_{i_{r'} - 1}(J)$ are dominated by the blue interval $y_{i_r - 1}(J)$.
  • Figure 2: The interval $w_{js}(J)$ does not dominate $J$.
  • Figure 3: The maps $a_n$ for $y = 1/2$ and $n = 2$ (blue), $n = 3$ (green) and $n = 4$ (red).

Theorems & Definitions (53)

  • Theorem 1
  • Theorem 2
  • Definition 2.1
  • Theorem 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Definition 2.5
  • Theorem 2.6
  • Lemma 2.7
  • ...and 43 more