The Poisson boundary of the mapping class group
type: Item
instance of: scholarly article
title: The Poisson boundary of the mapping class group
zbMATH Open document ID: 0864.57014
author: Howard Masur
author: Vadim A. Kaimanovich
published in: Inventiones Mathematicae
publication date: 1997-06-22T00:00:00Z
review text: Let \(M\) be a closed surface of genus \(\geq 2\), let \(\gamma\) be the mapping class group of \(M\), let \({\mathcal T}\) be its Teichmüller space and \({\mathcal P}{\mathcal M}{\mathcal F}\) Thurston's boundary of \({\mathcal T}\). In this paper, the authors develop the theory of random walks on \(\Gamma\) and on its subgroups. The main result is the following: Theorem. Let \(\mu\) be a probability measure on \(\Gamma\) such that the group generated by the support of \(\mu\) is non-elementary. Then there exists a unique \(\mu\)-stationary probability measure \(\nu\) on the space \({\mathcal P} {\mathcal M} {\mathcal F}\), which is purely non-atomic and is concentrated on the subset \({\mathcal U} {\mathcal E}\) of \({\mathcal P} {\mathcal M} {\mathcal F}\) whose elements are projective classes of uniquely ergodic measured foliations. The measure space \(({\mathcal U} {\mathcal E}, \nu)\) is a \(\mu\)-boundary. For almost every \(x\in {\mathcal T}\) and for almost every sample path \(g=(g_n)\) of the random walk \((\Gamma,\mu)\), the sequence \((g_nx)\) converges in \({\mathcal M} {\mathcal P} {\mathcal F}\) to a limit \(F=F(g)\in {\mathcal U} {\mathcal E}\), and the distribution of the limits \(F(g)\) is given by the measure \(\nu\). Furthermore, if the measure \(\mu\) has finite entropy and finite first logarithmic moment with respect to the Teichmüller distance, then the space \(({\mathcal P} {\mathcal M} {\mathcal F},\nu)\) is the Poisson boundary of \((\Gamma,\mu)\). The approach for proving these results uses of course the asymptotic geometry of Teichmüller space and contraction properties of the action of the mapping class group on Thurston's boundary, but also methods from ergodic theory, in particular entropy theory for random walks on groups and ideas which have been used by Furstenberg in his study of discrete subgroups of \(\text{SL} (2,\mathbb{R})\). The authors prove during their analysis that Teichmüller space is roughly isometric to a graph with un…
Mathematics Subject Classification ID: 57N05
Mathematics Subject Classification ID: 30F60
Mathematics Subject Classification ID: 60J05
zbMATH DE Number: 922465
zbMATH Keywords: mapping class group
zbMATH Keywords: Teichmüller space
zbMATH Keywords: Thurston's boundary
zbMATH Keywords: random walks
zbMATH Keywords: probability measure
zbMATH Keywords: entropy
reviewed by: Athanase Papadopoulos
MaRDI profile type: Publication
full work available at URL: s002220050074
OpenAlex ID: W2025967694
DOI: 10.1007/S002220050074
description: scientific article; zbMATH DE number 922465