The Poisson boundary of the mapping class group
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 uniformly bounded vertex degrees. As a by-product of this study, the authors prove that no non-elementary subgroup of the mapping class group can be a lattice in a higher rank semi-simple Lie group. In this paper, the authors propose also a list of interesting open questions related to this study.
scientific article; zbMATH DE number 922465