Random walks on mapping class groups

Abstract

This survey is concerned with random walks on mapping class groups. We illustrate how the actions of mapping class groups on Teichmüller spaces or curve complexes reveal the nature of random walks, and vice versa. Our emphasis is on the analogues of classical theorems, including laws of large numbers and central limit theorems, and the properties of harmonic measures, under optimal moment conditions. We also explain the geometric analogy between Gromov hyperbolic spaces and Teichmüller spaces that has been used to copy the properties of random walks from one to the other.

35 pages, 2 figures. The final version of this paper will appear on the Sullivan 80th birthday volume of the journal EMS Surveys in Mathematical Sciences which is edited by A. Basmajian, T. Koberda, A. Papadopoulos, J. Seade, and M. Zeinalian