The Poisson boundary of the mapping class group

Abstract.

A theory of random walks on the mapping class group and its non-elementary subgroups is developed. We prove convergence of sample paths in the Thurston compactification and show that the space of projective measured foliations with the corresponding harmonic measure can be identified with the Poisson boundary of random walks. The methods are based on an analysis of the asymptotic geometry of Teichmüller space and of the contraction properties of the action of the mapping class group on the Thurston boundary. We prove, in particular, that Teichmüller space is roughly isometric to a graph with uniformly bounded vertex degrees. Using our analysis of the mapping class group action on the Thurston boundary we prove that no non-elementary subgroup of the mapping class group can be a lattice in a higher rank semi-simple Lie group.