Boundary representations of mapping class groups

Abstract

Let $S=S_g$ be a closed orientable surface of genus $g\ge 2$ and $Mod(S)$ be the mapping class group of $S$. In this paper, we show that the boundary representation of $Mod(S)$ is ergodic using statistical hyperbolicity, which generalizes the classical result of Masur on ergodicity of the action of $Mod(S)$ on the projective measured foliation space $\mathcal{P}\mathcal{M}\mathcal{F}(S).$ As a corollary, we show that the boundary representation of $Mod(S)$ is irreducible.

The proof of Corollary 4.3 in the previous version is incorrect, hence section 4.3 has been rewritten. Section 3 and section 4.1 have also been reorganized. Comments welcome