Boundary representations of mapping class groups

Abstract

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