Boundary representations of mapping class groups

TL;DR

This paper demonstrates that the boundary representation of the mapping class group of a surface is ergodic and irreducible, extending classical results through the use of statistical hyperbolicity.

Contribution

It introduces a new approach to analyze boundary representations of mapping class groups using statistical hyperbolicity, generalizing classical ergodicity results.

Findings

Boundary representation of Mod(S) is ergodic.

Boundary representation of Mod(S) is irreducible.

Generalizes Masur's classical ergodicity result.

Abstract

Let $S = S_g$ be a closed orientable surface of genus $g \geq 2$ 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 $\mathcal{PMF}(S).$ As a corollary, we show that the boundary representation of $Mod(S)$ is irreducible.