Ergodic theory on moduli spaces

Abstract

Let $M$ be a compact surface with $\chi(M) < 0$ and let $G$ be a compact Lie group whose Levi factor is a product of groups locally isomorphic to $\mathrm{SU}(2)$ (for example $\mathrm{SU}(2)$ itself). Then the mapping class group $\Gamma_M$ of $M$ acts on the moduli space $X(M)$ of flat $G$-bundles over $M$ (possibly twisted by a fixed central limit of $G$). When $M$ is closed, then $\Gamma_M$ preserves a symplectic structure on $X(M)$ which has finite total volume on $M$. More generally, the subspace of $X(M)$ corresponding to flat bundles with fixed behavior over $\partial M$ carries of $\Gamma_M$-invariant symplectic structure. The main result is that $\Gamma_M$ acts ergodically on $X(M)$ with respect to the measure induced by the symplectic structure.