Mapping Class Group Dynamics on Surface Group Representations

Abstract

Deformation spaces Hom($\pi$,G)/G of representations of the fundamental group $\pi$ of a surface $\Sigma$ in a Lie group $G$ admit natural actions of the mapping class group $Mod_\Sigma$, preserving a Poisson structure. When $G$ is compact, the actions are ergodic. In contrast if $G$ is noncompact semisimple, the associated deformation space contains open subsets containing the Fricke-Teichm\"uller space upon which $Mod_\Sigma$ acts properly. Properness of the $Mod_\Sigma$-action relates to (possibly singular) locally homogeneous geometric structures on $\Sigma$. We summarize known results and state open questions about these actions.