Mapping Class Group Dynamics on Surface Group Representations

Abstract

Deformation spaces Hom($π$,G)/G of representations of the fundamental group $π$ of a surface $Σ$ in a Lie group $G$ admit natural actions of the mapping class group $Mod_Σ$, 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-Teichmüller space upon which $Mod_Σ$ acts properly. Properness of the $Mod_Σ$-action relates to (possibly singular) locally homogeneous geometric structures on $Σ$. We summarize known results and state open questions about these actions.