Dynamics of the mapping class group in character varieties of genus 2 surfaces

The following question was popularized by Goldman: given a genus $g$ surface $S$ and a group $G$, the mapping class group $\text{Mod}(S)$ acts on the character variety $X(S,G)$ of conjugacy classes of representations of the fundamental group of $S$ into $G$. When G is compact, Goldman and Xia–Pickrell showed that the action is ergodic whereas when $G=\text{PSL}_2(R)$, some component of $X(S,G)$ is the Teichmuller space, and the action is proper and discontinuous. In a work with M. Wolff we prove that the action on the remaining components is ergodic. We will prove one case with techniques mixing hyperbolic and symplectic geometry.