Action of the Johnson-Torelli group on representation varieties

Abstract:

Let $\Sigma$ be a compact orientable surface with genus $g$ and $n$ boundary components $B = (B_1,\dots , B_n)$. Let $c = (c_1,\dots ,c_n)\in [-2,2]^n$. Then the mapping class group $\mathsf {MCG}$ of $\Sigma$ acts on the relative $\mathsf {SU}(2)$-character variety $\mathfrak {X}_{\mathcal {C}}:=\mathsf {Hom}_\mathcal {C}(\pi ,\mathsf {SU}(2))/\mathsf {SU}(2)$, comprising conjugacy classes of representations $\rho$ with $\mathfrak {tr}(\rho (B_i)) = c_i$. This action preserves a symplectic structure on the smooth part of $\mathfrak {X}_{\mathcal {C}}$, and the corresponding measure is finite. Suppose $g =1$ and $n = 2$. Let $\mathcal {J} \subset \mathsf {MCG}$ be the subgroup generated by Dehn twists along null homologous simple loops in $\Sigma$. Then the action of $\mathcal {J}$ on $\mathfrak {X}_{\mathcal {C}}$ is ergodic for almost all $c$.