Action of the Johnson-torelli Group on Representation Varieties

Let Σ be a compact orientable surface with genus g and n boundary components B = (B1, . . . , Bn). Let c = (c1, . . . , cn) ∈ [−2, 2]n. Then the mapping class group MCG of Σ acts on the relative SU(2)-character variety XC := HomC(π,SU(2))/SU(2), comprising conjugacy classes of representations ρ with tr(ρ(Bi)) = ci. This action preserves a symplectic structure on the smooth part of XC , and the corresponding measure is finite. Suppose g = 1 and n = 2. Let J ⊂ MCG be the subgroup generated by Dehn twists along null homologous simple loops in Σ. Then the action of J on XC is ergodic for almost all c.