Action of the Johnson-Torelli group on Representation Varieties

Abstract

Let Σbe a compact orientable surface with genus g and n boundary components B = (B_1,..., B_n). Let c = (c_1,...,c_n) in [-2,2]^n. Then the mapping class group MCG of Σacts on the relative SU(2)-character variety X_c := Hom_C(π, SU(2))/SU(2), comprising conjugacy classes of representations ρwith tr(ρ(B_i)) = c_i. This action preserves a symplectic structure on the smooth part of X_c, and the corresponding measure is finite. Suppose g = 1 and n = 2. Let J be the subgroup of MCG generated by Dehn twists along null homologous simple loops in Σ. Then the action of J on X_c is ergodic for almost all c.

This new version includes a ribbon graph for clarity and corrects some typographic errors and a few misstatements between Remark 3.1 and Lemma 3.2 and between Remark 3.7 and Lemma 3.8 in the previous version