Action of the Johnson-Torelli group on Representation Varieties

Details Action of the Johnson-Torelli group on Representation Varieties Action of the Johnson-Torelli group on Representation Varieties

2010-03-14

arxiv.org/abs/1003.2825v2

Mathematics

Dynamical Systems

This new version includes a ribbon graph for clarity and corrects some typographic errors and a few misstatements between Rema

Scientific paper

Let \Sigma be 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 \Sigma acts on the relative SU(2)-character variety X_c := Hom_C(\pi, SU(2))/SU(2), comprising conjugacy classes of representations \rho with tr(\rho(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 \Sigma. Then the action of J on X_c is ergodic for almost all c.

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