Action of the Johnson-Torelli group on Representation Varieties
Abstract
Let be a compact orientable surface with genus g and n boundary components B = (B1,..., Bn). Let c = (c1,...,cn) in [-2,2]n. Then the mapping class group MCG of acts on the relative SU(2)-character variety Xc := HomC(π, SU(2))/SU(2), comprising conjugacy classes of representations with tr((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 be the subgroup of MCG generated by Dehn twists along null homologous simple loops in . Then the action of J on Xc is ergodic for almost all c.
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