Ergodic actions of mapping class groups on moduli spaces of representations of non-orientable surfaces

Abstract

Let M be a non-orientable surface with Euler characteristic χ(M) ≤ −2. We consider the moduli space of flat SU(2)-connections, or equivalently the space of conjugacy classes of representations $$\mathfrak{X} (M) = {\rm Hom} (\pi_1 (M), {\rm SU} (2)) / {\rm SU} (2).$$There is a natural action of the mapping class group of M on \({\mathfrak{X} (M)}\). We show here that this action is ergodic with respect to a natural measure. This measure is defined using the push-forward measure associated to a map defined by the presentation of the surface group. This result is an extension of earlier results of Goldman for orientable surfaces (see [8]).