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

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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]).

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  1. Institut Fourier, UMR5582, UFR Mathématiques, Université Grenoble I, 100 rue des Maths, BP74, 38402, St Martin d’Heres, France

    Frederic Palesi

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  1. Frederic Palesi
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Palesi, F. Ergodic actions of mapping class groups on moduli spaces of representations of non-orientable surfaces. Geom Dedicata 151, 107–140 (2011). https://doi.org/10.1007/s10711-010-9522-7

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