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

TL;DR

This paper proves that the mapping class group acts ergodically on the moduli space of SU(2) representations of non-orientable surface groups, extending Goldman’s results from orientable surfaces.

Contribution

It establishes ergodicity of the mapping class group action on the moduli space for non-orientable surfaces, using a measure derived from the surface group presentation.

Findings

The action is ergodic with respect to a natural measure.

The measure is constructed via the push-forward of Haar measure.

Extension of Goldman’s ergodicity results to non-orientable surfaces.

Abstract

The purpose of this paper is to study the action of the mapping class group on the moduli space of representations of the fundamental group of a non-orientable surface into SU(2). The action is shown to be 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. given by the pushforward of Haar measure. This result is an extension of earlier results of Goldman for orientable surfaces.