The action of the mapping class group on representation varieties of PSL(2,R). Case I: The one-holed torus

TL;DR

This paper investigates the action of the mapping class group on representation varieties of a one-holed torus into PSL(2,R), identifying regions where the action is ergodic, thus contributing to understanding Goldman’s conjecture.

Contribution

It extends the analysis of the mapping class group action to the one-holed torus case, providing explicit ergodic regions and mirroring Goldman's moduli space results.

Findings

Identified regions with ergodic action for the one-holed torus.

Extended Goldman’s results to surfaces with boundary.

Provided new insights into the dynamics of surface group representations.

Abstract

In this paper we consider the action of the mapping class group of a surface on the space of homomorphisms from the fundamental group of a surface into PSL(2,R). Goldman conjectured that when the surface is closed and of genus bigger than one, the action on non-Teichmuller connected components of the associated moduli space (i.e. the space of homomorphisms modulo conjugation) is ergodic. One approach to this question is to use sewing techniques which requires that one considers the action on the level of homomorphisms, and for surfaces with boundary. In this paper we consider the case of the one-holed torus with boundary condition, and we determine regions where the action is ergodic. Our main result mirrors a theorem of Goldman's at the level of moduli.