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

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.