Topology of representation spaces of surface groups in PSL(2,R) with assigned boundary monodromy

The aim of this paper is to determine the topology of the variety of representations of the fundamental group of a punctured surface in PSL(2,R) with prescribed behavior at the punctures. In order to do that, we follow the strategy employed by Hitchin in the unpunctured case and we exploit the correspondence between representations of the fundamental group of the surface and Higgs bundles in the parabolic case.