On the topology of the representation variety of a punctured surface group in PSL(2,R)

Goldman classified the connected components of the representation variety of a closed surface group in PSL(2,R) and Hitchin described the topology of the components with nonzero Euler number. In this talk I will describe how to perform a similar analysis for surfaces with punctures, thus detecting the topology of the representation variety in PSL(2,R) with assigned peripheral monodromy and nonzero Euler number. We follow Hitchin's strategy and we exploit Simpson's correspondence between representations of punctured surface groups and parabolic Higgs bundles.