Topology of representation spaces of surface groups in $\mathrm{PSL}_2 (\mathbb{R})$ with assigned boundary monodromy and nonzero Euler number

In this paper we complete the topological description of the space of representations of the fundamental group of a punctured surface in $\mathrm{SL}_2 (\mathbb{R})$ with prescribed behavior at the punctures and nonzero Euler number, following the strategy employed by Hitchin in the unpunctured case and exploiting Hitchin–Simpson correspondence between flat bundles and Higgs bundles in the parabolic case. This extends previous results by Boden–Yokogawa and Nasatyr–Steer. A relevant portion of the paper is intended to give an overview of the subject.

Keywords

representation spaces, Euler number, parabolic Higgs bundles, uniformization

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Received 15 July 2016

Published 9 February 2018