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

TL;DR

This paper completes the topological classification of surface group representations into PSL(2,R) with fixed boundary monodromy and nonzero Euler number, extending prior work to punctured surfaces using Higgs bundle techniques.

Contribution

It extends the topological description of representation spaces to punctured surfaces with boundary conditions, utilizing Hitchin-Simpson correspondence in the parabolic setting.

Findings

Complete topological description of the representation space

Extension of previous results to punctured surfaces

Application of Higgs bundle correspondence in this context

Abstract

In this paper we complete the topological description of the space of representations of the fundamental group of a punctured surface in SL(2,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.