Topological components of surface group representations and signature

TL;DR

This paper investigates the topological structure of surface group representations into SL(2,R) and PSL(2,R), using a signature formula to count connected components based on boundary holonomies.

Contribution

It provides a detailed analysis of the connected components of representation spaces for surface groups, extending understanding through the application of the signature formula.

Findings

Determined the number of connected components for various boundary holonomies.

Applied the signature formula to classify representation spaces.

Enhanced the understanding of surface group representation topology.

Abstract

We study the topological components of the surface group representations into $\mathrm{SL}(2,\mathbb{R})$ and $\mathrm{PSL}(2,\mathbb{R})$. Utilizing the signature formula established in [14], we determine the number of connected components of the representation spaces with boundary elliptic, hyperbolic, and parabolic holonomies.