Studying Deformations of Fuchsian Representations with Higgs Bundles

TL;DR

This survey explores how Higgs bundle theory helps understand the structure of character varieties of surface groups, focusing on Fuchsian representations and their relation to higher Teichmüller theory.

Contribution

It develops the general theory of Higgs bundles for real groups and illustrates their application to studying components of character varieties, especially Fuchsian representations.

Findings

Descriptions of deformation spaces related to Fuchsian representations

Connections between Higgs bundles and higher Teichmüller theory

Examples illustrating the use of Higgs bundles in character variety analysis

Abstract

This is a survey article whose main goal is to explain how many components of the character variety of a closed surface are either deformation spaces of representations into the maximal compact subgroup or deformation spaces of certain Fuchsian representations. This latter family is of particular interest and is related to the field of higher Teichm\"uller theory. Our main tool is the theory of Higgs bundles. We try to develop the general theory of Higgs bundles for real groups and indicate where subtleties arise. However, the main emphasis is placed on concrete examples which are our motivating objects. In particular, we do not prove any of the foundational theorems, rather we state them and show how they can be used to prove interesting statements about components of the character variety. We have also not spent any time developing the tools (harmonic maps) which define the bridge between Higgs bundles and the character variety. For this side of the story we refer the reader to the survey article of Q. Li [arXiv:1809.05747].