Various generalizations and deformations of $PSL(2,\mathbb{R})$ surface group representations and their Higgs bundles

TL;DR

This paper explores how classical surface group representations into $PSL(2,\mathbb{R})$ and their Higgs bundles extend to higher rank groups, identifying new components and their deformation properties.

Contribution

It introduces new connected components in higher rank character varieties and analyzes their deformation within larger groups, generalizing Hitchin's results.

Findings

Parameterization of $n(2g-2)$ new components in $SO_0(n,n+1)$-character variety.

Description of how these components deform in $SO_0(n,n+2)$-character variety.

Extension of classical $PSL(2,\mathbb{R})$ results to higher rank groups.

Abstract

Recall that the group $PSL(2,\mathbb R)$ is isomorphic to $PSp(2,\mathbb R),\ SO_0(1,2)$ and $PU(1,1).$ The goal of this paper is to examine the various ways in which Fuchsian representations of the fundamental group of a closed surface of genus $g$ into $PSL(2,\mathbb R)$ and their associated Higgs bundles generalize to the higher rank groups $PSL(n,\mathbb R),\ PSp(2n,\mathbb R),\ SO_0(2,n),\ SO_0(n,n+1)$ and $PU(n,n)$. For the $SO_0(n,n+1)$-character variety, we parameterize $n(2g-2)$ new connected components as the total space of vector bundles over appropriate symmetric powers of the surface and study how these components deform in the $SO_0(n,n+2)$-character variety. This generalizes results of Hitchin for $PSL(2,\mathbb R)$.