# 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.

## Key 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 $PSL(2,\mathbb R)$.

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Source: https://tomesphere.com/paper/1704.02486