# Representation Variety of Surface Groups

## TL;DR

This paper provides an exact formula for the dimension of the variety of homomorphisms from surface groups of genus g to any semisimple real algebraic group, advancing understanding of surface group representations.

## Contribution

It introduces a precise formula for the dimension of the representation variety of surface groups into semisimple real algebraic groups.

## Key findings

- Exact dimension formula for homomorphism varieties
- Applicable to any semisimple real algebraic group
- Enhances understanding of surface group representations

## Abstract

We give an exact formula for the dimension of the variety of homomorphisms from $S_g$ to $\mathit{any}$ semisimple real algebraic group, where $S_g$ is a surface group of genus $g \geq 2$.