A survey on spaces of homomorphisms to Lie groups

TL;DR

This survey explores the topological properties of spaces of homomorphisms from finitely generated discrete groups to Lie groups, linking them to classical representation theory and other mathematical structures.

Contribution

It provides a comprehensive exposition of the topological and algebraic properties of these homomorphism spaces, including cases involving small categories and right-angled Artin groups.

Findings

Topological properties of homomorphism spaces are characterized.

Connections to classical representation theory are elucidated.

Extensions to small categories and right-angled Artin groups are discussed.

Abstract

The purpose of this article is to give an exposition of topological properties of spaces of homomorphisms from certain finitely generated discrete groups to Lie groups $G$, and to describe their connections to classical representation theory, as well as other structures. Various properties are given when $G$ is replaced by a small category, or the discrete group is given by a right-angled Artin group.