On the character varieties of finitely generated groups

TL;DR

This paper investigates the dimensions of character varieties of finitely generated groups and explores how to realize certain algebraic varieties as character varieties, advancing understanding of their structure and realizability.

Contribution

It provides new results on the dimension growth of character varieties and methods for realizing specific algebraic varieties as character varieties.

Findings

Dimension of $X_n(\Gamma)$ as a function of $n$ is characterized.

Conditions for realizing a $\mathbb{Q}$-defined variety as a character variety are established.

Results contribute to understanding the structure and realizability of character varieties.

Abstract

We establish three results dealing with the character varieties of finitely generated groups. The first two are concerned with the behavior of $\dim X_n(\Gamma)$ as a function of $n$, and the third addresses the problem of realizing a $\mathbb{Q}$-defined complex affine algebraic variety as a character variety.