# Rank 1 character varieties of finitely presented groups

## TL;DR

This paper presents an algorithm and software implementations for computing the coordinate ring of rank 1 character varieties of finitely presented groups, facilitating research and experimentation in algebraic geometry.

## Contribution

It introduces a practical algorithm and implementations for deriving presentations of character varieties, including new descriptions for free groups.

## Key findings

- Algorithm successfully computes coordinate rings
- Implementations in Mathematica, SageMath, and Python available
- Provides new descriptions of relations and parameters for free groups

## Abstract

Let X(F,G) be the G-character variety of F where G is a rank 1 complex affine algebraic group and F is a finitely presentable discrete group. We describe an algorithm, which we implement in Mathematica, SageMath, and in Python, that takes a finite presentation for F and produces a finite presentation of the coordinate ring of X(F,G). We also provide a new description of the defining relations and local parameters of the coordinate ring when F is free. Although the theorems used to create the algorithm are not new, we hope that as a well-referenced exposition with a companion computer program it will be useful for computation and experimentation with these moduli spaces.