Varieties of Characters

TL;DR

This paper introduces the concept of G-characters for finitely generated groups and demonstrates how their quotient by trace-preserving automorphisms relates to the variety of G-characters in specific cases.

Contribution

It defines the variety of G-characters and establishes its normalization via quotient by trace-preserving outer automorphisms for certain classes of groups.

Findings

The G-character variety is well-defined for finitely generated groups.

The quotient by trace-preserving automorphisms normalizes the G-character variety.

Results apply to free, free abelian, and surface groups.

Abstract

Let G be a connected reductive affine algebraic group. In this short note we define the "variety of G-characters" of a finitely generated group F and show that the quotient of the G-character variety of F by the action of the trace preserving outer automorphisms of G normalizes the variety of G-characters when F is a free group, free abelian group, or a surface group.