Automorphisms of character varieties

TL;DR

This paper investigates the automorphism group of the SL(2,C) character variety of closed surfaces, revealing it as a finite extension of the mapping class group, and characterizes valuations from measured laminations.

Contribution

It establishes a precise relationship between automorphisms of character varieties and the mapping class group, and characterizes valuations from measured laminations.

Findings

Automorphism group is a finite extension of the mapping class group.

Provides a simple characterization of valuations from measured laminations.

Enhances understanding of the structure of character varieties.

Abstract

We show that the algebraic automorphism group of the SL(2,C) character variety of a closed orientable surface with negative Euler characteristic is a finite extension of its mapping class group. Along the way, we provide a simple characterization of the valuations on the character algebra coming from measured laminations.