Algebraic Mapping Class Group Rigidity

TL;DR

This paper explores the algebraic structure of the mapping class group of surfaces via the rigidity of the SL2-character variety, revealing new connections and descriptions of the group through geometric and algebraic methods.

Contribution

It establishes the rigidity of the relative automorphism group of the SL2-character variety and links it to the mapping class group, providing a novel algebraic perspective.

Findings

The relative automorphism group of the SL2-character variety is finite extension of the mapping class group.

The isomorphism with moduli of points on complex 3-sphere offers a new geometric description.

The automorphism group fixes monodromies along punctures, indicating rigidity.

Abstract

Let $g, n \geq 0$ and $\Sigma = \Sigma_{g, n}$ be a connected oriented surface of genus $g$ with $n$ punctures. The $\mathrm{SL}_2$-character variety of $\Sigma$ has a rigid relative automorphism group, whose elements fix each monodromies along punctures, and is a finite extension of the mapping class group. The exceptional isomorphism between the $\mathrm{SL}(2, \mathbb{C})$-character variety and moduli of points on complex $3$-sphere provides a new description of the mapping class group of certain $\Sigma$.