The Extended Mapping Class Group Can Be Generated by Two Torsions

TL;DR

This paper proves that for surfaces of genus at least 5, the extended mapping class group can be generated by just two torsion elements, one of order 2 and the other of order 4g+2.

Contribution

It establishes a minimal generating set of two torsion elements for the extended mapping class group of high-genus surfaces, extending previous understanding.

Findings

The extended mapping class group is generated by two torsions for genus ≥ 5.

One generator is an involution (order 2).

The other generator has order 4g+2.

Abstract

Let $S_g$ be the closed oriented surface of genus g and let $\text{Mod}^{\pm}(S_g)$ be the extended mapping class group of $S_g$. When the genus is at least 5, we prove that $\text{Mod}^{\pm}(S_g)$ can be generated by two torsion elements. One of these generators is an order 2 element, and the other one is an order 4g+2 element.