Generating the mapping class groups by torsions

TL;DR

This paper proves that for surfaces of genus at least 3, the mapping class group can be generated by a small set of torsion elements, with specific counts and types depending on the genus.

Contribution

It establishes minimal generating sets of torsion elements for mapping class groups, including explicit counts and types for genus 3 and higher.

Findings

For g ≥ 4, Mod(S_g) generated by 4 torsion elements.

For g = 3, Mod(S_3) generated by 5 torsion elements.

Specific generators include involutions and an order 3 element.

Abstract

Let $S_g$ be the closed oriented surface of genus g and let $\text{Mod}(S_g)$ be the mapping class group. When the genus is at least 3, $\text{Mod}(S_g)$ can be generated by torsion elements. We prove the follow results. For $g \geq 4$, $\text{Mod}(S_g)$ can be generated by 4 torsion elements. Three generators are involutions and the forth one is an order 3 element. $\text{Mod}(S_3)$ can be generated by 5 torsion elements. Four generators are involutions and the fifth one is an order 3 element.