Generating the mapping class group by torsion elements of small order

TL;DR

This paper demonstrates that the mapping class group of a closed oriented surface with genus at least three can be generated by a small set of torsion elements of orders 3 and 4, highlighting differences with lower genus cases.

Contribution

It establishes the minimal generating sets of the mapping class group using torsion elements of specific small orders for genus three and higher.

Findings

Generated by 3 elements of order 3 for genus ≥ 3

Generated by 4 elements of order 4 for genus ≥ 3

Cannot be generated by finitely many torsion elements of same order for genus 1 or 2

Abstract

We show that the mapping class group of a closed oriented surface of genus at least three is generated by 3 elements of order 3 and by 4 elements of order 4. Note that the mapping class group cannot be generated by finitely many torsion elements of same order if genus is equal to one or two.