The Torsion Generating Set Of The Extended Mapping Class Groups In Low Genus Cases

TL;DR

This paper investigates the minimal generating sets of extended mapping class groups for low genus surfaces, showing that for genus 3 and 4, two finite order elements suffice, while for genus 1, they do not.

Contribution

It establishes the minimal finite order generating sets for extended mapping class groups in low genus cases, filling a gap in the understanding of their algebraic structure.

Findings

For genus 3 and 4, the groups are generated by two finite order elements.

For genus 1, the group cannot be generated by two finite order elements.

Abstract

We prove that for genus $g=3,4$, the extended mapping class group $\text{Mod}^{\pm}(S_g)$ can be generated by two elements of finite orders. But for $g=1$, $\text{Mod}^{\pm}(S_1)$ cannot be generated by two elements of finite orders.