Generating Extended Mapping Class Groups with Two Periodic Elements

TL;DR

This paper proves that the extended mapping class group of an n-punctured sphere is generated by two finite order elements for all n except 4, and also for genus 2 surfaces, highlighting specific generating sets.

Contribution

It establishes the minimal generating set of two finite order elements for extended mapping class groups of certain surfaces, including genus 2 and n-punctured spheres (n≠4).

Findings

Generated extended mapping class group of n-punctured sphere with two elements for n≠4.

Extended mapping class group of genus 2 surface generated by two finite order elements.

Identified exceptions and specific cases for generating sets.

Abstract

The extended mapping class group of a surface $\Sigma$ is defined to be the group of isotopy classes of (not necessarily orientation-preserving) homeomorphisms of $\Sigma$. We are able to show that the extended mapping class group of an $n$-punctured sphere is generated by two elements of finite order exactly when $n\not=4$. We use this result to prove that the extended mapping class group of a genus 2 surface is generated by two elements of finite order.