Generating Extended Mapping Class Groups with Two Periodic Elements

Abstract

The extended mapping class group of a surface $\hat{I}\pounds$ is defined to be the group of isotopy classes of (not necessarily orientation-preserving) homeomorphisms of $\hat{I}\pounds$. 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 $n\ne 4$. 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.