Minimal Generation of Mapping Class Groups: A Survey of the Orientable Case

TL;DR

This survey reviews the development of minimal generating sets for orientable surface mapping class groups, highlighting recent advances such as generating these groups with just two elements or a small set of involutions.

Contribution

It provides a comprehensive overview of the evolution of minimal generating sets for mapping class groups, including new results on generating groups with three involutions for certain punctured surfaces.

Findings

Mapping class groups can be generated by two elements.

Recent results show generation by a small number of involutions.

New improvement: $ ext{Mod}( ext{Sigma}_{13,p})$ generated by three involutions for even p ≥ 8.

Abstract

The mapping class group of an orientable surface, which records its symmetries up to isotopy, plays a central role in low-dimensional topology. This chapter explores the foundational problem of determining minimal generating sets for these groups. We chart the development of this area from classical results involving Dehn twist generators to more recent breakthroughs showing that mapping class groups can be generated by just two elements, pairs of torsion elements, or a small collection of involutions. This chapter contains a discussion of the most current results for punctured surfaces, including a new improvement showing that for an even number of punctures $p\geq 8$ the group $\mathrm{Mod}(\Sigma_{13,p})$ is generated by three involutions. Throughout, we highlight the rich interplay between the algebraic features of these generating sets and the underlying geometric structures they encode. The chapter aims to provide a comprehensive account of the pursuit of algebraic and geometric efficiency within one of topology's most intricate and influential groups.