The mapping class group of a punctured surface is generated by three elements

TL;DR

This paper proves that the mapping class group of a punctured surface with genus at least one can be generated by three elements, extending previous results for fewer punctures.

Contribution

It establishes that for surfaces with two or more punctures, the mapping class group is generated by three elements, improving understanding of its algebraic structure.

Findings

For p=0,1, the group is generated by two elements.

For p≥2, the group is generated by three elements.

Extension of generation results to punctured surfaces with multiple punctures.

Abstract

Let $\Sigma_{g,p}$ be a closed oriented surface of genus $g\geq 1$ with $p$ punctures. Let $\rm Mod(\Sigma_{\textit{g,p}})$ be the mapping class group of $\Sigma_{g,p}$. Wajnryb proved in [Wa] that for $p=0, 1$ $\rm Mod({\Sigma_{\textit{g,p}}})$ is generated by two elements. Korkmaz proved in [Ko] that one of these generators can be taken as a Dehn twist. For $p\geq 2$, We proved that $\rm Mod(\Sigma_{\textit{g,p}})$ is generated by three elements.