On minimal generating sets for the mapping class group of a punctured surface

TL;DR

This paper proves that for surfaces with genus at least 3, the mapping class group and its extended version can be generated by just two elements, simplifying their algebraic structure.

Contribution

It establishes that both the mapping class group and the extended group are generated by two elements for surfaces with genus at least 3, improving understanding of their minimal generating sets.

Findings

Both groups are generated by two elements for g ≥ 3.

The result applies to surfaces with any number of punctures p.

Simplifies the algebraic structure of these groups.

Abstract

Let $\Sigma_{g,p}$ be a oriented connected surface of genus $g$ with $p$ punctures. We denote by $\mathcal{M}_{g,p}$ and $\mathcal{M}_{g,p}^\pm$ the mapping class group and the extended mapping class group of $\Sigma_{g,p}$, respectively. In this paper, we show that $\mathcal{M}_{g,p}$ and $\mathcal{M}_{g,p}^\pm$ are generated by two element for $g\geq 3$ and $p\geq 0$.