Generating the mapping class group of a punctured surface by involutions

TL;DR

This paper proves that the mapping class group of a punctured surface can be generated by a small, fixed number of involutions depending on the genus, advancing understanding of the group's torsion element structure.

Contribution

The authors establish new bounds showing that for genus at least 7, four involutions suffice, and for genus at least 5, five involutions suffice to generate the group.

Findings

Generated mapping class groups with 4 involutions for g≥7

Generated with 5 involutions for g≥5

Improved previous bounds on involution generators

Abstract

Let $\Sigma_{g,b}$ denote a closed orientable surface of genus $g$ with $b$ punctures and let $\rm Mod(\Sigma_{\textit{g,b}})$ denote its mapping class group. In [Luo] Luo proved that if the genus is at least 3, $\rm Mod(\Sigma_{\textit{g,b}})$ is generated by involutions. He also asked if there exists a universal upper bound, independent of genus and the number of punctures, for the number of torsion elements/involutions needed to generate $\rm Mod(\Sigma_{\textit{g,b}})$. Brendle and Farb [BF] gave an answer in the case of $g\geq 3, b=0$ and $g\geq 4, b=1$, by describing a generating set consisting of 6 involutions. Kassabov showed that for every $b$ $\rm Mod(\Sigma_{\textit{g,b}})$ can be generated by 4 involutions if $g\geq 8$, 5 involutions if $g\geq 6$ and 6 involutions if $g\geq 4$. We proved that for every $b$ $\rm Mod(\Sigma_{\textit{g,b}})$ can be generated by 4 involutions if $g\geq 7$ and 5 involutions if $g\geq 5$.