Generating the mapping class group of a punctured surface by involutions

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 $g\geq 5$.