Generating the Mapping Class Group of a Punctured Surface by Involutions

Abstract

Let $\Sigma_{g,b}$ denote a closed oriented surface of genus $g$ with $b$ punctures and let $\mathrm{Mod}(\Sigma_{g,b})$ denote its mapping class group. Kassabov showed that $\mathrm{Mod}(\Sigma_{g,b})$ is generated by 4 involutions if $g>7$ or $g=7$ and $b$ is even, 5 involutions if $g>5$ or $g=5$ and $b$ is even, and 6 involutions if $g>3$ or $g=3$ and $b$ is even. We proved that $\mathrm{Mod}(\Sigma_{g,b})$ is generated by 4 involutions if $g=7$ and $b$ is odd, and 5 involutions if $g=5$ and $b$ is odd.