Generating Mapping Class Groups by Involutions

TL;DR

This paper investigates the generation of mapping class groups of punctured surfaces using involutions, establishing bounds on the number of involutions needed, and extending previous results to multiple punctures.

Contribution

The authors generalize existing results by showing that for large genus, the mapping class group can be generated by four involutions, and they analyze bounds for smaller genus cases.

Findings

Mapping class groups can be generated by 4 involutions for large genus.

At least 3 involutions are necessary to generate the group.

The results extend previous work to surfaces with multiple punctures.

Abstract

Let $\Sigma_{g,b}$ denote a closed oriented surface genus $g$ with $b$ punctures and let $Mod_{g,b}$ denote its mapping class group. Luo proved that if the genus is at least 3, the group $Mod_{g,b}$ 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 $Mod_{g,b}$. Brendle and Farb gave a partial answer in the case of closed surfaces and surfaces with one puncture, by describing a generating set consisting of 7 involutions. Our main result generalizes the above result to the case of multiple punctures. We also show that the mapping class group can be generated by smaller number of involutions. More precisely, we prove that the mapping class group can be generated by 4 involutions if the genus $g$ is large enough. There is not a lot room to improve this bound because to generate this group we need at lest 3 involutions. In the case of small genus (but at least 3) to generate the whole mapping class group we need a few more involutions.