Generating the mapping class group of a non-orientable punctured surface by involutions

TL;DR

This paper proves that the mapping class group of a non-orientable punctured surface can be generated by a fixed small set of involutions, with the number depending on the genus parity and size.

Contribution

It establishes explicit bounds on the number of involutions needed to generate the mapping class group for various genus and puncture counts, improving previous linear bounds.

Findings

For odd genus g ≥ 13, 8 involutions suffice.

For even genus g ≥ 14, 11 involutions suffice.

Generation by involutions is independent of the number of punctures for large genus.

Abstract

Let $N_{g,n}$ denote the closed non-orientable surface of genus $g$ with $n$ punctures and let ${\mathcal N}_{g,n}$ denote the mapping class group of $N_{g,n}$. Szepietowski showed that ${\mathcal N}_{g,n}$ is generated by finitely many involutions. The number of elements in his generating set depends linearly on $g$ and $n$. In the case of $n=0$, Szepietowski found an involution generating set in such a way that the number of its elements does not depend on $g$, showing that ${\mathcal N}_{g,0}$ is generated by four involutions. In this thesis, for $n \geq 0$, we prove that ${\mathcal N}_{g,n}$ is generated by eight involutions if $g \geq 13$ is odd and by eleven involutions if $g \geq 14$ is even.