Generating the Mapping Class Group of a Nonorientable Surface by Two Elements or By Three Involutions

TL;DR

This paper proves that the mapping class group of a nonorientable surface of sufficiently high genus can be generated by two elements or three involutions, advancing understanding of its algebraic structure.

Contribution

It establishes new generating sets for the mapping class group of nonorientable surfaces using only two elements or three involutions for large genus.

Findings

For $g extgreater 18$, $ extrm{Mod}(N_g)$ is generated by two elements.

For $g extgreater 25$, $ extrm{Mod}(N_g)$ is generated by three involutions.

Provides explicit bounds on genus for generation by minimal elements.

Abstract

We prove that, for $g\geq19$ the mapping class group of a nonorientable surface of genus $g$, $\textrm{Mod}(N_g)$, can be generated by two elements, one of which is of order $g$. We also prove that for $g\geq26$, $\textrm{Mod}(N_g)$ can be generated by three involutions if $g\geq26$.