Generating the mapping class group of a nonorientable surface by three torsions

TL;DR

This paper demonstrates that the mapping class group of a nonorientable surface of genus g (not equal to 4) can be generated by just three torsion elements, with specific results for conjugate elements of certain orders.

Contribution

It establishes that the entire mapping class group of nonorientable surfaces can be generated by three torsion elements, including conjugates of a fixed order, extending understanding of its algebraic structure.

Findings

Generated by three torsion elements for most genera.

Existence of generating sets of three conjugate elements of order k.

Results also apply to subgroups generated by Dehn twists.

Abstract

We prove that the mapping class group $\mathcal{M}(N_g)$ of a closed nonorientable surface of genus $g$ different than 4 is generated by three torsion elements. Moreover, for every even integer $k\ge 12$ and $g$ of the form $g=pk+2q(k-1)$ or $g=pk+2q(k-1)+1$, where $p,q$ are non-negative integers and $p$ is odd, $\mathcal{M}(N_g)$ is generated by three conjugate elements of order $k$. Analogous results are proved for the subgroup of $\mathcal{M}(N_g)$ generated by Dehn twists.