On squares of Dehn twists about non-separating curves of a non-orientable closed surface

TL;DR

This paper investigates the structure of the level 2 mapping class group of non-orientable surfaces, showing that it cannot be generated solely by squares of Dehn twists about non-separating curves, and provides finite generating sets for related subgroups.

Contribution

It proves that the Dehn twist subgroup of the level 2 mapping class group of non-orientable surfaces cannot be generated by squares of Dehn twists, extending understanding of the group's generators.

Findings

Dehn twist subgroup cannot be generated by squares of Dehn twists.

Finite generating set for subgroup generated by separating curves and squares of non-separating twists.

Analysis of actions on non-separating simple closed curves.

Abstract

The level $2$ mapping class group of an orientable closed surface can be generated by squares of Dehn twists about non-separating curves. On the other hand, the level $2$ mapping class group $\mathcal{M}_2(N_g)$ of a non-orientable closed surface $N_g$ can not be generated by only Dehn twists, and so it can not be generated by squares of Dehn twists about non-separating curves. In this paper, we prove that the Dehn twist subgroup of $\mathcal{M}_2(N_g)$ can not be generated by squares of Dehn twists about non-separating curves either. As an application, we give a finite generating set for the subgroup of $\mathcal{M}_2(N_g)$ generated by Dehn twist about separating curves and squares of Dehn twists about non-separating curves. Moreover, we examine about actions on non-separating simple closed curves of $N_g$ by $\mathcal{M}_2(N_g)$.