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arXiv · 2108.13633

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

Nao Imoto·Ryoma Kobayashi
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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 $\mathcal{M}_2(N_g)$.

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Nao Imoto, Ryoma Kobayashi. 2023-03-09. On squares of Dehn twists about non-separating curves of a non-orientable closed surface. https://arxiv.org/abs/2108.13633

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