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arXiv2023

The level $d$ principal congruence subgroup of $\textrm{SL}(n;\mathbb{Z})$

The abelianization of the level $d$ principal congruence subgroup $Γ_d(n)$ of $\textrm{SL}(n;\mathbb{Z})$ was determined by Lee-Szczarba. By this result and a result of Tits, we can obtain a minimal generating set for $Γ_d(n)$. In this paper, we give a minimal generating set for $Γ_d(n)$ and determine the abelianization of $Γ_d(n)$, without using the results of Tits and Lee-Szczarba. Moreover, we give three theorems about $Γ_d(n)$.

math.GT↗

arXiv2023

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

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 $\mathcal{M}_2(N_g)$.

math.GT↗