The level $d$ mapping class group of a compact non-orientable surface

TL;DR

This paper studies the structure of level $d$ mapping class groups of non-orientable surfaces, providing generators and relations, and connecting them to principal congruence subgroups of special linear groups.

Contribution

It introduces new relations and generating sets for level $d$ mapping class groups of non-orientable surfaces, extending understanding of their algebraic structure.

Findings

Normal generating set for $ ext{Mod}_d(N_{g,n})$ when $g extgreater 4$

Finite generating sets for certain $d$, $g$, and $n$

Relations between mapping class groups and principal congruence subgroups

Abstract

Let $N_{g,n}$ be a genus $g$ compact non-orientable surface with $n$ boundaries. We explain about relations on the level $d$ mapping class group $\mathcal{M}_d(N_{g,0})$ of $N_{g,0}$ and the level $d$ principal congruence subgroup $\Gamma_d(g-1)$ of $\mathrm{SL}(g-1;\mathbb{Z})$. As applications, we give a normal generating set of $\mathcal{M}_d(N_{g,n})$ for $g\ge4$ and $n\ge0$, and finite generating sets of $\mathcal{M}_d(N_{g,n})$ for some $d$, any $g\ge4$ and $n\ge0$.