The torsion generating set of the mapping class groups and the Dehn twist subgroups of non-orientable surfaces of odd genus

Abstract

Let $N_g$ be the non-orientable surface of genus $g$, $\mathrm{MCG}(N_g)$ the mapping class group of $N_g$, $\mathscr{T} (N_g)$ the index $2$ subgroup generated by all Dehn twists of $\mathrm{MCG}(N_g)$. We prove that for odd genus, $(1)$ if $g = 4k + 3 (k\ge1), \mathrm{MCG}(N_g)$ can be generated by three elements of finite order; $(2)$ if $g = 4k + 1 (k\ge2)$, $\mathscr{T} (N_g)$ can be generated by three elements of finite order.