The Twist Subgroup is generated by two elements

Abstract

We show that the twist subgroup ${\mathcal{T}}_g$ of a nonorientable surface of genus $g$ can be generated by two elements for every odd $g\ge 27$ and even $g\ge 42$. Using these generators, we can also show that ${\mathcal{T}}_g$ can be generated by two or three commutators depending on $g$ modulo $4$. Moreover, we show that ${\mathcal{T}}_g$ can be generated by three elements if $g\ge 8$. For this general case, the number of commutator generators is either three or four depending on $g$ modulo $4$ again.

19 pages, 10 figures