The Twist Subgroup is generated by two elements

TL;DR

This paper proves that the twist subgroup of a nonorientable surface can be generated by two or three elements depending on the genus, providing new insights into its algebraic structure.

Contribution

It establishes minimal generating sets for the twist subgroup of nonorientable surfaces for various genus values, including explicit bounds and generator counts.

Findings

Twist subgroup generated by two elements for odd $g extgreater=27$ and even $g extgreater=42$

Generated by two or three commutators depending on $g$ modulo 4

Can be generated by three elements for $g extgreater=8$

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 $g\geq27$ and even $g\geq42$. 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 $g\geq 8$. For this general case, the number of commutator generators is either three or four depending on $g$ modulo $4$ again.