Commutator Subgroups of Singular Braid Groups

Soumya Dey; Krishnendu Gongopadhyay

paper

Commutator Subgroups of Singular Braid Groups

arXiv:1806.05902

Abstract

The singular braids with $n$ strands, $n \geq 3$ , were introduced independently by Baez and Birman. It is known that the monoid formed by the singular braids is embedded in a group that is known as singular braid group, denoted by $S G_{n}$ . There has been another generalization of braid groups, denoted by $G V B_{n}$ , $n \geq 3$ , which was introduced by Fang as a group of symmetries behind quantum quasi-shuffle structures. The group $G V B_{n}$ simultaneously generalizes the classical braid group, as well as the virtual braid group on $n$ strands. We investigate the commutator subgroups $S G_{n}^{'}$ and $G V B_{n}^{'}$ of these generalized braid groups. We prove that $S G_{n}^{'}$ is finitely generated if and only if $n \geq 5$ , and $G V B_{n}^{'}$ is finitely generated if and only if $n \geq 4$ . Further, we show that both $S G_{n}^{'}$ and $G V B_{n}^{'}$ are perfect if and only if $n \geq 5$ .

arXiv comment24 pages. Expanded version; singular braid groups added, structural changes done. Comments are welcome