Commutator Subgroups of Singular Braid Groups

TL;DR

This paper studies the algebraic properties of commutator subgroups of singular and generalized virtual braid groups, revealing conditions for their finite generation and perfection based on the number of strands.

Contribution

It provides the first detailed analysis of the finite generation and perfection of commutator subgroups of these generalized braid groups.

Findings

$SG_n'$ is finitely generated iff $n \\geq 5$

$GVB_n'$ is finitely generated iff $n \\geq 4$

Both $SG_n'$ and $GVB_n'$ are perfect iff $n \\geq 5$

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 $SG_n$. There has been another generalization of braid groups, denoted by $GVB_n$, $n \geq 3$, which was introduced by Fang as a group of symmetries behind quantum quasi-shuffle structures. The group $GVB_n$ simultaneously generalizes the classical braid group, as well as the virtual braid group on $n$ strands. We investigate the commutator subgroups $SG_n'$ and $GVB_n'$ of these generalized braid groups. We prove that $SG_n'$ is finitely generated if and only if $n \ge 5$, and $GVB_n'$ is finitely generated if and only if $n \ge 4$. Further, we show that both $SG_n'$ and $GVB_n'$ are perfect if and only if $n \ge 5$.