Commutator Subgroups of Singular Braid Groups

The singular braids with $n$ strands, $n\ge 3$, 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 $GV{B}_n$, $n\ge 3$, which was introduced by Fang as a group of symmetries behind quantum quasi-shuffle structures. The group $GV{B}_n$ simultaneously generalizes the classical braid group, as well as the virtual braid group on $n$ strands. We investigate the commutator subgroups $S{G}_n^{\mathrm{\prime }}$ and $GV{B}_n^{\mathrm{\prime }}$ of these generalized braid groups. We prove that $S{G}_n^{\mathrm{\prime }}$ is finitely generated if and only if $n\ge 5$, and $GV{B}_n^{\mathrm{\prime }}$ is finitely generated if and only if $n\ge 4$. Further, we show that both $S{G}_n^{\mathrm{\prime }}$ and $GV{B}_n^{\mathrm{\prime }}$ are perfect if and only if $n \ge 5$.