Commutator subgroups of singular braid groups

Abstract

The singular braids with [Formula: see text] strands, [Formula: see text], 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 [Formula: see text]. There has been another generalization of braid groups, denoted by [Formula: see text], [Formula: see text], which was introduced by Fang as a group of symmetries behind quantum quasi-shuffle structures. The group [Formula: see text] simultaneously generalizes the classical braid group, as well as the virtual braid group on [Formula: see text] strands. We investigate the commutator subgroups [Formula: see text] and [Formula: see text] of these generalized braid groups. We prove that [Formula: see text] is finitely generated if and only if [Formula: see text], and [Formula: see text] is finitely generated if and only if [Formula: see text]. Further, we show that both [Formula: see text] and [Formula: see text] are perfect if and only if [Formula: see text].