Commutator subgroups of singular braid groups

Abstract

The singular braids with n strands, n≥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 SGn. There has been another generalization of braid groups, denoted by GVBn, n≥3, which was introduced by Fang as a group of symmetries behind quantum quasi-shuffle structures. The group GVBn 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≥5, and GVB′n is finitely generated if and only if n≥4. Further, we show that both SG′n and GVB′n are perfect if and only if n≥5.