Commutator subgroups of virtual and welded braid groups

Abstract

Let [Formula: see text], respectively [Formula: see text] denote the virtual, respectively welded, braid group on [Formula: see text]-strands. We study their commutator subgroups [Formula: see text] and, [Formula: see text], respectively. We obtain a set of generators and defining relations for these commutator subgroups. In particular, we prove that [Formula: see text] is finitely generated if and only if [Formula: see text], and [Formula: see text] is finitely generated for [Formula: see text]. Also, we prove that [Formula: see text] and for [Formula: see text] the commutator subgroups [Formula: see text] and [Formula: see text] are perfect, i.e. the commutator subgroup is equal to the second commutator subgroup.