Commutator subgroups of virtual and welded braid groups

Let VBn, respectively WBn denote the virtual, respectively welded, braid group on n-strands. We study their commutator subgroups VBn' = [VBn , VBn] and, WBn' = (WBn,WBn], respectively. We obtain a set of generators and definng relations for these commutator subgroups. In particular, we prove that VBn' is finitely generated if and only if n >= 4, and WBn' is finitely generated for n >= 3. Also, we prove that VB3'/VB3 '' = Z(3)circle plus Z(3)circle plus Z(infinity), VB4'/VB4 '' = Z(3)circle plus Z(3)circle plus Z(3), WB3'/WB3 '' = Z(3)circle plus Z(3)circle plus Z(3)circle plus Z, WB4'/WB4 '' = Z3, and for n >= 5 the commutator subgroups VBn' and WBn' are perfect, i.e. the commutator subgroup is equal to the second commutator subgroup.