Commutator subgroups of virtual and welded braid groups

Abstract

Let VB𝑛, respectively WB𝑛 denote the virtual, respectively welded, braid group on 𝑛-strands. We study their commutator subgroups VB′𝑛=[VB𝑛,VB𝑛] and, WB′𝑛=[WB𝑛,WB𝑛], respectively. We obtain a set of generators and defining relations for these commutator subgroups. In particular, we prove that VB′𝑛 is finitely generated if and only if 𝑛≥4, and WB′𝑛 is finitely generated for 𝑛≥3. Also, we prove that VB′3/VB″3=ℤ3⊕ℤ3⊕ℤ3⊕ℤ∞,VB′4/VB″4=ℤ3⊕ℤ3⊕ℤ3,WB′3/WB″3=ℤ3⊕ℤ3⊕ℤ3⊕ℤ,WB′4/WB″4=ℤ3, and for 𝑛≥5 the commutator subgroups VB′𝑛 and WB′𝑛 are perfect, i.e. the commutator subgroup is equal to the second commutator subgroup.