Commutator subgroups of virtual and welded braid groups

Valeriy G. Bardakov, Krishnendu Gongopadhyay, Mikhail V. Neshchadim

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1 Citation (Scopus)


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 Z3Z3Z 8,VB 4/VB 4 = Z3Z3Z3,WB 3/WB 3 = Z3Z3Z3Z,WB 4/WB 4 = Z3, and for n = 5 the commutator subgroups VB n andWB n are perfect, i.e. the commutator subgroup is equal to the second commutator subgroup.

Original languageEnglish
Pages (from-to)507-533
Number of pages27
JournalInternational Journal of Algebra and Computation
Issue number3
Publication statusPublished - 1 May 2019


  • commutator subgroup
  • perfect group
  • Virtual braid
  • welded braid

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