TY - JOUR

T1 - Commutator subgroups of virtual and welded braid groups

AU - Bardakov, Valeriy G.

AU - Gongopadhyay, Krishnendu

AU - Neshchadim, Mikhail V.

PY - 2019/5/1

Y1 - 2019/5/1

N2 - 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.

AB - 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.

KW - commutator subgroup

KW - perfect group

KW - Virtual braid

KW - welded braid

UR - http://www.scopus.com/inward/record.url?scp=85058222429&partnerID=8YFLogxK

U2 - 10.1142/S0218196719500127

DO - 10.1142/S0218196719500127

M3 - Article

AN - SCOPUS:85058222429

VL - 29

SP - 507

EP - 533

JO - International Journal of Algebra and Computation

JF - International Journal of Algebra and Computation

SN - 0218-1967

IS - 3

ER -