Research output: Contribution to journal › Article › peer-review
Commutator subgroups of virtual and welded braid groups. / Bardakov, Valeriy G.; Gongopadhyay, Krishnendu; Neshchadim, Mikhail V.
In: International Journal of Algebra and Computation, Vol. 29, No. 3, 01.05.2019, p. 507-533.Research output: Contribution to journal › Article › peer-review
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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 -
ID: 20346410