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- Title
Commutator subgroups of virtual and welded braid groups.
- Authors
Bardakov, Valeriy G.; Gongopadhyay, Krishnendu; Neshchadim, Mikhail V.
- Abstract
Let VB n , respectively WB n denote the virtual, respectively welded, braid group on n -strands. We study their commutator subgroups VB n ′ = [ VB n , VB n ] and, WB n ′ = [ WB n , WB n ] , 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 ″ = ℤ 3 ⊕ ℤ 3 ⊕ ℤ 3 ⊕ ℤ ∞ , VB 4 ′ / VB 4 ″ = ℤ 3 ⊕ ℤ 3 ⊕ ℤ 3 , WB 3 ′ / WB 3 ″ = ℤ 3 ⊕ ℤ 3 ⊕ ℤ 3 ⊕ ℤ , WB 4 ′ / WB 4 ″ = ℤ 3 , 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.
- Subjects
BRAID group (Knot theory); COMMUTATION (Electricity); COMMUTATORS (Operator theory); BRAID; GROUPS
- Publication
International Journal of Algebra & Computation, 2019, Vol 29, Issue 3, p507
- ISSN
0218-1967
- Publication type
Article
- DOI
10.1142/S0218196719500127