Commutator subgroups of welded braid groups

Abstract

The singular braids with n strands, n ≥ 3, were introduced independently by Baez and Birman. It is known that the monoid formed by the singular braids is embedded in a group that is known as singular braid group, denoted by SGn. There has been another generalization of braid groups, denoted by GV Bn, n ≥ 3, which was introduced by Fang as a group of symmetries behind quantum quasi-shuffle structures. The group GV Bn simultaneously generalizes the classical braid group, as well as the virtual braid group on n strands.We investigate the commutator subgroups SG ′ n and GV B ′ n of these generalized braid groups. We prove that SG ′ n is finitely generated if and only if n ≥ 5, and GV B ′ n is finitely generated if and only if n ≥ 4. Further, we show that both SG ′ n and GV B ′ n are perfect if and only if n ≥ 5.

Virtual and universal braid groups, their quotients and representations

“…The map Ψ : V B n → Aut(U ), defined by ( 30) and ( 31) is a representation. The images Ψ(σ i ) and Ψ(ρ i ) satisfy both forbidden relations (23) for V B n .…”

Commutator subgroups of virtual and welded braid groups

“…Zaremsky proved this result using discrete Morse theory, without constructing explicit finite presentation. Dey and Gongopadhyay [6] also proved that W B ′ n is finitely generated for all n ≥ 3. In the present paper we have found a better bound on the number of generators than in [6].…”

“…Dey and Gongopadhyay [6] also proved that W B ′ n is finitely generated for all n ≥ 3. In the present paper we have found a better bound on the number of generators than in [6]. We prove the following result.…”

“…The presentation of W B ′ n obtained in this paper is slightly different from the one obtained in [6]. We obtain this presentation using the presentation of V B n , while computing that we use successive conjugation rule in the rewritting process, see Lemma 3.1.…”

Commutator subgroups of twin groups and Grothendieck's cartographical groups

“…This method is a well-known technique to obtain presentations for subgroups, for details see [MKS04]. This algorithm has been used to obtain presentations for certain classes of generalized braid groups and Artin groups in [BGN18], [DG18], [L 10], [Man97]. We obtain a presentation for T W n , n ≥ 3, using this approach and then remove some of the generators using Tietze transformations.…”