Commutator subgroups of virtual and welded braid groups

Abstract

Let V Bn, resp. W Bn denote the virtual, resp. welded, braid group on n strands. We study their commutator subgroups V B ′ n = [V Bn, V Bn] and, W B ′ n = [W Bn, W Bn] respectively. We obtain a set of generators and defining relations for these commutator subgroups. In particular, we prove that V B ′ n is finitely generated if and only if n ≥ 4, and W B ′ n is finitely generated for n ≥ 3. Also we prove that, and for n ≥ 5 the commutator subgroups V B ′ n and W B ′ n are perfect, i. e. the commutator subgroup is equal to the second commutator subgroup.

Commutator subgroups of welded braid groups

“…Recently commutator subgroups of the virtual braid groups V B n have been investigated in [BGN18]. Theorem 1.2 shows that algebraically GV B ′ n has similar properties as compared to V B ′ n , as obtained in [BGN18]. On the other hand, Theorem 1.1 shows a contrast between SG ′ n and the groups V B ′ n and GV B ′ n , if we compare finite generation for n = 4.…”

“…Virtual braids are subject of curiosity due to their connection with knot invariants, see [FKV05]. Recently commutator subgroups of the virtual braid groups V B n have been investigated in [BGN18]. Theorem 1.2 shows that algebraically GV B ′ n has similar properties as compared to V B ′ n , as obtained in [BGN18].…”

“…The proofs of the theorems are obtained by application of the Reidemeister-Schreier algorithm. Similar tools have been used in [BGN18] and [DG18] to obtain finite generations of the virtual and the welded braid groups. However, there are certain technical differences between the approaches of this paper and the above ones.…”

“…Note that the initial presentation for V B ′ n in [BGN18], or W B ′ n in [DG18], had generating set indexed by Z × Z 2 . Due to the presence of the simple order two group Z 2 in the index set, the elimination process in [BGN18] or [DG18] was an one-variable process.…”

“…Note that the initial presentation for V B ′ n in [BGN18], or W B ′ n in [DG18], had generating set indexed by Z × Z 2 . Due to the presence of the simple order two group Z 2 in the index set, the elimination process in [BGN18] or [DG18] was an one-variable process. On the other hand, the initial generating sets for both GV B ′ n and SG ′ n obtained by the Reidemeister-Schreier algorithm are indexed by Z × Z. Consequently, the elimination process of the generators is a two-variable process here and seemingly more complicated.…”