Commutator Subgroups of Virtual and Welded Braid Groups

TL;DR

This paper investigates the structure of commutator subgroups of virtual and welded braid groups, providing generators, relations, and conditions for finite generation, along with detailed algebraic properties for various n.

Contribution

It offers the first explicit generators, relations, and structural properties of the commutator subgroups of virtual and welded braid groups.

Findings

VB_n' is finitely generated iff n ≥ 4

WB_n' is finitely generated iff n ≥ 3

Certain quotient groups are direct sums of cyclic groups

Abstract

Let $VB_n$, resp. $WB_n$ denote the virtual, resp. 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 $n \geq 4$, and $WB_n'$ is finitely generated for $n \geq 3$. Also we prove that $VB_3'/VB_3'' =\mathbb{Z}_3 \oplus \mathbb{Z}_3 \oplus\mathbb{Z}_3 \oplus \mathbb{Z}^{\infty}$, $VB_4' / VB_4'' = \mathbb{Z}_3 \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_3$, $WB_3'/WB_3'' = \mathbb{Z}_3 \oplus \mathbb{Z}_3 \oplus\mathbb{Z}_3 \oplus \mathbb{Z},$ $WB_4'/WB_4'' = \mathbb{Z}_3,$ and for $n \geq 5$ the commutator subgroups $VB_n'$ and $WB_n'$ are perfect, i.e. the commutator subgroup is equal to the second commutator subgroup.