Commutator Subgroups of Virtual and Welded Braid Groups

Abstract

Let $V{B}_n$, resp. $W{B}_n$ denote the virtual, resp. welded, braid group on $n$ strands. We study their commutator subgroups $V{B}_n^{\prime }=[V{B}_n,V{B}_n]$ and, $W{B}_n^{\prime }=[W{B}_n,W{B}_n]$ respectively. We obtain a set of generators and defining relations for these commutator subgroups. In particular, we prove that $V{B}_n^{\prime }$ is finitely generated if and only if $n\ge 4$, and $W{B}_n^{\prime }$ is finitely generated for $n\ge 3$. Also we prove that $V{B}_3^{\prime }/V{B}_3^{\prime \prime }={\mathbb{Z}}_3\oplus {\mathbb{Z}}_3\oplus {\mathbb{Z}}_3\oplus {\mathbb{Z}}^{\infty }$, $V{B}_4^{\prime }/V{B}_4^{\prime \prime }={\mathbb{Z}}_3\oplus {\mathbb{Z}}_3\oplus {\mathbb{Z}}_3$, $W{B}_3^{\prime }/W{B}_3^{\prime \prime }={\mathbb{Z}}_3\oplus {\mathbb{Z}}_3\oplus {\mathbb{Z}}_3\oplus \mathbb{Z},$ $W{B}_4^{\prime }/W{B}_4^{\prime \prime }={\mathbb{Z}}_3,$ and for $n\ge 5$ the commutator subgroups $V{B}_n^{\prime }$ and $W{B}_n^{\prime }$ are perfect, i.e. the commutator subgroup is equal to the second commutator subgroup.

24 pages