Let $PB_n$ be the pure braid group on $n$ strands. The group $PB_n$ has every conceivable finiteness property. Also, it has a large abelianization. My question is whether the commutator subgroup $[PB_n,PB_n]$ is finitelygenerated or not.
No, the group $PB_n$, $n\ge 3$ has an epimorphism onto the free group $F_2$. Since the commutator subgroup $[F_2,F_2]$ is not finitely generated, the commutator subgroup $[PB_n,PB_n]$ is not finitely generated either. 

