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 finitely-generated or not.



Finite generation of the commutator subgroup of the pure braid group 
No, the group $PB\_n$, $n\ge 3$ has an [epimorphism](http://arxiv.org/pdf/1106.4602v1.pdf) 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. 


