I’m interested in computing the cohomological dimension of the commutator subgroup $ [P_n,P_n]$ of the pure braid group $ P_n$ . I wasn’t able to find a reference in the literature.

Because $ [P_n,P_n]$ has an abelian subgroup of rank $ \lfloor(n-1)/2\rfloor$ we have$ $ (n-2)/2\le\text{cd}([P_n,P_n])\le n-2.$ $ My guess is that in fact $ \text{cd}([P_n,P_n])=n-2$ .

There is a right split short exact sequence $ $ 1\to[F_n,F_n]\to[P_{n+1},P_{n+1}]\to[P_n,P_n]\to1,$ $ which implies that $ [P_n,P_n]$ is an iterated semidirect product of infinitely generated free groups. This suggests an inductive spectral sequence argument but I’m having problems understanding the cohomology groups $ H^{n-2}\left([P_n,P_n];H^1([F_n,F_n])\right)$ .