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 $H^{n-2}\left([P_n,P_n];H^1([F_n,F_n])\right)$.

EDIT: The commutator subgroup $[B_n,B_n]$ of the full braid group has been studied by Gorin and Lin in "Algebraic equations with continuous coefficients and some problems of the algebraic theory of braids" (1969) Math. USSR Sb. 7 569-596. In particular, it follows from their results that $\text{cd}([B_3,B_3])=1$ and $\text{cd}([B_4,B_4])=2$. However, this is not immediately helpful because $[P_n,P_n]$ is not a finite index subgroup of $[B_n,B_n]$.


Here is the answer: for $n\geq 2$ we have $\mathrm{cd}([P_n,P_n])=n-2$.



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