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](http://iopscience.iop.org/article/10.1070/SM1969v007n04ABEH001104/meta). 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 $[B\_n,B\_n]$.



What is the cohomological dimension of the commutator subgroup of the pure braid group?
Here is the answer: for $n\geq 2$ we have $\mathrm{cd}([P\_n,P\_n])=n-2$.


<https://arxiv.org/abs/1905.05099>


