What is the lower central series $\Gamma\_k(P\_n)$, where $P\_n$ is the pure braid group with $n$ strands? We know that $P\_n$ is generated by elements $A\_{i,j}$; do we know the generators of $\Gamma\_k(P\_n)$?



Lower Central Series of Pure Braid Groups?
As Jim Conant says, the structure of the lower central series of the pure braid group $P\_n$ can be deduced from the fact that $P\_n$ is an iterated semidirect product of free groups, $P\_n=F\_{n-1}\rtimes F\_{n-2}\cdots F\_2\rtimes F\_1$, though that's not quite enough: one also needs to know that the action of $P\_{n-1}$ on $F\_{n-1}$ is by basis-conjugating automorphisms, and thus it's trivial in homology. 


One can then apply Theorem 3.1 from Michael Falk and Richard Randell, *The lower central series of a fiber-type arrangement*, Invent. Math. **82** (1985), no. 1, 77--88, <http://www.ams.org/mathscinet-getitem?mr=0808110>, which says the following:


Let $1\to A\to B\to C \to 1$ be a split exact sequence of groups, and suppose the induced conjugation action of $C$ on $H\_1(A)$ is trivial (that is, $[A, C] \subseteq [A, A]$). Then the sequence of induced maps $1\to \Gamma\_k A\to \Gamma\_k B\to \Gamma\_k C \to 1$ 
is split exact, for every $k\ge 1$.


