The cohomology of canonical quotients of free groups and Lyndon words

For a prime number p and a free profinite group S, let S^(n,p) be the nth term of its lower p-central filtration, and S^[n,p] the corresponding quotient. Using tools from the combinatorics of words, we construct a canonical basis of the cohomology group H²(S^[n,p], Z/p), which we call the Lyndon basis, and use it to obtain structural results on this group. We show a duality between the Lyndon basis and canonical generators of S^(n,p)/S^(n+1,p). We prove that the cohomology group satisfies shuffle relations, which for small values of n fully describe it.