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 $n$th 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 $H^2(S^{[n,p]},\mathbb{Z}/p)$, 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.