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.
Categories: math.NT math.GR