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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 $S^{[n,p]}$ thecorresponding quotient. Using tools from the combinatorics of words, weconstruct a canonical basis of the cohomology group$H^2(S^{[n,p]},\mathbb{Z}/p)$, which we call the Lyndon basis, and use it toobtain structural results on this group. We show a duality between the Lyndonbasis and canonical generators of $S^{(n,p)}/S^{(n+1,p)}$. We prove that thecohomology group satisfies shuffle relations, which for small values of $n$fully describe it.

The Cohomology of canonical quotients of free groups and Lyndon words