The lower $p$-central series of a free profinite group and the shuffle algebra

Abstract

For a prime number $p$ and a free profinite group $S$ on the basis $X$, let $S^{(n,p)}$, $n=1,2,\ldots$ be the lower $p$-central filtration of $S$. For $p>n$, we give a combinatorial description of $H^2(S/S^{(n,p)},\mathbb{Z}/p)$ in terms of the Shuffle algebra on $X$.