THE p-ZASSENHAUS FILTRATION OF A FREE PROFINITE GROUP AND SHUFFLE RELATIONS

For a prime number p and a free profinite group S on the basis X, let $S_{\left (n,p\right )}$ , $n=1,2,\dotsc ,$ be the p-Zassenhaus filtration of S. For $p>n$ , we give a word-combinatorial description of the cohomology group $H^2\left (S/S_{\left (n,p\right )},\mathbb {Z}/p\right )$ in terms of the shuffle algebra on X. We give a natural linear basis for this cohomology group, which is constructed by means of unitriangular representations arising from Lyndon words.