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

Abstract

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.

Cohomology and the Combinatorics of Words for Magnus Formations

“…p)) ⊕ Sh(X) indec,n ⊗ (Z/p) ∼ − → H 2 (G/K (n,p) ).When t = 1, the subgroups K (n,p) are closely related to the p-Zassenhaus filtration of G (see the Introduction). Namely, in[Efr22, Th. 4.6] it is shown using p-restricted Lie algebra techniques that K (n,p) , G (n,p) coincide modulo (G (n,p) ) p [G, G (n,p) ].…”