Cohomology and the Combinatorics of Words for Magnus Formations

Abstract

For a prime number p and a free pro-p group G on a totally ordered basis X, we consider closed normal subgroups G Φ of G which are generated by p-powers of iterated commutators associated with Lyndon words in the alphabet X. We express the profinite cohomology group H 2 (G/G Φ ) combinatorically, in terms of the shuffle algebra on X. This partly extends existing results for the lower p-central and p-Zassenhaus filtrations of G.