The shuffle algebra and Galois cohomology

Consider a prime number p and a free profinite group S on basis X.

We describe the quotients of S by the lower p-central filtration in terms of the shuffle algebra on X.  This description is obtained by combining tools from the combinatorics of words with Galois cohomology methods.

In the context of absolute Galois groups, this machinery gives a new general perspective on recent arithmetical results on Massey products and other cohomological operations.