The Zassenhaus filtration, Massey products, and representations of profinite groups

We consider the p-Zassenhaus filtration (_{G n}) of a profinite group G. Suppose that G = S/N for a free profinite group S and a normal subgroup N of S contained in _{S n}. Under a cohomological assumption on the n-fold Massey products (which holds, e.g., if G has p-cohomological dimension ≤ 1), we prove that _{G n +1} is the intersection of all kernels of upper-triangular unipotent (n + 1)-dimensional representations of G over Fp. This extends earlier results by Mináč, Spira, and the author on the structure of absolute Galois groups of fields.