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

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