The Zassenhaus filtration, Massey Products, and Representations of Profinite Groups

TL;DR

This paper explores the structure of the p-Zassenhaus filtration of profinite groups, linking it to Massey products and representations, and extends known results on absolute Galois groups.

Contribution

It establishes a new connection between the Zassenhaus filtration and unipotent representations under cohomological conditions, generalizing previous work.

Findings

G_{n+1} equals the intersection of kernels of certain unipotent representations.

The result applies when the p-cohomological dimension of G is at most 1.

Extends earlier results on the structure of absolute Galois groups.

Abstract

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 the p-cohomological dimension of G is at most 1), we prove that G_{n+1} is the intersection of all kernels of upper-triangular unipotent (n+1)-dimensional representations of G over \mathbb F_p. This extends earlier results by Minac, Spira, and the author on the structure of absolute Galois groups of fields.