Filtrations of free groups arising from the lower central series

TL;DR

This paper systematically studies filtrations of free groups derived from the lower central series, characterizing them through algebraic and representation-theoretic methods, and relating them to cohomological Massey products.

Contribution

It generalizes classical and recent results on group filtrations, providing new characterizations and recursive definitions, and connects these filtrations to Massey products in cohomology.

Findings

Characterization of filtrations via group algebra and Magnus algebra

Recursive definitions extending Lazard's results

Relation of filtrations to Massey products in cohomology

Abstract

We make a systematic study of filtrations of a free group F defined as products of powers of the lower central series of F. Under some assumptions on the exponents, we characterize these filtrations in terms of the group algebra, the Magnus algebra of non-commutative power series, and linear representations by upper-triangular unipotent matrices. These characterizations generalize classical results of Grun, Magnus, Witt, and Zassenhaus from the 1930's, as well as later results on the lower p-central filtration and the p-Zassenhaus filtrations. We derive alternative recursive definitions of such filtrations, extending results of Lazard. Finally, we relate these filtrations to Massey products in group cohomology.