Filtrations of free groups arising from the lower central series

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 Grün, Magnus, Witt, and Zassenhaus from the 1930s, 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.

THE p-ZASSENHAUS FILTRATION OF A FREE PROFINITE GROUP AND SHUFFLE RELATIONS

“…The following result gives similar restrictions on the Magnus coefficients of elements of S (n,p) . In the discrete case it was proved in [2,Example 4.6], where it was further shown that these restrictions in fact characterize S (n,p) . While it is possible to derive the proposition from the discrete case using a density argument, we provide a direct proof.…”

“…The Lyndon words of length ≤ 2 are the words w = (x) and w = (xy), with x,y ∈ X, x < y. Then σ w is τ p j 2 (1) w = x p and τ p j 2 (2) w = [x,y], respectively. In [8, §10] it is shown that the value of w,w , where w,w are Lyndon words of lengths ≤ 2, is 1 if w = w and is 0 otherwise.…”

On the stable Andreadakis problem

“…IV.1.9]. We also refer to [CE16] for a nice discussion of filtrations defined by induction. This description has a nice consequence, similar to Proposition 1.9 (we abbreviate L(Γ…”

The kernel generating condition and absolute Galois groups

“…Then, for a more general profinite group G (such as G F ), one takes a profinite presentation, i.e., a continuous epimorphism π : S → G, where S is a free profinite group, and transfers the equality (1.1) from S to G. The first part is purely group-theoretic, and is usually proved using Magnus theory, i.e., by viewing the elements of G = S as formal power series. A general machinery to obtain such results in the free profinite case is given in [Efr14b] (see also [CE16]).…”