Filtrations of free groups as intersections

TL;DR

This paper characterizes the n-th terms of various natural filtrations of free groups as intersections of kernels of homomorphisms into upper-triangular unipotent matrix groups, extending classical results.

Contribution

It generalizes Grun's classical result by expressing filtration terms as intersections of kernels of specific homomorphisms for multiple filtrations.

Findings

Expressed filtration terms as intersections of kernels of homomorphisms.

Extended classical results to new filtrations.

Applied to lower p-central filtration of free groups.

Abstract

For several natural filtrations of a free group S we express the n-th term of the filtration as the intersection of all kernels of homomorphisms from S to certain groups of upper-triangular unipotent matrices. This generalizes a classical result of Grun for the lower central filtration. In particular, we do this for the n-th term in the lower p-central filtration of S.