Lower central subgroups of a free group and its subgroup

TL;DR

This paper investigates the relationship between lower central subgroups of a free group and its subgroups, establishing conditions under which one cannot contain the other, with implications for 3D topology invariants.

Contribution

It provides a criterion based on normal generation for when lower central subgroups of a subgroup can contain those of the parent free group.

Findings

If G does not normally generate F, then G's lower central subgroup cannot contain that of F.

The result applies to free groups of arbitrary rank, including infinite.

Implications for Hirzebruch-type invariants in 3D topology.

Abstract

For a given free group $F$ of arbitrary rank (possibly infinite), and its subgroup $G$, we address the question whether a lower central subgroup of $G$ can contain a lower central subgroup of $F$. We show that the answer is no if $G$ does not normally generate $F$. The question comes from a study of Hirzebruch-type invariants from iterated $p$-covers for 3-dimensional homology cylinders.