On some of the residual properties of finitely generated nilpotent groups

TL;DR

This paper investigates the residual properties of finitely generated nilpotent groups, establishing that such groups are either virtually abelian or not virtually RFRS, with implications for the structure of RFRS groups.

Contribution

It proves that finitely generated nilpotent groups are either virtually abelian or not virtually RFRS, highlighting their residual properties and relation to the RFRS condition.

Findings

Finitely generated nilpotent groups are either virtually abelian or not virtually RFRS.

Any RFRS group cannot contain a nonabelian torsion-free nilpotent subgroup.

The result clarifies the relationship between residual torsion-free nilpotence and the RFRS condition.

Abstract

In recent years, the RFRS condition has been used to analyze virtual fibering in 3-manifold topology. Agol's work shows that any 3-manifold with zero Euler characteristic satisfying the RFRS condition on its fundamental group virtually fibers over the circle. In this note we will show that a finitely generated nilpotent group is either virtually abelian or is not virtually RFRS, a result which may be of independent interest though not directly applicable to 3-manifold topology. As a corollary, we deduce that any RFRS group cannot contain a nonabelian torsion-free nilpotent group. This result also illustrates some of the interplay between residual torsion-free nilpotence and the RFRS condition.