The nilpotent genus of finitely generated residually nilpotent groups

TL;DR

This paper investigates the relationships between finitely generated residually nilpotent groups, focusing on conditions for para-$G$ groups and their properties, including Hirsch length and pro-nilpotent completions.

Contribution

It establishes sufficient conditions for a group to be para-$G$ and explores properties of polycyclic groups related to their nilpotent genus and pro-nilpotent completions.

Findings

Conditions for a group to be para-$G$ are identified.

Para-$G$ groups share the same Hirsch length in certain cases.

Pro-nilpotent completions of specific polycyclic groups are locally polycyclic.

Abstract

If $G$ and $H$ are finitely generated residually nilpotent groups, then $G$ and $H$ are in the same nilpotent genus if they have the same lower central quotients (up to isomorphism). A stronger condition is that $H$ is para-$G$ if there exists a monomorphism of $G$ into $H$ which induces isomorphisms between the corresponding quotients of their lower central series. We first consider residually nilpotent groups and find sufficient conditions on the monomorphism so that $H$ is para-$G.$ We then prove that for certain polycyclic groups, if $H$ is para-$G$, then $G$ and $H$ have the same Hirsch length. We also prove that the pro-nilpotent completions of these polycyclic groups are locally polycyclic.