Torsion-free nilpotent groups of small Hirsch length with isomorphic finite quotients

TL;DR

This paper classifies finitely generated torsion-free nilpotent groups of small Hirsch length based on their finite quotients and demonstrates the existence of multiple non-isomorphic groups sharing the same finite quotients for higher Hirsch lengths.

Contribution

It explicitly describes the sets of groups with identical finite quotients for groups of Hirsch length up to 5 and shows the potential for infinitely many such groups at higher lengths.

Findings

Explicit classification for Hirsch length ≤ 5.

Existence of arbitrarily many groups with identical finite quotients for lengths ≥ 4.

Abstract

Let $\mathcal{T}$ denote the class of finitely generated torsion-free nilpotent groups. For a group $G$ let $F(G)$ be the set of isomorphism classes of finite quotients of $G$. Pickel proved that if $G \in \mathcal{T}$, then the set $\mathfrak{g}(G)$ of isomorphism classes of groups $H \in \mathcal{T}$ with $F(G)=F(H)$ is finite. We give an explicit description of the sets $\mathfrak{g}(G)$ for the $\mathcal{T}$-groups $G$ of Hirsch length at most $5$. Based on this, we show that for each Hirsch length $n\geq 4$ and for each $m \in \mathbb{N}$ there is a $\mathcal{T}$-group $G$ of Hirsch length $n$ with $\vert\mathfrak{g}(G)\vert\geq m$.