Nilpotent groups with balanced presentations. II

TL;DR

This paper investigates the structure of nilpotent groups with balanced presentations, establishing bounds on their properties and classifying certain cases based on their abelian subgroups and rational Betti numbers.

Contribution

It provides new structural results and classifications for nilpotent groups with balanced presentations, especially regarding their torsion-freeness, Hirsch length, and specific semi-direct product forms.

Findings

If G has an abelian normal subgroup with G/A ≅ Z^2, then G is torsion-free with Hirsch length ≤ 4.

For G with β₁(G;Q)=1 and an abelian normal subgroup with G/A ≅ Z, G is a semi-direct product of a cyclic group and a finite cyclic group.

Groups not isomorphic to Z^3 with balanced presentations have rational first Betti number ≤ 2.

Abstract

If $G$ is a nilpotent group with a balanced presentation and $G\not\cong\mathbb{Z}^3$ then $\beta_1(G;\mathbb{Q})\leq2$ \cite{Hi22}. We show that if such a group $G$ has an abelian normal subgroup $A$ such that $G/A\cong\mathbb{Z}^2$ then $G$ is torsion-free and has Hirsch length $h(G)\leq4$. On the other hand, if $\beta_1(G;\mathbb{Q})=1$ and $G$ has an abelian normal subgroup $A$ such that $G/A\cong\mathbb{Z}$ then $G\cong\mathbb{Z}/m\mathbb{Z}\rtimes_n\mathbb{Z}$, for some $m,n\not=0$ such that $m$ divides a power of $n-1$.