Nilpotent groups with balanced presentations

TL;DR

This paper investigates properties of torsion-free nilpotent groups with balanced presentations, revealing constraints on their Betti numbers based on Hirsch length and constructing examples with specific homological features.

Contribution

It establishes new restrictions on Betti numbers for such groups and provides explicit examples with particular homological properties.

Findings

If h(G)>3 and G has a balanced presentation, then β₁(G;Q)=2.

There is exactly one torsion-free nilpotent group with h=4 and balanced presentation.

Constructed a torsion-free nilpotent group with h=6 where β₂(G;F)=β₁(G;F) for all fields F.

Abstract

We show that if a torsion free nilpotent group $G$ has a balanced presentations and Hirsch length $h(G)>3$ then $\beta_1(G;\mathbb{Q})=2$. There is just one such group which is torsion-free and of Hirsch length $h=4$, and none with $h=5$. We also construct a torsion-free nilpotent group $G$ with $h=6$ and such that $\beta_2(G;F)=\beta_1(G;F)$ for all fields $F$.