nilpotent groups with balanced presentations. II

J. A. Hillman

Abstract

Let \(G\) be a nilpotent group with a balanced presentation. Then either \(G\cong\mathbb{Z}^3\) or \(\beta_1(G;\mathbb{Q})\leq2\). We show that if \(G\) has an abelian normal subgroup \(A\) such that \(G/A\cong\mathbb{Z}^2\) then \(G\) is torsion-free and has Hirsch length \(h(G)\leq4\). We also consider the torsion subgroup of \(G\) when \(h(G)\leq2\).

Keywords: balanced, nilpotent, Hirsch length, metabelian.

AMS Subject Classification: Primary 20F18; secondary 20J05, 57N13.