Nilpotent groups with balanced presentations

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 $F$.

arXiv commentv2: New final section giving examples with $h=6$. v3: New Theorem 5. v4: Introduction and \S3 rewritten. Simpler proof of main theorem, with stronger result. v5: The discussion of Lie analogues has been moved from the introduction to a new \S6, while \S5 has been rewritten to clarify the exposition

References in corpus (1)

• 3-Manifolds with nilpotent embeddings in $S^4$