## 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.

: Primary 20F18; secondary 20J05, 57N13.

This paper is available as a pdf (324kB) file. It is also on the arXiv: arxiv.org/abs/2107.09985.

 Tuesday, May 31, 2022