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Nilpotent groups with balanced presentations. II

J. A. Hillman

Abstract

If $G$ is a nilpotent group with a balanced presentation and $G\not\cong\mathbb{Z}^3$ then $β_1(G;\mathbb{Q})\leq2$ \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 $β_1(G;\mathbb{Q})=1$ 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 $n-1$.

Nilpotent groups with balanced presentations. II

Abstract

If is a nilpotent group with a balanced presentation and then \cite{Hi22}. We show that if such a group has an abelian normal subgroup such that then is torsion-free and has Hirsch length . On the other hand, if and has an abelian normal subgroup such that then , for some such that divides a power of .

Paper Structure

This paper contains 6 sections, 22 theorems, 30 equations.

Table of Contents

  1. notation and generalities
  2. unipotent automorphisms
  3. unipotent actions on abelian groups
  4. wang sequence estimates
  5. $h=1$: virtually $\mathbb{Z}$
  6. metabelian nilpotent groups with hirsch length $>1$

Key Result

Lemma 1

Let $\psi$ be an automorphism of a finitely generated nilpotent group $N$. Then $G=N\rtimes_\psi\mathbb{Z}$ is nilpotent if and only if $\psi^{ab}$ is unipotent. ∎

Theorems & Definitions (42)

  • Lemma : Hall
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Corollary 5
  • ...and 32 more