Elementary amenable groups of cohomological dimension 3

Jonathan A. Hillman

Abstract

We show that torsion-free elementary amenable groups of Hirsch length $\le 3$ are solvable, of derived length $\le 3$. This class includes all solvable groups of cohomological dimension $3$. We show also that groups in the latter subclass are either polycyclic, semidirect products $BS(1,n)⋊\mathbb{Z}$, or properly ascending HNN extensions with base ${\mathbb{Z}}^2$ or ${\pi }_1(Kb)$.

Keywords: cohomological dimension, elementary amenable, finitely presentable, Hirsch length, solvable, torsion-free.

AMS Subject Classification: Primary 20J05.