Elementary amenable groups of cohomological dimension 3

Jonathan A. Hillman


We show that torsion-free elementary amenable groups of Hirsch length \(\leq3\) are solvable, of derived length \(\leq3\). This class includes all solvable groups of cohomological dimension \(3\). We show also that groups in the latter subclass are either polycyclic, semidirect products \(BS(1,n)\rtimes\mathbb{Z}\), 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.

This paper is available as a pdf (252kB) file. It is also on the arXiv: [math.GR].

Thursday, June 17, 2021