Elementary amenable groups of cohomological dimension 3

TL;DR

This paper classifies torsion-free elementary amenable groups of Hirsch length at most 3, showing they are solvable with derived length at most 3, and describes their structure in detail.

Contribution

It establishes that such groups are solvable with bounded derived length and characterizes their specific algebraic structures.

Findings

Groups of this class are solvable with derived length ≤ 3.

They include all solvable groups of cohomological dimension 3.

Such groups are either polycyclic, certain semidirect products, or ascending HNN extensions.

Abstract

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 $\pi_1(Kb)$.