Elementary amenable groups of cohomological dimension 3

Abstract

Abstract We show that torsion-free elementary amenable groups of Hirsch length at most 3 are solvable, of derived length at most 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 ) ⋊ Z \mathrm{BS}(1,n)\rtimes\mathbb{Z} , or properly ascending HNN extensions with base Z 2 \mathbb{Z}^{2} or π 1 ⁢ ( Kb ) \pi_{1}(\mathrm{Kb}) .