On Bredon homology of elementary amenable groups

Abstract

We show that for elementary amenable groups the Hirsch length is equal to the Bredon homological dimension. This also implies that countable elementary amenable groups admit a finite-dimensional model for $\underline {E}G$ of dimension less than or equal to the Hirsch length plus one. Some remarks on groups of type ${\operatorname {FP}}_{\infty }$ are also made.