  The goal of this article is to study results and examples concerning finitely
presented covers of finitely generated amenable groups. We collect examples of
groups $G$ with the following properties: (i) $G$ is finitely generated, (ii)
$G$ is amenable, e.g. of intermediate growth, (iii) any finitely presented
group $E$ with a quotient isomorphic to $G$ contains non-abelian free
subgroups, or the stronger (iii') any finitely presented group with a quotient
isomorphic to $G$ is large.
