Amenable groups without finitely presented amenable covers
  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. 
Mustafa Gokhan Benli, Rostislav Grigorchuk and Pierre De La Harpe
math.GR