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.

Some non-amenable groups

We generalize a result of R. Thomas to establish the non-vanishing of the first l2-Betti number for a class of finitely generated groups.

Amenable actions, invariant means and bounded cohomology

We show that topological amenability of an action of a countable discrete group on a compact space is equivalent to the existence of an invariant mean for the action. We prove also that this is equivalent to vanishing of bounded cohomology for a class of Banach G-modules associated to the action, as well as to vanishing of a specific cohomology class. In the case when the compact space is a point our result reduces to a classic theorem of B.E. Johnson characterising amenability of groups. In the case when the compact space is the Stone-v{C}ech compactification of the group we obtain a cohomological characterisation of exactness for the group, answering a question of Higson.

Amenable uniformly recurrent subgroups and lattice embeddings

We study lattice embeddings for the class of countable groups $Gamma$ defined by the property that the largest amenable uniformly recurrent subgroup $A_Gamma$ is continuous. When $A_Gamma$ comes from an extremely proximal action and the envelope of $A_Gamma$ is co-amenable in $Gamma$, we obtain restrictions on the locally compact groups $G$ that contain a copy of $Gamma$ as a lattice, notably regarding normal subgroups of $G$, product decompositions of $G$, and more generally dense mappings from $G$ to a product of locally compact groups.

Leptin densities in amenable groups

We define a notion of uniform density on translation bounded measures in unimodular amenable locally compact Hausdorff groups, which is based on a group invariant introduced by Leptin in 1966. We show that this density notion coincides with the well-known Banach density. We use Leptin densities for a new geometric proof of the model set density formula, which expresses the density of a uniform regular model set in terms of the volume of its window, and for a proof of uniform mean almost periodicity of such model sets.