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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.
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
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.
Suppose an amenable group $G$ is acting freely on a simply connected simplicial complex $tilde X$ with compact quotient $X$. Fix $n geq 1$, assume $H_n(tilde X, mathbb{Z})=0$ and let $(H_i)$ be a Farber chain in $G$. We prove that the torsion of the
Let $X$ be a locally compact Hadamard space and $G$ be a totally disconnected group acting continuously, properly and cocompactly on $X$. We show that a closed subgroup of $G$ is amenable if and only if it is (topologically locally finite)-by-(virtua
The Cohn-Umans group-theoretic approach to matrix multiplication suggests embedding matrix multiplication into group algebra multiplication, and bounding $omega$ in terms of the representation theory of the host group. This framework is general enoug