Isometry groups and countable groups with the Lévy property

Abstract

A topological group $G$ is said to have the Lévy property if it admits a dense subgroup which is decomposed as the union of an increasing sequence of compact subgroups $\mathcal{G}=\{G_i:i\in \mathbb{N}\}$ of $G$ which exhibits concentration of measure in the sense of Gromov and Milman. We say that $G$ has the strong Lévy property whenever the sequence $\mathcal{G}$ is comprised of finite subgroups. In this paper we give several new classes of isometry groups and countable topological groups with the strong Lévy property. We prove that if $\hat{I}”$ is a countable distance value set with arbitrarily small values, then $\mbox{Iso}(\mathbb{U}_Δ)$, the isometry group of the Urysohn $\hat{I}”$-metric space equipped with the pointwise convergence topology, where ${\mathbb{U}}_{\hat{I}}”$ is equipped with the metric topology, has the strong Lévy property. We also prove that if $\mathcal{L}$ is a Lipschitz continuous signature, then $\mbox{Iso}(\mathbb{U}_{\mathcal{L}})$, the isometry group of the unique separable Urysohn $\mathcal{L}$-structure, has the strong Lévy property. In addition, our approach shows that any countable omnigenous locally finite group can be given a topology with the Lévy property. As a consequence to our results, we obtain at least continuum many pairwise nonisomorphic countable topological groups or isometry groups with the strong Lévy property.