Isometry groups and countable groups with the L\'{e}vy property

Abstract

A topological group $G$ is said to have the L\'evy 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 L\'evy 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 L\'evy property. We prove that if $\Delta$ is a countable distance value set with arbitrarily small values, then $\mbox{Iso}(\mathbb{U}_\Delta)$, the isometry group of the Urysohn $\Delta$-metric space equipped with the pointwise convergence topology, where $\mathbb{U}_\Delta$ is equipped with the metric topology, has the strong L\'evy 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 L\'evy property. In addition, our approach shows that any countable omnigenous locally finite group can be given a topology with the L\'evy property. As a consequence to our results, we obtain at least continuum many pairwise nonisomorphic countable topological groups or isometry groups with the strong L\'evy property.