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

TL;DR

This paper introduces new classes of countable topological groups and isometry groups with the strong Lévy property, demonstrating measure concentration and providing numerous nonisomorphic examples.

Contribution

It establishes the strong Lévy property for isometry groups of Urysohn spaces with various metrics and structures, and shows how to topologize certain groups to have the Lévy property.

Findings

Isometry groups of Urysohn spaces with small distance sets have the strong Lévy property.

Countable omnigenous locally finite groups can be topologized to exhibit the Lévy property.

There are at least continuum many nonisomorphic countable groups with the strong Lévy 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.