Isometry groups and countable groups with the Lévy property
Wei Dai, Su Gao, Víctor Hugo Yañez
TL;DR
The work systematically expands the catalog of Lévy groups by introducing two new countable constructions and by demonstrating strong Lévy properties for isometry groups of Urysohn spaces with both discrete and continuous metric frameworks. It develops novel $\,\\Delta$-metric techniques and two independent proofs (Fraïssé-based and omnigenous-embedding) to show $\\mathrm{Iso}(\\mathbb{U}_{\\Delta})$ and related groups possess the strong Lévy property when $\\inf\\Delta=0$, with extensions to Lipschitz continuous signatures via continuous logic. The results imply a rich landscape of pairwise nonisomorphic strong Lévy groups, including continuum-many topologies on Hall’s locally finite group and dense embeddings of omnigenous groups into isometry groups. Technically, the paper blends concentration-of-measure, model-theoretic Fraïssé theory, and continuous logic to produce new examples and a finer understanding of the isomorphism types of Lévy groups, highlighting both the breadth and complexity of Lévy phenomena in topological groups.
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 $Δ$ is a countable distance value set with arbitrarily small values, then $\mbox{Iso}(\mathbb{U}_Δ)$, the isometry group of the Urysohn $Δ$-metric space equipped with the pointwise convergence topology, where $\mathbb{U}_Δ$ 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.