Subgroups of isometries of Urysohn-Katetov metric spaces of uncountable density

TL;DR

This paper investigates the structure of isometry groups of uncountable Urysohn-Katetov metric spaces, revealing they are not universal and have specific functional properties, contrasting with the classical Urysohn space.

Contribution

It demonstrates that for uncountable densities, the isometry groups are not universal and identifies conditions under which subgroups are functionally balanced.

Findings

The isometry group of uncountable Urysohn-Katetov spaces is not universal.

Subgroups with density less than the space's density and property (OB) are functionally balanced.

Contrasts with Uspenskij's result on the universality of the isometry group of the classical Urysohn space.

Abstract

According to Kat\vetov (1988), for every infinite cardinal $\mathfrak m$ satisfying ${\mathfrak m}^{\mathfrak n}\leq {\mathfrak m}$ for all ${\mathfrak n}<{\mathfrak m}$, there exists a unique $\mathfrak m$-homogeneous universal metric space $\Ur_{\mathfrak m}$ of weight $\mathfrak m$. This object generalizes the classical Urysohn universal metric space $\Ur = \Ur_{\aleph_0}$. We show that for $\mathfrak m$ uncountable, the isometry group $\Iso(\Urm)$ with the topology of simple convergence is not a universal group of weight $\mathfrak m$: for instance, it does not contain $\Iso(\Ur)$ as a topological subgroup. More generally, every topological subgroup of $\Iso(\Urm)$ having density $<{\mathfrak m}$ and possessing the bounded orbit property $(OB)$ is functionally balanced: right uniformly continuous bounded functions are left uniformly continuous. This stands in sharp contrast with Uspenskij's 1990 result about the group $\Iso(\Ur)$ being a universal Polish group.