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

Abstract

According to Kat\vetov (1988), for every infinite cardinal $\mathfrak{m}$ satisfying ${\mathfrak{m}}^{\mathfrak{n}}\le \mathfrak{m}$ 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.

14 pages, LaTeX 2e with Elsevier macros, the final submission to Top. Appl. with referee's remarks taken into account

References in corpus (1)

• A dichotomy for the number of ultrapowers

Cited by in corpus (2)

• Distance sets of universal and Urysohn metric spaces
• Actions des groupes topologiques sur les objets universels