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

Abstract

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