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

According to Kat\vetov (1988), for every infinite cardinal $\mathfrak{m}$ satisfying ${\mathfrak{m}}^{\mathfrak{n}}\le \mathfrak{m}$ for all ${\mathfrak n}<{\mathfrak m}$,thereexistsaunique$\mathfrak m$-homogeneous universal metric space ${\text{Ur}}_{\mathfrak{m}}$ of weight $\mathfrak{m}$. This object generalizes the classical Urysohn universal metric space $\Ur = \Ur_{\aleph_0}$\mathrm{.}Weshowthatfor$\mathfrak m$uncountable,theisometrygroup$\Iso(\Urm)$ with the topology of simple convergence is not a universal group of weight $\mathfrak{m}$: for instance, it does not contain $\text{Iso}(\text{Ur})$ as a topological subgroup. More generally, every topological subgroup of $\text{Iso}(\text{Urm})$ 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 $\text{Iso}(\text{Ur})$ being a universal Polish group.