Subgroups of isometries of Urysohn–Katětov metric spaces of uncountable density

According to Katětov (1988) [15], for every infinite cardinal m satisfying m n ⩽ m for all n < m , there exists a unique m -homogeneous universal metric space U m of weight m . This object generalizes the classical Urysohn universal metric space U = U ℵ 0 . We show that for m uncountable, the isometry group Iso ( U m ) with the topology of simple convergence is not a universal group of weight m : for instance, it does not contain Iso ( U ) as a topological subgroup. More generally, every topological subgroup of Iso ( U m ) having density < 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 Uspenskijʼs (1990) [35] result about the group Iso ( U ) being a universal Polish group.