Polish ultrametric Urysohn spaces and their isometry groups

Abstract

In this paper we give some new constructions of Polish ultrametric Urysohn spaces and investigate the universality properties of their isometry groups. It is shown that all isometry groups of Polish ultrametric Urysohn spaces, regardless of their distance sets, are embeddable into each other, and in particular universal for all isometry groups of Polish ultrametric spaces. We also consider a strengthening notion, called extensive isometric embedding, and show that any isometric embedding from a compact ultrametric space into a Polish ultrametric Urysohn space is extensive. It is shown that every isometry between two compact subsets of a Polish ultrametric Urysohn space can be extended to an isometry of the entire space. We introduce a notion of generalized trees to study Polish ultrametric spaces and prove a duality theorem between the categories of Polish ultrametric spaces and their generalized tree representations. Finally we draw some conclusions about the descriptive complexity of embeddability, biembeddability and isometry relations among Polish ultrametric spaces.