Unitary Representations of the Isometry Groups of Urysohn Spaces

TL;DR

This paper classifies all continuous unitary representations of the isometry group of the rational Urysohn space, establishing property (T) and deriving ergodic theoretic consequences for actions and invariant measures.

Contribution

It provides a complete classification of unitary representations of the isometry group of the rational Urysohn space and proves property (T) for this group.

Findings

Isom$(Q extbf{U})$ has property (T)

Actions are either essentially free or transitive

Invariant measures are product measures

Abstract

We obtain a complete classification of the continuous unitary representations of the isometry group of the rational Urysohn space $\mathbb{Q}\mathbb{U}$. As a consequence, we show that Isom$(\mathbb{Q}\mathbb{U})$ has property (T). We also derive several ergodic theoretic consequences from this classification: $(i)$ every probability measure-preserving action of Isom$(\mathbb{Q}\mathbb{U})$ is either essentially free or essentially transitive, $(ii)$ every ergodic Isom$(\mathbb{Q}\mathbb{U})$-invariant probability measure on $[0,1]^{\mathbb{Q}\mathbb{U}}$ is a product measure. We obtain the same results for isometry groups of variations of $\mathbb{Q}\mathbb{U}$, such as the rational Urysohn sphere $\mathbb{Q}\mathbb{U}_1$, the integral Urysohn space $\mathbb{Z}\mathbb{U}$, etc.