Unitary Representations of the Isometry Groups of Urysohn Spaces

2024/10/02 by Barritault, Rémi, Jahel, Colin, Joseph, Matthieu
#Dynamical Systems (math.DS) #FOS: Mathematics #Group Theory (math.GR) #Logic (math.LO) #Representation Theory (math.RT)

paper · doi:10.48550/arxiv.2410.01725

Abstract

We obtain a complete classification of the continuous unitary representations of the isometry group of the rational Urysohn space ℚ\mathbbU. As a consequence, we show that Isom(ℚ\mathbbU) has property (T). We also derive several ergodic theoretic consequences from this classification: (i) every probability measure-preserving action of Isom(ℚ\mathbbU) is either essentially free or essentially transitive, (ii) every ergodic Isom(ℚ\mathbbU)-invariant probability measure on [0,1]^ℚ\mathbbU is a product measure. We obtain the same results for isometry groups of variations of ℚ\mathbbU, such as the rational Urysohn sphere ℚ\mathbbU₁, the integral Urysohn space ℤ\mathbbU, etc.

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