Unitary Representations of the Isometry Groups of Urysohn Spaces
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866913527366680576 |
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| author | Barritault, Rémi Jahel, Colin Joseph, Matthieu |
| author_facet | Barritault, Rémi Jahel, Colin Joseph, Matthieu |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2410_01725 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Unitary Representations of the Isometry Groups of Urysohn Spaces Barritault, Rémi Jahel, Colin Joseph, Matthieu Group Theory Dynamical Systems Logic Representation Theory 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. |
| title | Unitary Representations of the Isometry Groups of Urysohn Spaces |
| topic | Group Theory Dynamical Systems Logic Representation Theory |
| url | https://arxiv.org/abs/2410.01725 |