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Metric model theory, Polish groups & diversities

Abstract : We study metric model theory and Polish groups as automorphism groups of separable metric structures. We expand upon the infinitary continuous logic treated in [12, 11, 23, 27] and give a new proof of the Omitting Types Theorem of infinitary continuous logic. We also find a new way of calculating the type distance of infinitary continuous logic. Furthermore, we show an infinitary version of the Ryll-Nardzewski Theorem. We also study the Roelcke completion of a Polish group and give amodel theoretical characterisation of locally Roelcke precompact Polish groups. We do this by showing that the Roelcke completion of a Polish group can be considered as a certain set of types in metric model theory. Furthermore, we develop the model theory of the Urysohn metric space U. We show that its theory TU eliminates quantifiers, that U is a prime model and that any separable model of TU is a disjoint union of isomorphic copies of U. Moreover, we show that the isometry group of U is locally Roelcke precompact by applying our result above. This was already known, but our proof is new. Finally, we study the Urysohn diversity U. Diversities are a natural generalisation of metric spaces, where positive values are assigned not just to pairs, but to all finite subsets. We develop the model theory ofU and show, among other things, that its automorphism group Aut(U) is locally Roelcke precompact, again by applying our result above. We also show that Aut(U) is a universal Polish group and that it has a dense conjugacy class. Lastly, we study the automorphism group of the rational Urysohn diversity UQ and show that Aut(UQ) has ample generics - a property with many strong implications.
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https://tel.archives-ouvertes.fr/tel-03216638
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Submitted on : Tuesday, May 4, 2021 - 11:18:37 AM
Last modification on : Wednesday, May 5, 2021 - 3:38:44 AM

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  • HAL Id : tel-03216638, version 1

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Andreas Hallbäck. Metric model theory, Polish groups & diversities. Logic [math.LO]. Université de Paris, 2020. English. ⟨NNT : 2020UNIP7078⟩. ⟨tel-03216638⟩

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