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Automorphism groups of randomized structures. (English) Zbl 1404.22051

Summary: We study automorphism groups of randomizations of separable structures, with focus on the \(\aleph_0\)-categorical case. We give a description of the automorphism group of the Borel randomization in terms of the group of the original structure. In the \(\aleph_0\)-categorical context, this provides a new source of Roelcke precompact Polish groups, and we describe the associated Roelcke compactifications. This allows us also to recover and generalize preservation results of stable and NIP formulas previously established in the literature, via a Banach-theoretic translation. Finally, we study and classify the separable models of the theory of beautiful pairs of randomizations, showing in particular that this theory is never \(\aleph_0\)-categorical (except in basic cases).

MSC:

22F10 Measurable group actions
22F50 Groups as automorphisms of other structures
03C30 Other model constructions
03C45 Classification theory, stability, and related concepts in model theory
03C60 Model-theoretic algebra
03E15 Descriptive set theory
22A05 Structure of general topological groups
22A15 Structure of topological semigroups
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