Automorphism groups of randomized structures

TL;DR

This paper investigates the automorphism groups of randomized structures, especially in the countably categorical case, providing new insights into their properties and connections to Roelcke precompact groups.

Contribution

It characterizes automorphism groups of randomized structures, introduces new Roelcke precompact groups, and generalizes preservation results for stable and NIP formulas.

Findings

Description of automorphism groups of Borel randomizations

Construction of new Roelcke precompact Polish groups

Generalization of stability and NIP preservation results

Abstract

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).