Automorphism groups of prime models, and invariant measures

TL;DR

This paper explores automorphism groups of prime models, focusing on invariant measures and their uniqueness, especially in stable theories, and connects these concepts with Galois cohomology and differential Galois theory.

Contribution

It extends the notion of definable subsets of automorphism groups to atomic, strongly omega-homogeneous models and analyzes invariant measures in this context.

Findings

Invariant measures exist and are unique in stable theories.

The definability notions align with Galois cohomology.

Connections established with differential Galois theory.

Abstract

We adapt the notion of a (relatively) definable subset of Aut(M) when M is a saturated model to the case Aut(M/A) when M is atomic and strongly omega-homogeneous over A. We discuss the existence and uniqueness of invariant measures on the Boolean algebra of definable subsets of Aut(M/A). For example when Th(M) is stable we have existence and uniqueness. We also discuss the compatibility of our definability notions with definable Galois cohomology and differential Galois theory.