Submission date: 17 April 2020

Abstract:

This paper contains the material in Section 4 of our preprint Amenability and definability (arXiv:1901.02859v1). Following the advice of editors and referees we have divided that preprint into two papers, the current paper being the second.

We introduce the notion of first order [extreme] amenability, as a property of a first order theory T: every complete type over ∅, in possibly infinitely many variables, extends to an automorphism-invariant global Keisler measure [type] in the same variables. [Extreme] amenability of T will follow from [extreme] amenability of the (topological) group Aut(M) for all sufficiently large ℵ_0-homogeneous countable models M of T (assuming T to be countable), but is radically less restrictive. First, we study basic properties of amenable theories, giving many equivalent conditions. Then, applying a version of the stabilizer theorem from [5], we prove that if T is amenable, then T is G-compact, namely Lascar strong types and Kim-Pillay strong types over ∅ coincide. This extends and essentially generalizes a similar result proved via different methods for \omega-categorical theories in [14]. In the special case when amenability is witnessed by ∅-definable global Keisler measures (which is for example the case for amenable ω-categorical theories), we also give a different proof, based on stability in continuous logic.

Mathematics Subject Classification: 03C45, 43A07

Keywords and phrases:

Full text arXiv 2004.08306: pdf, ps.