On first order amenability

Journal article

We introduce the notion of first order 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 in the same variables. Amenability of T follows from amenability of the (topological) group Aut(M) for all sufficiently large ℵ0-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 Selecta Math. (N.S.) 28, 16 (2022), 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 ω-categorical theories in Adv. Math. 345, 1253–1299 (2019). 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. Parallel (but easier) results hold for the notion of extreme amenability.

Authors: Hrushovski, E; Krupiński, K; Pillay, A

• Alternative title: On first order amenability
• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Springer
• Journal: Selecta Mathematica (New Series)
• Volume: 32
• Issue: 2
• Article number: 23
• Publication date: 2026-02-25
• Acceptance date: 2025-11-13
• DOI: 10.1007/s00029-026-01125-1
• EISSN: 1420-9020
• ISSN: 1022-1824
• Language: English
• Keywords: 43A07; 03C45; Amenable theory; G-compactness