Approximate equivalence relations

Journal article

Generalizing results for approximate subgroups, we study approximate equivalence relations up to commensurability, in the presence of a definable measure.
As a basic framework, we give a presentation of probability logic based on continuous logic. Hoover’s normal form is valid here; if one begins with a discrete logic structure, it reduces arbitrary formulas of probability logic to correlations between quantifier-free formulas. We completely classify binary correlations in terms of the Kim–Pillay space, leading to strong results on the interpretative power of pure probability logic over a binary language. Assuming higher amalgamation of independent types, we prove a higher stationarity statement for such correlations.
We also give a short model-theoretic proof of a categoricity theorem for continuous logic structures with a measure of full support, generalizing theorems of Gromov–Vershik and Keisler, and often providing a canonical model for a complete pure probability logic theory. These results also apply to local probability logic, providing in particular a canonical model for a local pure probability logic theory with a unique 1-type and geodesic metric.
For sequences of approximate equivalence relations with an “approximately unique” probability logic 1-type, we obtain a structure theorem generalizing the “Lie model” theorem for approximate subgroups (Theorem 5.5). The models here are Riemannian homogeneous spaces, fibered over a locally finite graph.
Specializing to definable graphs over finite fields, we show that after a finite partition, a definable binary relation converges in finitely many self-compositions to an equivalence relation of geometric origin. This generalizes the main lemma for strong approximation of groups.
For NIP theories, pursuing a question of Pillay’s, we prove an archimedean finite-dimensionality statement for the automorphism groups of definable measures, acting on a given type of definable sets. This can be seen as an archimedean analogue of results of Macpherson and Tent on NIP profinite groups.

Authors: Hrushovski, E

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Mathematical Sciences Publishers
• Journal: Model Theory More from this journal
• Volume: 3
• Issue: 2
• Pages: 317-416
• Publication date: 2024-07-19
• Acceptance date: 2024-06-25
• DOI: 10.2140/mt.2024.3.317
• EISSN: 2832-9058
• ISSN: 2832-904X
• Language: English
• Keywords: NIP; probability logic; approximate equivalence relations