Definably amenable NIP groups
Artem Chernikov, Pierre Simon

TL;DR
This paper explores definably amenable NIP groups, establishing a unified theory of generics, analyzing invariant measures, and proving key conjectures linking model-theoretic and topological properties.
Contribution
It develops a comprehensive theory of generics in NIP groups, proves the Petrykowski conjecture, and connects the Ellis group conjecture with model-theoretic connected components.
Findings
Unified definitions of generics in NIP groups
Characterization of regular ergodic measures
Proof of the Petrykowski conjecture and Ellis group conjecture
Abstract
We study definably amenable NIP groups. We develop a theory of generics, showing that various definitions considered previously coincide, and study invariant measures. Applications include: characterization of regular ergodic measures, a proof of the conjecture of Petrykowski connecting existence of bounded orbits with definable amenability in the NIP case, and the Ellis group conjecture of Newelski and Pillay connecting the model-theoretic connected component of an NIP group with the ideal subgroup of its Ellis enveloping semigroup.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsAdvanced Topology and Set Theory · Advanced Operator Algebra Research · Computability, Logic, AI Algorithms
