Definably amenable NIP groups

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.