Definably amenable NIP groups

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.

The introduction was reworked to make it more accessible to the general mathematical audience; the argument in Proposition 3.15 was clarified; discussion of the unique ergodicity was moved to Section 3.4, Section 4 now has no subsections; minor presentation improvements and clarifications were made throughout the article; accepted to the Journal of the American Mathematical Society