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 we study invariant measures. As applications, we characterize ergodic measures, give a proof of the conjecture of Petrykowski connecting existence of bounded orbits with definable amenability in the NIP case, and prove 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.

On minimal flows, definably amenable groups, and o-minimality

“…Following the notation of [2] we call a type p as in the right hand side a (global) strongly f -generic, over M, type of G.…”

“…The recent paper [2] gives, among other things, a fairly comprehensive account of the relations betweeen different notions of genericity in the NIP definably amenable context. Although we will only be using "easy" directions of their observations, there is no harm in describing some of their results.…”

“…Call a complete type p (over some set of parameters) f -generic iff every formula in p is f -generic. The following appears in [2]. Fact 1.14.…”

“…Newelski's question had nothing to do with definable amenability in itself, but there is no harm in looking within the class of definably amenable groups for a counterexample. In fact Newelski's question is essentially restated in [2] (Question 3.35) in the special case of definably amenable groups in NIP theories.…”

“…This question was raised in an early draft of [2]. Pierre Simon recently told us that he had solved it in the general NIP context.…”