External definability and groups in NIP theories

Abstract

We prove that many properties and invariants of definable groups in NIP theories (i.e. theories without the independence property), such as definable amenability, G / G 00 , etc., are preserved when passing to the theory of the Shelah expansion by externally definable sets, M ext , of a model M. In the light of these results, we continue the study of the ‘definable topological dynamics’ of groups in NIP theories. In particular, we prove the Ellis group conjecture relating the Ellis group to G / G 00 in some new cases, including definably amenable groups in o-minimal structures.