External definability and groups in NIṔ theories

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^{\operatorname {ext} }, of a model M. In the light of these results, we continue the study of the 'definable topological dynamics' of groups in \operatorname {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. © 2014 London Mathematical Society.