External definability and groups in NIP theories

28 pages. Introduction was expanded and some minor mistakes were corrected. Journal of the London Mathematical Society, acceptedWe prove that many properties and invariants of definable groups in NIP theories, 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