External definability and groups in NIP theories

Abstract

Abstract. We 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.

Some model theory and topological dynamics of $p$-adic algebraic groups

“…This account of the generalized Bohr compactification was discussed in [16] and studied further there in the model-theoretic context. 4 The action of SL(2, Q p ) on the type space of the projective line over Q p…”

Generalised Bohr compactification and model-theoretic connected components

“…It would be natural to make a definability of types assumption in (i), but the question makes sense without it, and a positive answer was given in [12] when T has NIP. But the NIP case can also be deduced from Corollary 0.4 and Theorem 0.6 (proved in the next section), bearing in mind the invariance of the relevant connected components when passing to M ext which was proved in [2] (see also the beginning of the next section for a short discussion on M ext ).…”

“…We will deduce below Proposition 6.3 from Theorem 2.7 of [2] which proves definability of measures assuming definability of types, in the NIP environment.…”

“…Therefore, if G is a group definable in a model M of a NIP theory, then externally definable strong amenability of G is equivalent to definable strong amenability of G treated as a group definable in M ext . On the other hand, by [2,Theorem 3.17], we know that in a NIP theory definable amenability is preserved under passing to M ext and coincides with externally definable amenability. That is why, replacing M by M ext , we can assume that all types in S G (M) are definable and we can rewrite Theorem 0.6 in the following equivalent form.…”

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

“…In Section 5.8 of [3], the "Ellis group conjecture" is proved in the ominimal situation and the analysis given there will be useful for the current paper. So we revisit Lemma 5.11 of [3].…”