External definability and groups in NIP theories

Artem Chernikov (1, 2) , Anand Pillay (3) , Pierre Simon (4)

1 IMJ - Institut de Mathématiques de Jussieu
2 ELM - Équipe de Logique Mathématique
3 UND - University of Notre Dame [Indiana]
4 AGL - Algèbre, géométrie, logique

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.

Mots clés: connected components; topological dynamics; Ellis group conjecture; model theory; NIP; amenability; externally definable sets; o-minimality; definable groups

Domaines: Logique [math.LO]; Systèmes dynamiques [math.DS]; Théorie des groupes [math.GR]

Artem Chernikov, Anand Pillay, Pierre Simon. External definability and groups in NIP theories. 2014. ⟨hal-00952501⟩