Submission date: 22 December 2016

Abstract:

We prove several theorems relating amenability of groups in various categories (discrete, definable, topological, automorphism group) to model-theoretic invariants (quotients by connected components, Lascar Galois group, G-compactness, ...). For example, if M is a countable, ω-categorical structure and Aut(M) is amenable, as a topological group, then the Lascar Galois group Gal_L(T) of the theory T of M is compact, Hausdorff (also over any finite set of parameters), that is T is G-compact. An essentially special case is that if Aut(M) is extremely amenable, then Gal_L(T) is trivial, so, by a theorem of Lascar, the theory T can be recovered from its category Mod(T) of models. On the side of definable groups, we prove for example that if G is definable in a model M, and G is definably amenable, then the connected components (G^*)^{00}_{M} and (G^*)^{000}_{M} coincide, answering positively a question from an earlier paper of the authors.
We also take the opportunity to further develop the model-theoretic approach to topological dynamics, obtaining for example some new invariants for topological groups, as well as allowing a uniform approach to the theorems above and the various categories.

Mathematics Subject Classification: 03C45, 54H20, 54H11, 43A07

Keywords and phrases: Amenability, model-theoretic connected components, G-compactness

Full text arXiv 1612.07560: pdf, ps.