Amenability, definable groups, and automorphism groups

Abstract

We prove several theorems relating amenability of groups in various categories (discrete, definable, topological, automor phism group) to model-theoretic invariants (quotients by connected components, Lascar Galois group, G-compactness, ...). Among the main tools, which is possibly of independent interest, is the adaptation and generalization of Theorem 12 of [27] to other settings including amenable topological groups whose topology is generated by a family of subgroups.On the side of definable groups, we prove that if G is definable in a model M and G is definably amenable, then the connected components G⁎M00 and G⁎M000 coincide, answering positively a question from [22]. We also prove some natural counterparts for topological groups, using our generalizations of [27]. By finding a dictionary relating quotients by connected components and Galois groups of ω-categorical theories, we conclude that if M is countable and ω-categorical, and Aut(M) is amenable as a topological group, then T:=Th(M) is G-compact, i.e. the Lascar Galois group GalL(T) is compact, Hausdorff (equivalently, the natural epimorphism from GalL(T) to GalKP(T) is an isomorphism).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.