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Preprint Number 1137
1137. Krzysztof Krupinski and Anand Pillay Amenability, definable groups, and automorphism groups E-mail: , 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. Mathematics Subject Classification: 03C45, 54H20, 54H11, 43A07 Keywords and phrases: Amenability, model-theoretic connected components, G-compactness |

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