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Preprint Number 1551

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1551. Ehud Hrushovski, Krzysztof Krupinski, and Anand Pillay
Amenability and definability

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We study amenability of definable groups and topological groups, as well as a new notion of first order amenability of a theory T.
Among our main tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version of the stabilizer theorem, and also some results about measures and measure-like functions.
As an application, we show that if G is an amenable topological group, then the Bohr compactification of G coincides with a certain “weak Bohr compactification” introduced in [27]. In other words, the conclusion says that certain connected components of G coincide. We also prove wide generalizations of this result, implying in particular its extension to a “definable-topological” context, confirming the main conjectures from [27].
We study the relationship between definability of an action of a definable group on a compact space, weakly almost periodic actions of G, and stability. We conclude that for every group G definable in a sufficiently saturated structure, any definable action of G on a compact space supports a G-invariant probability measure, which answers negatively some questions and conjectures raised in [23] and [27].
We introduce the notion of first order [extreme] amenability, as a property of a first order theory T: every complete type over ∅ extends to an invariant global Keisler measure [type]. [Extreme] amenability of T will follow from [extreme] amenability of the (topological) group Aut(M) for all sufficiently large ℵ_0-homogeneous countable models M of T (assuming T to be countable), but is radically less restrictive. A further adaptation of the technical tools mentioned above is used to prove that if T is amenable, then T is G-compact. This extends and essentially generalizes results in [27].

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

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Full text arXiv 1901.02859: pdf, ps.

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