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

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].