Amenability, connected components, and definable actions

DOI10.1007/S00029-021-00735-1arXiv1901.02859MaRDI QIDQ6312227

Author name not available (Why is that?)

Publication date: 9 January 2019

Abstract: We study amenability of definable and topological groups. Among our main technical tools is an elaboration on and strengthening of the Massicot-Wagner version of the stabilizer theorem, and some results around measures. 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 [24]. Formally,

G_{t o p o}^{00} = G_{t o p o}^{000}

. We also prove wide generalizations of this result, implying in particular its extension to a "definable-topological" context, confirming the main conjectures from [24]. We introduce

-definable group topologies on a given

e m p t y s e t

-definable group

G

(including group topologies induced by type-definable subgroups as well as uniformly definable group topologies), and prove that the existence of a mean on the lattice of closed, type-definable subsets of

G

implies (under some assumption) that

c l (G_{M}^{00}) = c l (G_{M}^{000})

for any model

M

. We study the relationship between definability of an action of a definable group on a compact space, weakly almost periodic actions, and stability. We conclude that for any group

G

definable in a sufficiently saturated structure, every definable action of

G

on a compact space supports a

G

-invariant probability measure. This gives negative solutions to some questions and conjectures from [22] and [24]. We give an example of a

e m p t y s e t

-definable approximate subgroup

X

in a saturated extension of the group

m a t h b b F_{2} i m e s m a t h b b Z

in a suitable language for which the

-definable group

H := l a n g l e X a n g l e

contains no type-definable subgroup of bounded index. This refutes a conjecture by Wagner and shows that the Massicot-Wagner approach to prove that a locally compact "model" exists for each approximate subgroup does not work in general.

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