Amenability, connected components, and definable actions

We study amenability of definable groups and topological groups, and prove various results, briefly described below. Among our main technical tools, of interest in its own right, is an elaboration on and strengthening of the Massicot-Wagner version (Massicot and Wagner in J Ec Polytech Math 2:55–63, 2015) of the stabilizer theorem (Hrushovski in J Am Math Soc 25:189–243, 2012), and also some results about measures and measure-like functions (which we call means and pre-means). 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 Krupiński and Pillay (Adv Math 345:1253–1299, 2019). In other words, the conclusion says that certain connected components of G coincide: $$G^{00}_{{{\,\mathrm{{top}}\,}}} = G^{000}_{{{\,\mathrm{{top}}\,}}}$$ G top 00 = G top 000 . We also prove wide generalizations of this result, implying in particular its extension to a “definable-topological” context, confirming the main conjectures from Krupiński and Pillay (2019). We also introduce $$\bigvee $$ ⋁ -definable group topologies on a given $$\emptyset $$ ∅ -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 $${{\,\mathrm{{cl}}\,}}(G^{00}_M) = {{\,\mathrm{{cl}}\,}}(G^{000}_M)$$ cl ( G M 00 ) = cl ( G M 000 ) for any model M. Secondly, we study the relationship between (separate) definability of an action of a definable group on a compact space [in the sense of Gismatullin et al. (Ann Pure Appl Log 165:552–562, 2014)], weakly almost periodic (wap) actions of G [in the sense of Ellis and Nerurkar (Trans Am Math Soc 313:103–119, 1989)], and stability. We conclude that any group G definable in a sufficiently saturated structure is “weakly definably amenable” in the sense of Krupiński and Pillay (2019), namely any definable action of G on a compact space supports a G-invariant probability measure. This gives negative solutions to some questions and conjectures raised in Krupiński (J Symb Log 82:1080–1105, 2017) and Krupiński and Pillay (2019). Stability in continuous logic will play a role in some proofs in this part of the paper. Thirdly, we give an example of a $$\emptyset $$ ∅ -definable approximate subgroup X in a saturated extension of the group $${{\mathbb {F}}}_2 \times {{\mathbb {Z}}}$$ F 2 × Z in a suitable language (where $${{\mathbb {F}}}_2$$ F 2 is the free group in 2-generators) for which the $$\bigvee $$ ⋁ -definable group $$H:=\langle X \rangle $$ H : = ⟨ X ⟩ 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 (and in consequence also Lie) “model” exists for each approximate subgroup does not work in general (they proved in (Massicot and Wagner 2015) that it works for definably amenable approximate subgroups).

DEFINABLE TOPOLOGICAL DYNAMICS

For a group G definable in a first order structure M we develop basic topological dynamics in the category of definable G-flows. In particular, we give a description of the universal definable G-ambit and of the semigroup operation on it. We find a natural epimorphism from the Ellis group of this flow to the definable Bohr compactification of G, that is to the quotient ${G^{\rm{*}}}/G_M^{{\rm{*}}00}$ (where G* is the interpretation of G in a monster model). More generally, we obtain these results locally, i.e., in the category of Δ-definable G-flows for any fixed set Δ of formulas of an appropriate form. In particular, we define local connected components $G_{{\rm{\Delta }},M}^{{\rm{*}}00}$ and $G_{{\rm{\Delta }},M}^{{\rm{*}}000}$, and show that ${G^{\rm{*}}}/G_{{\rm{\Delta }},M}^{{\rm{*}}00}$ is the Δ-definable Bohr compactification of G. We also note that some deeper arguments from [14] can be adapted to our context, showing for example that our epimorphism from the Ellis group to the Δ-definable Bohr compactification factors naturally yielding a continuous epimorphism from the Δ-definable generalized Bohr compactification to the Δ-definable Bohr compactification of G. Finally, we propose to view certain topological-dynamic and model-theoretic invariants as Polish structures which leads to some observations and questions.

Generalised Bohr compactification and model-theoretic connected components

For a group G first order definable in a structure M, we continue the study of the “definable topological dynamics” of G (from [9] for example). The special case when all subsets of G are definable in the given structure M is simply the usual topological dynamics of the discrete group G; in particular, in this case, the words “externally definable” and “definable” can be removed in the results described below.Here we consider the mutual interactions of three notions or objects: a certain model-theoretic invariant G*/(G*)000M of G, which appears to be “new” in the classical discrete case and of which we give a direct description in the paper; the [externally definable] generalised Bohr compactification of G; [externally definable] strong amenability. Among other things, we essentially prove: (i) the “new” invariant G*/(G*)000M lies in between the externally definable generalised Bohr compactification and the definable Bohr compactification, and these all coincide when G is definably strongly amenable and all types in SG(M) are definable; (ii) the kernel of the surjective homomorphism from G*/(G*)000M to the definable Bohr compactification has naturally the structure of the quotient of a compact (Hausdorff) group by a dense normal subgroup; (iii) when Th(M) is NIP, then G is [externally] definably amenable iff it is externally definably strongly amenable.In the situation when all types in SG(M) are definable, one can just work with the definable (instead of externally definable) objects in the above results.

Bohr density of simple linear group orbits

We show that any non-zero orbit under a non-compact, simple, irreducible linear group is dense in the Bohr compactification of the ambient space.