The Borel cardinality of Lascar strong types
Abstract: Abstract. We show that if the restriction of the Lascar equivalence relation to a KP-strong type is non-trivial, then it is non-smooth (when viewed as a Borel equivalence relation on an appropriate space of types).
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Cited by 14 publications
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We generalise the main theorems from the paper "The Borel cardinality of Lascar strong types" by I. Kaplan, B. Miller and P. Simon to a wider class of bounded invariant equivalence relations. We apply them to describe relationships between fundamental properties of bounded invariant equivalence relations (such as smoothness or type-definability) which also requires finding a series of counterexamples. Finally, we apply the generalisation mentioned above to prove a conjecture from a paper by the first author and J. Gismatullin, showing that the key technical assumption of the main theorem (concerning connected components in definable group extensions) from that paper is not only sufficient but also necessary to obtain the conclusion. §1. Introduction.1.1. Preface. This paper will concern the Borel cardinalities of bounded, invariant equivalence relations, as well as some weak analogues in an uncountable case. More precisely, we are concerned with the connection between typedefinability and smoothness of these relations-type-definable equivalence relations are always smooth (cf. Fact 2.14), while the converse is not true in general. We also apply this to the study of connected components in definable group extensions.The general motivation for the use of Borel cardinality in the context of bounded invariant equivalence relations is a better understanding of "spaces" of strong types (i.e., "spaces" of classes of such relations). For a bounded type-definable equivalence relation, its set of classes, equipped with the so-called logic topology, forms a compact Hausdorff topological space. However, for relations which are only invariant, but not type-definable, the logic topology is not necessarily Hausdorff, so it is not so useful. The question arises how to measure the complexity of the spaces of classes of such relations. One of the ideas is to investigate their Borel cardinalities, which was formalised in [4], wherein the authors asked whether the Lascar strong type must be nonsmooth if it is not equal to the Kim-Pillay strong type. This question was answered in the positive in [4], and in this paper, we generalise its methods to a more general class of invariant equivalence relations, and we find an important application in the context of definable group extensions.
We generalise the main theorems from the paper "The Borel cardinality of Lascar strong types" by I. Kaplan, B. Miller and P. Simon to a wider class of bounded invariant equivalence relations. We apply them to describe relationships between fundamental properties of bounded invariant equivalence relations (such as smoothness or type-definability) which also requires finding a series of counterexamples. Finally, we apply the generalisation mentioned above to prove a conjecture from a paper by the first author and J. Gismatullin, showing that the key technical assumption of the main theorem (concerning connected components in definable group extensions) from that paper is not only sufficient but also necessary to obtain the conclusion. §1. Introduction.1.1. Preface. This paper will concern the Borel cardinalities of bounded, invariant equivalence relations, as well as some weak analogues in an uncountable case. More precisely, we are concerned with the connection between typedefinability and smoothness of these relations-type-definable equivalence relations are always smooth (cf. Fact 2.14), while the converse is not true in general. We also apply this to the study of connected components in definable group extensions.The general motivation for the use of Borel cardinality in the context of bounded invariant equivalence relations is a better understanding of "spaces" of strong types (i.e., "spaces" of classes of such relations). For a bounded type-definable equivalence relation, its set of classes, equipped with the so-called logic topology, forms a compact Hausdorff topological space. However, for relations which are only invariant, but not type-definable, the logic topology is not necessarily Hausdorff, so it is not so useful. The question arises how to measure the complexity of the spaces of classes of such relations. One of the ideas is to investigate their Borel cardinalities, which was formalised in [4], wherein the authors asked whether the Lascar strong type must be nonsmooth if it is not equal to the Kim-Pillay strong type. This question was answered in the positive in [4], and in this paper, we generalise its methods to a more general class of invariant equivalence relations, and we find an important application in the context of definable group extensions.
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).
We develop topological dynamics for the group of automorphisms of a monster model of any given theory. In particular, we find strong relationships between objects from topological dynamics (such as the generalized Bohr compactification introduced by Glasner) and various Galois groups of the theory in question, obtaining essentially new information about them, e.g. we present the closure of the identity in the Lascar Galois group of the theory as the quotient of a compact, Hausdorff group by a dense subgroup.We apply this to describe the complexity of bounded, invariant equivalence relations, obtaining comprehensive results, subsuming and extending the existing results and answering some open questions from earlier papers. We show that, in a countable theory, any such relation restricted to the set of realizations of a complete type over ∅ is type-definable if and only if it is smooth. Then we show a counterpart of this result for theories in an arbitrary (not necessarily countable) language, obtaining also new information involving relative definability of the relation in question. As a final conclusion we get the following trichotomy. Let C be a monster model of a countable theory, p ∈ S(∅), and E be a bounded, (invariant) Borel (or, more generally, analytic) equivalence relation on p(C). Then, exactly one of the following holds:(1) E is relatively definable (on p(C)), smooth, and has finitely many classes, (2) E is not relatively definable, but it is type-definable, smooth, and has 2 ℵ0 classes, (3) E is not type definable and not smooth, and has 2 ℵ0 classes. All the results which we obtain for bounded, invariant equivalence relations carry over to the case of bounded index, invariant subgroups of definable groups. IntroductionGenerally speaking, this paper concerns applications of topological dynamics and the "descriptive set theory" of compact topological groups to model theory.2010 Mathematics Subject Classification. 03C45, 54H20, 03E15, 54H11.
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