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.

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 obtain several fundamental results on finite index ideals and additive subgroups of rings as well as on model-theoretic connected components of rings, which concern generating in finitely many steps inside additive groups of rings.Let R be any ring equipped with an arbitrary additional first order structure, and A a set of parameters. We show that whenever H is an A-definable, finite index subgroup of pR, `q, then H `RH contains an A-definable, two-sided ideal of finite index. As a corollary, we obtain a positive answer to Question 3.9 ofA " R00 A , which also implies that R00A " R000 A , where R ą R is a sufficiently saturated elementary extension of R, p R, `q00A [resp. R00 A ] denotes the smallest A-type-definable, bounded index additive subgroup [resp. ideal] of R, and R000A is the smallest invariant over A, bounded index ideal of R. If R is of finite characteristic (not necessarily unital), we get a sharper result: p R, `q00A `R ¨p R, `q00 A " R00 A . We obtain a similar result (but with more steps required) for finitely generated (not necessarily unital) rings. We obtain analogous results for topological rings. The above result for unital rings implies that the simplified descriptions of the definable (and so also classical) Bohr compactifications of triangular groups over unital rings obtained in Corollary 3.5 of [GJK20] are valid for all unital rings.We also analyze many concrete examples, where we compute the number of steps needed to generate a group by p R Y t1uq ¨p R, `q00A and study related aspects, showing "optimality" of some of our main results and yielding answers to some natural questions.

We extend some recent results about bounded invariant equivalence relations and invariant subgroups of definable groups: we show that type-definability and smoothness are equivalent conditions in a wider class of relations than heretofore considered, which includes all the cases for which the equivalence was proved before.As a by-product, we show some analogous results in purely topological context (without direct use of model theory).

We present the (Lascar) Galois group of any countable theory as a quotient of a compact Polish group by an F σ normal subgroup: in general, as a topological group, and under NIP, also in terms of Borel cardinality. This allows us to obtain similar results for arbitrary strong types defined on a single complete type over ∅. As an easy conclusion of our main theorem, we get the main result of [KPR15] which says that for any strong type defined on a single complete type over ∅, smoothness is equivalent to type-definability.We also explain how similar results are obtained in the case of bounded quotients of type-definable groups. This gives us a generalization of a former result from [KPR15] about bounded quotients of type-definable subgroups of definable groups.Date: November 9, 2018. 2010 Mathematics Subject Classification. 03C45, 54H20, 22C05, 03E15, 54H11.