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Structurable Equivalence Relations

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Structurable Equivalence Relations Structurable equivalence relations Ruiyuan Chen∗ and Alexander S. Kechrisy Abstract For a class of countable relational structures, a countable Borel equivalence relation E is said to be -structurableK if there is a Borel way to put a structure in on each E-equivalence class. We studyK in this paper the global structure of the classes of -structurableK equivalence relations for various . We show that -structurability interactsK well with several kinds of Borel homomorphismsK and reductions commonlyK used in the classification of countable Borel equivalence relations. We consider the poset of classes of -structurable equivalence relations for various , under inclusion, and show that it is a distributiveK lattice; this implies that the Borel reducibilityK preordering among countable Borel equivalence relations contains a large sublattice. Finally, we consider the effect on -structurability of various model-theoretic properties of . In particular, we characterize the Ksuch that every -structurable equivalence relation is smooth,K answering a question of Marks.K K Contents 1 Introduction 3 2 Preliminaries 7 2.1 Theories and structures..................................7 2.2 Countable Borel equivalence relations..........................8 2.3 Homomorphisms......................................9 2.4 Basic operations...................................... 10 2.5 Special equivalence relations................................ 11 2.6 Fiber products....................................... 12 2.7 Some categorical remarks................................. 12 3 Structures on equivalence relations 13 3.1 Examples of elementary classes.............................. 14 3.2 Classwise pullback structures............................... 15 4 Basic universal constructions 15 4.1 The universal σ-structured equivalence relation..................... 15 4.2 The \Scott sentence" of an equivalence relation..................... 18 4.3 Structurability and class-bijective homomorphisms................... 20 4.4 Class-bijective products.................................. 21 4.5 Categorical limits in ( ; cb)............................... 24 E !B ∗Research partially supported by NSERC PGS D yResearch partially supported by NSF Grants DMS-0968710 and DMS-1464475 1 5 Structurability and reducibility 25 5.1 Embeddings and class-injective homomorphisms..................... 27 5.2 Reductions, smooth homomorphisms, and class-surjectivity.............. 28 5.3 Elementary reducibility classes.............................. 30 5.4 Reductions and compressibility.............................. 31 6 The poset of elementary classes 33 6.1 Projections and closures.................................. 34 6.2 The lattice structure.................................... 35 6.3 Closure under independent joins............................. 38 6.4 Embedding the poset of Borel sets............................ 40 6.5 A global picture....................................... 40 7 Free actions of a group 41 8 Structurability and model theory 44 8.1 Smoothness of E1σ .................................... 44 8.2 Universality of E1σ .................................... 49 8.3 Classes axiomatizable by a Scott sentence........................ 53 9 Some open problems 54 9.1 General questions...................................... 54 9.2 Order-theoretic questions................................. 54 9.3 Model-theoretic questions................................. 55 A Appendix: Fiber spaces 57 A.1 Fiber spaces......................................... 57 A.2 Fiber spaces and cocycles................................. 59 A.3 Equivalence relations as fiber spaces........................... 60 A.4 Countable Borel quotient spaces............................. 61 A.5 Factorizations of fiber space homomorphisms...................... 62 A.6 Structures on fiber spaces................................. 63 B Appendix: The category of theories 65 B.1 Interpretations....................................... 65 B.2 Products and coproducts of theories........................... 66 B.3 Interpretations and structurability............................ 67 B.4 !1!-Boolean algebras................................... 69 C Appendix: Representation of !1-distributive lattices 72 References 73 Index 76 2 1 Introduction (A) A countable Borel equivalence relation on a standard Borel space X is a Borel equivalence 2 relation E X with the property that every equivalence class [x]E, x X, is countable. We ⊆ 2 denote by the class of countable Borel equivalence relations. Over the last 25 years there has E been an extensive study of countable Borel equivalence relations and their connection with group actions and ergodic theory. An important aspect of this work is an understanding of the kind of countable (first-order) structures that can be assigned in a uniform Borel way to each class of a given equivalence relation. This is made precise in the following definitions; see [JKL], Section 2.5. Let L = Ri i I be a countable relational language, where Ri has arity ni, and a class f j 2 g K of countable structures in L closed under isomorphism. Let E be a countable Borel equivalence A relation on a standard Borel space X. An L-structure on E is a Borel structure A = (X; Ri )i2I n of L with universe X (i.e., each RA X i is Borel) such that for i I and x1; : : : ; xn X, i ⊆ 2 i 2 RA(x1; : : : ; xn ) = x1 E x2 E E xn . Then each E-class C is the universe of the countable i i ) ··· i L-structure A C. If for all such C; A C , we say that A is a -structure on E. Finally if E j j 2 K K admits a -structure, we say that E is -structurable. K K Many important classes of countable Borel equivalence relations can be described as the - K structurable relations for appropriate . For example, the hyperfinite equivalence relations are K exactly the -structurable relations, where is the class of linear orderings embeddable in Z. The K K treeable equivalence relations are the -structurable relations, where is the class of countable K K trees (connected acyclic graphs). The equivalence relations generated by a free Borel action of a countable group Γ are the -structurable relations, where is the class of structures corresponding K K to free transitive Γ-actions. The equivalence relations admitting no invariant probability Borel measure are the -structurable relations, where L = R; S , R unary and S binary, and consists K f g K of all countably infinite structures A = (A; RA;SA), with RA an infinite, co-infinite subset of A and SA the graph of a bijection between A and RA. For L = Ri i I as before and countable set X, we denote by ModX (L) the standard Borel f j 2 g space of countable L-structures with universe X. Clearly every countable L-structure is isomorphic to some A ModX (L), for X 1; 2;:::; N . Given a class of countable L-structures, closed 2 2 f g K under isomorphism, we say that is Borel if ModX (L) is Borel in ModX (L), for each countable K K\ set X. We are interested in Borel classes in this paper. For any L! !-sentence σ, the class of K 1 countable models of σ is Borel. By a classical theorem of Lopez-Escobar [LE], every Borel class K of L-structures is of this form, for some L!1!-sentence σ. We will also refer to such σ as a theory. Adopting this model-theoretic point of view, given a theory σ and a countable Borel equivalence relation E, we put E = σ j if E is -structurable, where is the class of countable models of σ, and we say that E is σ- K K structurable if E = σ. We denote by σ the class of σ-structurable countable Borel equiva- j E ⊆ E lence relations. Finally we say that a class of countable Borel equivalence relations is elementary C if it is of the form σ, for some σ (which axiomatizes ). In some sense the main goal of this E C paper is to study the global structure of elementary classes. First we characterize which classes of countable Borel equivalence relations are elementary. We need to review some standard concepts from the theory of Borel equivalence relations. Given equivalence relations E; F on standard Borel spaces X; Y , resp., a Borel homomorphism of E to F is a Borel map f : X Y with x E y = f(x) F f(y). We denote this by f : E B F . ! ) ! 3 If moreover f is such that all restrictions f [x]E :[x]E [f(x)]F are bijective, we say that f j ! is a class-bijective homomorphism, in symbols f : E cb F . If such f exists we also write !B E cb F . We similarly define the notion of class-injective homomorphism, in symbols ci. !B !B A Borel reduction of E to F is a Borel map f : X Y with x E y f(x) F f(y). We ! () denote this by f : E B F . If f is also injective, it is called a Borel embedding, in symbols ≤ f : E B F . If there is a Borel reduction of E to F we write E B F and if there is a Borel v ≤ embedding we write E B F . An invariant Borel embedding is a Borel embedding f as above v i i with f(X) F -invariant. We use the notation f : E B F and E B F for these notions. By the i v i v usual Schroeder-Bernstein argument, E F and F E E =B F , where =B is Borel vB vB () ∼ ∼ isomorphism. Kechris-Solecki-Todorcevic [KST, 7.1] proved a universality result for theories of graphs, which was then extended to arbitrary theories by Miller; see Corollary 4.4. Theorem 1.1 (Kechris-Solecki-Todorcevic, Miller). For every theory σ, there is an invariantly i universal σ-structurable countable Borel equivalence relation E1σ, i.e., E1σ = σ, and F E1σ j vB for any other F = σ. j Clearly E1σ is uniquely determined up to Borel isomorphism. In fact in Theorem 4.1 we formulate a \relative" version of this result and its proof that allows us to capture more information. Next we note that clearly every elementary class is closed downwards under class-bijective Borel homomorphisms. We now have the following characterization of elementary classes (see Corollary 4.12). Theorem 1.2. A class of countable Borel equivalence relations is elementary iff it is (downwards-)closed underC class-bijective ⊆ E Borel homomorphisms and contains an invariantly uni- versal element E .
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