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i∃x ∈ [x0 ]G ∃∗ g ∈ G (g · x ∈ Nλ(u,v) ∩ Ns(i ) ).

Granting such a sequence, let x = i s(i). Then ∀u ∈ P ∃v∃i∀i > i (Ns(i ) ∩ Nλ(u,v) = ∅). It follows that ∀u ∈ P ∃v (x is a limit point of Nλ(u,v) ). Since Nλ(u,v) is closed, we have that ∀u ∈ P ∃v (x ∈ Nλ(u,v) ), and hence x ∈ [x0 ]G . To define the sequence we use induction and simultaneously define a sequence c(i) of Borel codes for closed sets. To begin with let s(0) = ∅ and Bc(0) = X. In general suppose s(i) and c(i) have been defined and suppose Γ(s(i), c(i)) is the winning strategy for Player II in the game G(s(i), c(i)) defined in the preceding paragraph. Let Player I play ui and apply Γ(s(i), c(i)) to obtain s(i + 1)  s(i) as a part of Player II’s response. Let v be the other part of Player II’s moves. Then let Bc(i+1) = Bc ∩Nλ(ui ,v) . To verify that this is as required, let u ∈ P . Let ui = u and v be played by Player II according to the construction. Then for all i > i, Bc(i ) ⊆ Nλ(u,v) . Since Γ(s(i), c(i)) is winning for Player II, we have that ∃x ∈ [x0 ]G ∃∗ g ∈ G (g ·x ∈ Nλ(u,v) ∩Ns(i ) ). This finishes the proof of the theorem.

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Corollary 5.2.4 Let G be a Polish group, X a Borel G-space, Y a standard Borel space, and F a Borel equivalence relation on Y . Suppose f : X → Y is a Borel reduction X X to F such that [f (X)]F = Y . Then EG ∼B F . from EG Proof. The above proof gives rise to a Borel function g : Y → X such that for any y ∈ [f (X)]F , letting x = g(y), then f (x)F y. Note that the definition of x does not depend on the choice of x0 . It is easy to see that g is a reduction X from F to EG . Exercise 5.2.1 Show that the Borel reduction constructed in Corollary 5.2.4 is also faithful. Exercise 5.2.2 Let E, F be Borel orbit equivalence relations on Polish spaces X, Y , respectively. Show that if E ≤B F then there is an invariant Borel set Z ⊆ Y such that E ∼B F  Z.

5.3

Perfect set theorems for equivalence relations

In this and the next sections we study Borel reductions involving the identity relations. Recall from Proposition 5.1.12 that reduction from the identity relation on 2ω is equivalent to the existence of perfectly many equivalence classes. In this section we give various sufficient conditions for there being perfectly many equivalence classes for an equivalence relation on a Polish space. Again category arguments are playing the vital role in this study. Theorem 5.3.1 (Mycielski) Let X be a Polish space and E a meager equivalence relation on X. Then id(2ω ) c E. Proof. First note that X is perfect. In fact, otherwise X must contain at least one isolated point x, and any equivalence relation on X must contain the open set {(x, x)}, and therefore is nonmeager.

Let d < 1 be a compatible complete metric on X. Let E ⊆ n Fn with Fn ⊆ X 2 closed and nowhere dense for all n ∈ ω. We may assume that id(X) ⊆ F0 and Fn ⊆ Fn+1 for all n ∈ ω. We define a sequence (Us )s∈2<ω of open sets in X by induction on lh(s) such that, for all s ∈ 2<ω , (i) diam(Us ) ≤ 2−lh(s) ; (ii) Us 0 , Us 1 ⊆ Us ; Us 0 ∩ Us 1 = ∅;

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(iii) for all s, t ∈ 2<ω with lh(s) = lh(t) = n+1 and s = t, (Us ×Ut )∩Fn = ∅. To begin let U∅ = X. Assume now that for all s with lh(s) = n the set Us has been defined. We define all Us 0 and Us 1 . For notational convenience we enumerate all Us for lh(s) = n as U1 , . . . , Ui , . . . , Uk with 1 ≤ i ≤ k = 2n . Since X is perfect there are disjoint open subsets Vi , Vk+i ⊆ Ui for all 1 ≤ i ≤ k with diam(Vi ), diam(Vk+i ) ≤ 2−n−1 , and by regularity we may assume that Vi , Vk+i ⊆ Ui and Vi ∩Vk+i = ∅. Thus we have obtained a collection {Vi }1≤i≤2k of disjoint open sets satisfying (i) and (ii). To guarantee (iii) we make the following observation. Since Fn is nowhere dense, for any disjoint nonempty open sets W1 and W2 , W1 × W2 ⊆ Fn . It follows that there are nonempty W1 ⊆ W1 and W2 ⊆ W2 with (W1 × W2 ) ∩ Fn = ∅. Applying this observation successively to all pairs (i, j) with 1 ≤ i, j ≤ 2k and i = j, we obtain open sets Vi ⊆ Vi such that, for all such pairs (i, j), (Vi × Vj ) ∩ Fn = ∅. Now condition  (iii) is satisfied if we let Us 0 = Vi and Us 1 = Vk+i where Ui = Us . ω For u ∈ 2 we now let f (u) to be the unique element of the set n Uun . Then f is a continuous function from 2ω into X. To finish the proof of the theorem we only need to verify that (f (u), f (v)) ∈ E for u = v. For this let u, v ∈ 2ω with u = v. Let n be the least integer with u(n) = v(n). Then for all m > n, u  m = v  m. By (iii), (Uum × Uvm ) ∩ Fm = ∅. Thus (f (u), f (v)) ∈ Fm for all m > n. Therefore (f (u), f (v)) ∈ E as required. Corollary 5.3.2 Let X be a Polish space and E an equivalence relation on X. If there is a nonempty open set U ⊆ X such that E is meager on U × U , then id(2ω ) c E. Corollary 5.3.3 Let X be a Polish space and E an equivalence relation on X. Suppose E has the Baire property. If every E-equivalence class is meager, then there are perfectly many E-equivalence classes. Proof. The condition can be stated as ∀x ∀∗ y (x, y) ∈ E. By Kuratowski– Ulam Theorem 3.2.1 it follows that E is meager. Another corollary of Theorem 5.3.1 concerning the Gandy–Harrington topology is going to play an important role in the proof of Silver’s theorem later in this section. Corollary 5.3.4 Let τ be the Gandy–Harrington topology on the Baire space ω ω and E an equivalence relation on ω ω . If there is a Σ11 set V ⊆ ω ω such that E is τ × τ -meager in V × V , then there are perfectly many E-equivalence classes. Proof. Let Y = Xlow be the Polish space of low elements with the Gandy– Harrington topology (Theorem 1.8.5). By the Gandy Basis Theorem 1.8.4

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U = Y ∩ V is a nonempty open subset of Y . Now E is meager on U × U , and thus by Corollary 5.3.2 there is a continuous reduction f : 2ω → Y witnessing id(2ω )  E. Since τ is finer than the usual topology f as a function from 2ω into ω ω is still continuous. Thus there are perfectly many E-equivalence classes. The following theorem, often referred to as the Silver dichotomy theorem, is the first of a series of dichotomy theorems undoubtedly occupying the center of the invariant descriptive set theory of equivalence relations. It is also important because of its proof presented here, due to Harrington, which is an elegant example of the use of effective descriptive set theoretic methods in establishing an important result of classical character. Theorem 5.3.5 (Silver) Let X be a standard Borel space and E a coanalytic equivalence relation on X. Then either there are countably many E-equivalence classes or there are perfectly many E-equivalence classes. Proof. Without loss of generality we may assume X = ω ω . By relativization we also assume that E is Π11 . Let τ be the Gandy–Harrington topology on ω ω . First we define V = {x ∈ X : there is no Δ11 set U such that x ∈ U ⊆ [x]E }. We claim that if V = ∅ then there are only countably many E-equivalence classes. This is because, if V = ∅, then every E-equivalence class contains some nonempty Δ11 set. Since there are only countably many Δ11 sets, it follows that there are only countably many E-equivalence classes. For the rest of the proof we assume V = ∅. We claim that V ∈ Σ11 . Indeed, x ∈ V ⇐⇒ ∀U ∈ Δ11 ( x ∈ U → ∃y ∈ U (x, y) ∈ E ). With the coding of Δ11 sets given by Theorem 1.7.4 we get that x ∈ V ⇐⇒ ∀n [(n ∈ D ∧ x ∈ Pn+ ) → ∃y (y ∈ Pn− ∧ (x, y) ∈ E)]. This shows that V ∈ Σ11 . To finish the proof we claim that E is τ × τ -meager in V × V . Then by the above corollary there are perfectly many E-equivalence classes. We establish the claim in several steps. As the first step, we check that for every x ∈ V there is no Σ11 set U such that x ∈ U ⊆ [x]E . To see this, assume toward a contradiction that x ∈ V and U ∈ Σ11 and x ∈ U ⊆ [x]E . Then note that [x]E is Π11 since y ∈ [x]E ⇐⇒ ∀z(z ∈ U → (y, z) ∈ E). Now by the separation property for Σ11 sets (see Theorem 1.7.1 and the remarks following it) there is W ∈ Δ11 such that U ⊆ W ⊆ [x]E , thus x ∈ V .

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From this it follows immediately that every nonempty Σ11 subset U of V meets more than one E-equivalence class. Next we note that every set involved has the Baire property in the Gandy– Harrington topology or its products. Specifically, the equivalence relation E as a subset of ω ω × ω ω has the Baire property in τ × τ , and each equivalence class [x]E has the Baire property in τ . This is because, by a theorem of Nikodym (see the remarks preceeding Proposition 2.3.1), the collection of all sets with the Baire property in any topological space is closed under the Suslin operation. Now both E and [x]E are Π11 , and therefore coanalytic, and therefore their complements are results of the Suslin operation applied to sequences of closed sets in the usual topology of ω ω . Since the Gandy–Harrington topology is coarser, they continue to be the results of Suslin operation, and thus have the Baire property in the sense of τ . This allows us to use the Kuratowski– Ulam Theorem 3.2.1. Now to show that E is τ × τ -meager in V × V , by the Kuratowski–Ulam theorem it suffices to show that for all x ∈ V , [x]E is τ -meager in V . Thus it suffices to show that for all x ∈ V , [x]E is not τ -comeager in any U ⊆ V where U ∈ Σ11 . Again toward a final contradiction assume that for some x ∈ V and U ⊆ V , U ∈ Σ11 , [x]E is τ -comeager in U . Now by Louveau’s lemma Exercise 1.8.3, if [x]E is τ -comeager in U ∈ Σ11 , then [x]E × [x]E is comeager in U × U in the Gandy–Harrington topology of ω ω × ω ω . Since (U × U ) ∩ (ω ω × ω ω − E) is nonempty and Σ11 , we have that ([x]E × [x]E ) ∩ (U × U ) ∩ (ω ω × ω ω − E) = ∅, which is a contradiction. Exercise 5.3.1 Let X be a Polish space and E an equivalence relation on X with the Baire property. Suppose every E-equivalence class is countable. Show that there are perfectly many E-equivalence classes. Exercise 5.3.2 Let X be a Polish space, U ⊆ X open, and E an equivalence relation on X with the Baire property. Suppose τ is a strong Choquet topology on X finer than the Polish topology. Show that if E is τ × τ -meager on U × U , then there are perfectly many E-equivalence classes. Exercise 5.3.3 Let E be a Π11 equivalence relation on ω ω . Show that the following are equivalent: (i) There are only countably many E-equivalence classes. (ii) For every x ∈ ω ω there exists a Δ11 set U with x ∈ U ⊆ [x]E . (iii) For every x ∈ ω ω there exists a Σ11 set U with x ∈ U ⊆ [x]E . (iv) Every E-equivalence class is Π11 .

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Exercise 5.3.4 Let E be a Π11 equivalence relation on ω ω with only countably many classes. Show that E is a union of countably many Δ11 relations. In particular E is Borel.

5.4

Smooth equivalence relations

Definition 5.4.1 Let E be an equivalence relation on a standard Borel space X. We call E smooth or concretely classifiable if E ≤B id(2ω ). It is clear from the definition that smooth equivalence relations are Borel. Since any two uncountable standard Borel spaces are Borel isomorphic, E is smooth iff E ≤B id(X) for any uncountable Polish space X. Among the countable Polish spaces we consider ω with the discrete topology and let id(ω) denote its identity relation. The following is an immediate corollary of the Silver dichotomy theorem. Theorem 5.4.2 Let E be a smooth equivalence relation. Then exactly one of the following holds: (I) E ∼B id(2ω ); (II) E ≤B id(ω). Proof. By the Silver dichotomy theorem either there are perfectly many E-equivalence classes or else there are only countably many E-equivalence classes. In the first case we have both id(2ω ) ≤B E and, by smoothness, E ≤B id(2ω ). Hence E ∼B id(2ω ). For the second case let C0 , C1 , . . . enumerate all the E-equivalence classes. Then the function θ defined by θ(x) = n iff x ∈ Cn gives a Borel reduction from E to id(ω). The converse of this theorem is trivial. Easily we obtain a complete classification of smooth equivalence relations in terms of Borel bireducibility (see Exercise 5.4.2). Definition 5.4.3 Let E be an equivalence relation on a space X and S a family of subsets of X. We say that E is generated by S if for any x, y ∈ X, xEy ⇐⇒ for any S ∈ S, x ∈ S iff y ∈ S. In this situation S is also called a generating family or a separating family for E. If X is a Borel space then a generating family S is called Borel if every element of S is Borel.

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Proposition 5.4.4 Let E be an equivalence relation on a standard Borel space X. Then E is smooth iff there is a countable Borel generating family for E. Proof. Let S = {S0 , S1 , . . . , Sn , . . . } be a countable Borel generating family for E. Define θ : X → 2ω by θ(x)(n) = 1 ⇐⇒ x ∈ Sn . Then θ is a Borel reduction from E to id(2ω ). Conversely, suppose θ : X → 2ω is a Borel reduction. Then define Sn = {z ∈ 2ω : θ(z)(n) = 1}. We have that S = {S0 , S1 , . . . , Sn , . . . } is a countable Borel generating family for E. Corollary 5.4.5 Let E be a smooth equivalence relation on a Polish space X. Then there is a finer Polish topology σ on X such that E is closed as a subset of (X 2 , σ × σ). Proof. Let S0 , S1 , . . . , Sn , . . . enumerate a countable Borel generating family for E. By Theorem 4.2.3 there is a finer Polish topology σ on X such that each of Sn is σ-clopen. Now xEy ⇐⇒ ∀n ( x ∈ Sn ⇐⇒ y ∈ Sn ), and thus E is closed in (X 2 , σ × σ). Next we give some sufficient conditions for smoothness. First we show that closed equivalence relations are smooth. To do this we establish the following useful general lemma which is reminiscent of the Luzin separation theorem for Σ11 sets. Lemma 5.4.6 Let E be a Σ11 equivalence relation on a standard Borel space X. Let A, B ⊆ X be disjoint Σ11 E-invariant sets. Then there is an E-invariant Borel set C with A ⊆ C and B ∩ C = ∅. Proof. Let A0 = A and C0 be a Borel set separating A0 and B by the Luzin separation theorem. That is, A0 ⊆ C0 and B ∩ C0 = ∅. Let A1 = [C0 ]E . Then A1 is Σ11 and A1 ∩ B = ∅. By the separation theorem again let C1 be a Borel set separating A1 and B. Continue this way, and we obtain A = A0 ⊆ C0 ⊆ A1 ⊆ C1 ⊆ A2 ⊆ . . . Cn ⊆ An ⊆ . . . 1 where Cn is Borel

and An is E-invariant, Σ1 and disjoint from B. Let now C = n Cn = n An . Then C is invariant Borel, A ⊆ C, and B ∩ C = ∅ as required.

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Proposition 5.4.7 Let E be a closed equivalence relation on a Polish space X. Then E is smooth.

Proof. Since E is closed we can write X 2 − E = n Un × Vn where Un , Vn are basic open sets. Let An = [Un ]E and Bn = [Vn ]E . Then we have that An and Bn are disjoint Σ11 E-invariant sets. By Lemma 5.4.6 there is E-invariant Borel Cn such that An ⊆ Cn and Bn ∩ Cn = ∅. We claim that {Cn : n ∈ ω} is a generating family for E, and thus E is smooth. To see this, let x, y ∈ X be such that x ∈ Cn ⇐⇒ y ∈ Cn for all n ∈ ω. Assume (x, y) ∈ E. Then for some n ∈ ω we have that x ∈ Un and y ∈ Vn . However, this implies that x ∈ An ⊆ Cn and y ∈ Bn ⊆ X − Cn , a contradiction. In the next chapter we will show that any Borel equivalence relation whose equivalence classes are Gδ is smooth. For now we turn to the following easy fact. Proposition 5.4.8 Let E be an equivalence relation on a standard Borel space. If E has a Borel selector then E is smooth. Proof.

The selector function is a reduction.

Note that not every closed equivalence relation has a Borel selector (Exercise 5.4.6), hence the converse of the above proposition is not true. Next we give a characterization for the existence of Borel selectors for smooth equivalence relations. Definition 5.4.9 Let X be a standard Borel space and E an equivalence relation on X. We call E idealistic if there is an assignment C → IC that associates with each E-equivalence class C a σ-ideal IC of subsets of C such that (i) C ∈ IC ; (ii) for each Borel set A ⊆ X 2 the set AI defined by x ∈ AI ⇐⇒ {y ∈ [x] : (x, y) ∈ A} ∈ I[x] is Borel. Clause (ii) in the above definition is often referred to as Borelness of the map C → IC . The following proposition helps motivate the definition. Proposition 5.4.10 X is idealistic. Let G be a Polish group and X a Borel G-space. Then EG

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Let x ∈ X and C = [x]G . Then define IC on C by S ∈ IC ⇐⇒ {g ∈ G : g · x ∈ S} is meager in G.

It is easy to check that IC is a σ-ideal and C ∈ IC . To see that C → IC is Borel, let A ⊆ X 2 be a Borel set. Then x ∈ AI ⇐⇒ {y ∈ [x] : (x, y) ∈ A} ∈ I[x] ⇐⇒ {g ∈ G : (x, g · x) ∈ A} is meager in G ⇐⇒ ∀∗ g ∈ G (x, g · x) ∈ A, hence AI is Borel. Theorem 5.4.11 (Kechris) Let E be an equivalence relation on a standard Borel space X. Then E has a Borel selector iff E is smooth and idealistic.

Proof. (⇒) Let s be a Borel selector for E. We show that E is idealistic. For any x ∈ X and C = [x]E , define IC on C by S ∈ IC ⇐⇒ s(x) ∈ S. Then IC is a σ-ideal and C ∈ IC . To see that C → IC is Borel, let A ⊆ X 2 be a Borel set. Then x ∈ AI ⇐⇒ {y ∈ [x] : (x, y) ∈ A} ∈ I[x] ⇐⇒ (x, s(x)) ∈ A is Borel. (⇐) Suppose E is smooth and idealistic. Let f : X → 2ω witness that E ≤B 2ω . By Theorem 4.2.3 we may assume X is Polish and f is continuous. Let F be the graph of f , that is, (x, y) ∈ F ⇐⇒ f (x) = y. Then F ⊆ X × 2ω is closed. By the usual argument we can obtain a sequence

(Fs )s∈ω<ω of nonempty closed subsets of X × 2ω so that F∅ = F , Fs = n∈ω Fs n , and diam(Fs ) < 2−lh(s) . For each y ∈ 2ω and s ∈ ω <ω , let Fsy = {x ∈ X : (x, y) ∈ Fs }. Then for any E-equivalence class C = [x]E , if f (x) = y then F∅y = C and for any

<ω y s ∈ ω , Fs = n∈ω Fsy n . Now let C → IC witness that E is idealistic. Then F∅y ∈ IC , and moreover if Fsy ∈ IC then for some n ∈ ω, Fsy n ∈ IC . Thus if we define T y = {s ∈ ω <ω : Fsy ∈ IC }, then T y is a tree on ω. Let g(y) be the leftmost branch of T y . It is straightforward to check that g is y a Borel function. Furthermore, n∈ω Fg(y)n is a singleton, and we name its unique element h(y). It is clear that (h(y), y) ∈ F and therefore h(y) ∈ C. Let H be the set of all h(y) thus obtained. It is clear that H is a transversal for E on X. To finish the proof it suffices to show that H is Borel. For this, simply note that y ). x ∈ H ⇐⇒ ∀n ∈ ω (x ∈ Fg(y)n

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Corollary 5.4.12 (Burgess) X Let G be a Polish group and X a Borel G-space. If EG is smooth then it has a Borel selector. Exercise 5.4.1 Show that for any equivalence relation E on a Polish space, id(ω) ≤B E iff there are infinitely many E-equivalence classes. Exercise 5.4.2 Show that if an equivalence relation E is smooth then either E ∼B id(2ω ), or E ∼B id(ω), or for some finite n ∈ ω, E ∼B id(n), where id(n) denotes the identity relation on any finite set of size n. Exercise 5.4.3 Let E be an equivalence relation on a Polish space X. Show that E is smooth iff there is a finer Polish topology σ on X such that E is closed in (X 2 , σ × σ). Exercise 5.4.4 Show that an intersection of countably many smooth equivalence relations on a Polish space is smooth. Exercise 5.4.5 Let G be a compact Polish group and X a Borel G-space. X is smooth. Show that EG Exercise 5.4.6 Let F ⊆ ω ω × ω ω be closed so that {x : ∃y (x, y) ∈ F } is Σ11 but not Borel. Let X = F and define the equivalence relation E on X by (x1 , y1 )E(x2 , y2 ) ⇐⇒ x1 = x2 . Show that E is closed but has no Borel selector. Exercise 5.4.7 Show that the orbit equivalence relation induced by the conjugacy action of S∞ on itself is smooth. Exercise 5.4.8 Let G be a Polish group and X be a Polish G-space. Suppose X X is Gδ . Show that EG is smooth. (Hint: The map x → [x]G every orbit of EG is a Borel reduction.)

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Chapter 6 The Glimm–Effros Dichotomy

The Glimm–Effros dichotomy is undoubtedly the most important theoretical result that helped form the subject of invariant descriptive set theory. In the context of operator algebra Glimm and Effros have obtained the dichotomy for locally compact group actions and more generally Fσ orbit equivalence relations. Nowadays the terminology usually refers to the remarkable theorem for all Borel equivalence relations proved by Harrington, Kechris, and Louveau. The theorem and its proof were so influential that for a while the main activity of the field was to prove new dichotomy theorems. For orbit equivalence relations the Glimm–Effros dichotomy have been obtained by Solecki, Hjorth, Becker, and others. Another remarkable connection with other fields of mathematics is the study of hyperfinite equivalence relations. To this date there are still intriguing open problems around which exciting research is actively going on.

6.1

The equivalence relation E0

Definition 6.1.1 The equivalence relation E0 is the relation of eventual agreement on 2ω , that is, it is defined by xE0 y ⇐⇒ ∃m ∀n ≥ m x(n) = y(n). We also consider the eventual agreement relation on ω ω , denoted by E0 (ω). Proposition 6.1.2 E0 ∼B E0 (ω). Proof. The identity embedding from 2ω into ω ω is a reduction from E0 to E0 (ω). To define a reduction f to witness E0 (ω) ≤B E0 , fix a computable bijection ·, · : ω × ω → ω. Given x ∈ ω ω , let f (x) ∈ 2ω be defined by f (x)(n, k) = the (k + 1)-th least significant digit of the binary expansion of x(n).

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f is actually continuous, even computable. To see that it is a desired reduction, assume that xE0 (ω)y. Suppose x(n) = y(n) for n ≥ m. Then for all n ≥ m and k ∈ ω, f (x)(n, k) = f (y)(n, k). For n < m, since the binary expansions of x(n) and y(n) are of finite lengths, there is kn such that for all k ≥ kn , f (x)(n, k) = 0 = f (y)(n, k). Thus for any n, k ∈ ω, if f (x)(n, k) = f (y)(n, k) we must have that n < m and k < kn , and there are only finitely many such pairs n, k. This shows that f (x)E0 f (y). Conversely, if x E0 y, then the set A = {n ∈ ω : x(n) = y(n)} is infinite. For each n ∈ A, let kn be such that the (kn + 1)-th least significant digits of binary expansions of x(n) and y(n) differ. Then we have that f (x)(n, kn ) = f (y)(n, kn ) for all n ∈ A. This means that f (x) and f (y) are not E0 (ω)equivalent. Definition 6.1.3 The Vitali equivalence relation Ev is the equivalence relation on R defined by xEv y ⇐⇒ x − y ∈ Q. The terminology is motivated by the theorem of Vitali in real analysis that any transversal of Ev is not Lebesgue measurable. Proposition 6.1.4 E0 ∼B Ev . Proof.

We first show that Ev ≤B E0 . Given x ∈ R, express x as x = a0 +

a2 an a1 + + ··· + + ··· 2! 3! (n + 1)!

where a0 = x and for n > 0, an ∈ {0, 1, . . . , n}. Note that such an expression is uniquely determined. In fact, one can inductively define that an = (x − a0 −

an−1 a1 − ···− )(n + 1)!. 2! n!

This allows us to define a function f : R → ω ω by f (x)(n) = an . It is clear that f is Borel (but not continuous). To see that f is a reduction, let x, y ∈ R. First assume that xEv y. Without loss of generality assume x > y and let k for some k ∈ ω. r = x − y. Let m ∈ ω be the least such that r = (m + 1)! Then for all n > m, f (x)(n) = f (y)(n) from the definition of f . Conversely, assume f (x)E0 f (y). Then the expressions for x and y differ by only finitely many terms, and it is clear that x − y is rational. Thus we have shown that Ev ≤B E0 (ω). But since E0 (ω) ∼B E0 , we have that Ev ≤B E0 .

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To show the converse, that E0 ≤B Ev , we follow the same idea. Let x ∈ 2ω be given. We define g(x) = x(0) +

x(1) x(2) x(n) + + ··· + + ··· . 2! 3! (n + 1)!

The series is absolutely convergent, and thus g(x) is well-defined. It follows from the above argument that xE0 y iff g(x)− g(y) ∈ Q. Therefore g is a Borel reduction from E0 to Ev . Note that every E0 -equivalence class is countable. In fact, E0 can be viewed as an orbit equivalence relation of a countable group action, where the acting group is Z<ω 2 , the direct sum of infinitely many copies of the two-element group Z2 . For this reason we also refer to the E0 -equivalence classes as E0 -orbits. Also note that every E0 -orbit is dense. It follows that every E0 -orbit is not Gδ . In fact, a dense Gδ orbit would be comeager, whereas every E0 -orbit is countable, and therefore meager. These obvious facts distinguish E0 from all smooth equivalence relations. In the following we give two proofs of the nonsmoothness of E0 . Definition 6.1.5 Let X be a standard Borel space and E an equivalence relation on X. A σ-finite nonzero Borel measure μ on X is (E−)ergodic if for any E-invariant Borel set A ⊆ X, either μ(A) = 0 or μ(X − A) = 0. The measure μ is (E−)nonatomic if μ(C) = 0 for any E-equivalence class C. Proposition 6.1.6 Let E be an equivalence relation on a standard Borel space. If there is an E-nonatomic, E-ergodic measure on X, then E is not smooth. Proof. Let μ be E-nonatomic and E-ergodic. Assume E is smooth, and {Sn }n∈ω a countable Borel generating family for E. Since each Sn is Einvariant, either μ(Sn ) = 0 or μ(X − Sn ) = 0. Consider the set   C = {X − Sn : μ(Sn ) = 0} ∩ {Sn : μ(X − Sn ) = 0}. Then μ(X − C) = 0. But C is an equivalence class, hence μ(C) = 0. It follows that μ(X) = 0, a contradiction. Proposition 6.1.7 The usual product measure on 2ω is E0 -nonatomic and E0 -ergodic. Hence E0 is not smooth.

Proof. Let μ be the product measure on 2ω . Since μ is atomless, that is, μ({x}) = 0 for any x ∈ 2ω , and every E0 -orbit is countable, μ is E0 -nonatomic.

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To see that μ is also E0 -ergodic, let A ⊆ 2ω be an E0 -invariant Borel set. Suppose μ(A) > 0. Let  > 0. Then there exists a basic open set Ns such that μ(A ∩ Ns )/μ(Ns ) > 1 − . We claim then that for any t ∈ 2<ω with lh(t) = lh(s), μ(A ∩ Nt )/μ(Nt ) > 1 − . In fact, let ϕs,t : 2ω → 2ω be given by ⎧ if s ⊆ x and t ⊆ x, ⎨ x, ϕs,t (x) = t y, if x = s y, ⎩  s y, if x = t y. Then ϕs,t (Ns ) = Nt and ϕs,t (A ∩ Ns ) = A ∩ Nt , and moreover ϕs,t preserves the measure μ. It follows that μ(A) > 1 −  for any  > 0, and therefore indeed μ(A) = 1. Definition 6.1.8 Let X be a Polish space and E an equivalence relation on X. We say that E is generically ergodic if every E-invariant Borel set is either meager or comeager. Proposition 6.1.9 Let G be a group of homeomorphisms on a Polish space X. Then the following are equivalent: (i) EG is generically ergodic; (ii) Every nonempty EG -invariant open set is dense; (iii) There is an invariant dense Gδ set Y ⊆ X all of whose orbits are dense in X; (iv) There exists a dense orbit; (v) Every invariant set with the Baire property is meager or comeager. Proof. The implications (i)⇒(ii), (iii)⇒(iv), and (v)⇒(i) are obvious. The implication (iv)⇒(v) follows from Exercise 3.3.2. It remains to show (ii)⇒(iii). Let {Un }n∈ω be a countable base for the topology of X. Every saturation [Un ]G is EG -invariant open, and therefore dense. Let Y = n [Un ]G . Then Y is an EG -invariant dense Gδ set. If x ∈ Y , then [x]G ⊆ [Un ]G and therefore [x]G ∩ Un = ∅. Thus every orbit in Y is dense. Now we have an analog of Proposition 6.1.6 for generically ergodic equivalence relations, whose proof we leave as an exercise (Exercise 6.1.4). Proposition 6.1.10 Let E be an equivalence relation on a Polish space X. If E is generically ergodic and has no comeager orbits, then E is not smooth.

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Exercise 6.1.1 Define the tail equivalence relation Et on 2ω by xEt y ⇐⇒ ∃n∃m∀k ( x(n + k) = y(m + k) ). Show that Et is not smooth. Exercise 6.1.2 Show that the Lebesgue measure on R is Ev -nonatomic and Ev -ergodic. Exercise 6.1.3 Show that if E0 ≤B E then there is an E-nonatomic, Eergodic measure. Exercise 6.1.4 Prove Proposition 6.1.10. Exercise 6.1.5 Show that both E0 and Ev are generically ergodic. Exercise 6.1.6 Let X, Y be Polish spaces and G, H be groups of homeoX morphisms on X, Y respectively. Show that if EG is generically ergodic and X Y Z EG ≤c EH then there is a closed invariant subspace Z of Y so that EH is generically ergodic. Exercise 6.1.7 Let G be a countably infinite group acting by shift on 2G . Let EG be the induced orbit equivalence relation. Show that EG is generically ergodic.

6.2

Orbit equivalence relations embedding E0

In this section we give a construction of an embedding of E0 into a given equivalence relation. In particular we will derive the original Glimm–Effros dichotomy theorems. Most theorems about E0 -embeddability require that the equivalence relation is dense and meager. However, we first note that more needs to be assumed. Take the example of the following equivalence relation E on R: xEy ⇐⇒ x = y ∨ (x, y ∈ Q). Then E is dense and meager as a subset of R2 , but E is smooth. The following theorem assumes a condition weaker than a continuous action of a Polish group. Theorem 6.2.1 (Becker–Kechris) Let X be a Polish space and G a group of homeomorphisms of X. Suppose EG is meager and that there is a dense orbit. Then E0 c EG .

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Proof. Let D be a dense orbit and {Wn }n∈ω be a decreasing sequence of dense open subsets of X 2 with EG ∩ n Wn = ∅. Without loss of generality assume that W0 ∩ {(x, x) : x ∈ X} = ∅. We construct a sequence {Us }s∈2<ω of open sets and elements gs,t ∈ G for s, t ∈ 2<ω and lh(s) = lh(t), such that, for all s, t, u ∈ 2<ω with lh(s) = lh(t) = lh(u): (a) U∅ = X; diam(Us ) < 2−lh(s) ; Us 0 , Us 1 ⊆ Us ; Us 0 ∩ Us 1 = ∅; (b) If lh(s) = lh(t) = n and s(n − 1) = t(n − 1), then Us × Ut ⊆ Wn ; −1 (c) gs,t · Us = Ut ; gs,s = 1G ; gt,s = gs,t ; gs,u = gt,u gs,t ;

(d) If n ≤ lh(s) = lh(t) is the largest with s(n − 1) = t(n − 1), then gs,t = gsn,tn . Granting the construction, we may continuously embed E0 into E G as follows. For x ∈ 2ω , we let f (x) be the unique element of the singleton n Uxn . Then f is a continuous embedding of 2ω into X. We claim that f is also a reduction from E0 into EG . Suppose first x  E 0 y. Then for infinitely many n, x(n − 1) = y(n − 1). By (c), Uxn × Uyn ⊆ Wn for infinitely many n, and therefore (f (x), f (y)) ∈ n Wn ⊆ X 2 − EG . On the other hand, suppose xE0 y and n is the largest such that x(n − 1) = y(n − 1). Let g = gxn,yn . Then by (d) gxm,ym = g for any m ≥ n. Since g ·Uxm = Uym for all m ≥ n, we have that g · f (x) = f (y), and thus f (x)EG f (y). The construction is by induction on lh(s). We start with lh(s) = 1. At this stage we need to find disjoint open sets U0 , U1 and g0,1 ∈ G such that diam(U0 ), diam(U1 ) < 1/2, g0,1 · U0 = U1 , and U0 × U1 ⊆ W1 . To do this we first let V0 , V1 ⊆ X be open sets such that V0 × V1 ⊆ W1 . Without loss of generality we may assume diam(V0 ), diam(V1 ) < 1/2. By the density of D we can find x0 ∈ D ∩ V0 and x1 ∈ D ∩ V1 . Let g0,1 be an arbitrary g ∈ G such that g ·x0 = x1 . Let now U0 = g −1 (g ·V0 ∩V1 ) and U1 = g ·U0 . Since U0 ⊆ V0 and U1 ⊆ V1 , we have the required properties. In the inductive step assume Us and gs,t have been defined for all s, t with lh(s) = lh(t) ≤ n. We need to define Us i and gs i,t j for all s, t with lh(s) = lh(t) = n and i, j ∈ {0, 1}. First note that it is required by the condition (d) that gs i,t i = gs,t for all s, t with lh(s) = lh(t) = n and i ∈ {0, 1}. Therefore in view of (c) all gs i,t j will be determined once we have defined g0 0,0 1 , where 0 here stands for the element s with lh(s) = n and s(k) = 0 for all k < n. Similarly, all Us i will be determined once we have defined U0 0 . As in the base step we start with choosing disjoint open sets V0 0 and V0 1 , both of diameter < 2−n−1 , whose closures are contained in U0 , and so that V0 0 × V0 1 ⊆ Wn+1 . The last requirement can be fulfilled since Wn+1 is dense open. For any other s with lh(s) = n, we can then let Vs 0 = g0,s · V0 0 and Vs 1 = g0,s · V0 1 . Note that Vs 0 and Vs 1 are disjoint, open, and their closures are contained in Us . By shrinking we may assume also that each of

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them has diameter < 2−n−1 . Note that in the shrinking process we always maintain that g0,s · V0 i = Vs i . We next go through a finite number of steps to shrink the sets Vs i so as to guarantee that Vs 0 × Vt 1 ⊆ Wn+1 . Again in this shrinking process we always maintain that g0,s ·V0i = Vs i . To illustrate the argument we consider V0 0 and V0 1 . Since Wn+1 is dense open, it meets the set V0 0 × V0 1 . Thus we could shrink both of them so as to guarantee that V0 0 × V0 1 ⊆ Wn+1 . Remember this could result in other sets being shrunken too. In general, as we go through all possible pairs Vs 0 and Vt 1 , each of the sets in the collection could be shrunken, but the requirements fulfilled in the earlier process are maintained. Thus at the end of this shrinking process, we obtained open sets satisfying the properties prescribed by (a) and (b). Now by the density of D we may find x0 0 ∈ V0 0 and x0 1 ∈ V0 1 . Let g0 0,0 1 be an arbitrary g ∈ G such that g · x0 0 = x0 1 . Finally let U0 0 = g −1 (g · V0 0 ∩ V0 1 ). As remarked above, all other sets Us i and elements gs i,t j are determined. This finishes the proof of the theorem. Corollary 6.2.2 Let G be a Polish group and X a Polish G-space. Suppose there is no Gδ X orbit. Then E0 c EG . Proof. Let x ∈ X be arbitrary and consider Y = [x]G . Then Y is a Polish Y X G-space and EG c EG via the identity embedding. In Y there is obviously a dense orbit. Since there is no Gδ orbit in X, the same is true for Y . It follows from Effros’ theorem that every orbit is meager in itself, and in Y is analytic, it has the Baire property. particular meager in Y . Since EG Y Y By the Kuratowski–Ulam theorem EG is meager. Thus E0 c EG and so X E0 c EG . The corollary is very useful since it explicitly relates the descriptive complexity of orbits to the complexity of the entire equivalence relation. It is usually easier to investigate the complexity of a particular orbit. For instance, in a Baire space a dense Fσ set with a dense complement is not Gδ , since otherwise both the set and its complement would be dense Gδ , hence having a nonempty intersection. Using this it is easy to show that E0 and Ev have no Gδ orbits. The following theorem establishes a strong dichotomy if the Polish action meets some strong requirement. From it we can deduce the original Glimm– Effros dichotomy theorems.

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Theorem 6.2.3 Let G be a Polish group and X a Polish G-space. Suppose every Gδ orbit is X . In particular, either also Fσ . Then either every orbit is Gδ or else E0 c EG X X EG is smooth or else E0 c EG . X Proof. Suppose E0 c EG . We show that every orbit is Gδ . Let x ∈ X. Consider Y = {y ∈ X : [y]G = [x]G }.

It is easy to see that Y is invariant. We also note that Y is Gδ . To see this, let U be a countable base for the topology of X. Then  ∅)∧ y ∈ Y ⇐⇒ ∀U ∈ U ( U ∩ [x]G = ∅ ⇒ U ∩ [y]G = ∀U ∈ U ( U ∩ [y]G = ∅ ⇒ U ∩ [x]G =  ∅)  ∅ ⇒ y ∈ [U ]G )∧ ⇐⇒ ∀U ∈ U ( U ∩ [x]G = ∀U ∈ U ( U ∩ [x]G = ∅ ⇒ y ∈ [U ]G ). Since [U ]G is an open set, the first condition is Gδ and the second is closed. It follows that Y is Polish, and in fact a Polish G-space with the inherited action Y X Y c EG via the identity embedding, we have that E0 c EG . of G. Since EG Now by Corollary 6.2.2 there is a Gδ orbit in Y . Let [y]G ⊆ Y be Gδ . It is also true that every orbit of Y is dense. Hence Y cannot contain more than one orbit. In fact, by our assumption Y − [y]G is also Gδ , hence if it were nonempty then it would be dense Gδ and meeting [y]G , a contradiction. It follows that Y = [x]G , and hence [x]G is Gδ . X X Finally by Exercise 5.4.8 if every orbit of EG is Gδ then EG is smooth. Corollary 6.2.4 (Effros) X is Fσ . Then Let G be a Polish group and X a Polish G-space. Suppose EG X X either EG is smooth or else E0 c EG .

Proof. Since the equivalence relation is Fσ every orbit is Fσ too; in particular, every Gδ orbit is also Fσ . Corollary 6.2.4 is the original Glimm–Effros dichotomy theorem, proved by Effros, strengthening earlier work of Glimm for locally compact Polish group actions (see Exercise 6.2.2). Exercise 6.2.1 Consider the following equivalence relation E on Rω : xEy ⇐⇒ x − y ∈ 1 ⇐⇒

∞  n=0

Show that E0 c E.

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Exercise 6.2.2 (Glimm) Let G be a locally compact Polish group and X a X X is smooth or else E0 c EG . Polish G-space. Then either EG Exercise 6.2.3 Let G be a Polish group and X a Polish G-space. Define an equivalence relation F on X by xF y ⇐⇒ [x]G = [y]G . X ⊆ F. (a) Verify that F is an equivalence relation and show that EG

(b) Show that F is Gδ . X (c) Show that xF y ⇐⇒ for all EG -invariant open (or closed) sets C, x ∈ C iff y ∈ C.

6.3

The Harrington–Kechris–Louveau theorem

In this section we prove the Glimm–Effros dichotomy theorem for Borel equivalence relations [64]. This is again a theorem of classical character for which the only known proof uses effectively descriptive set theory. Theorem 6.3.1 (Harrington–Kechris–Louveau) Let E be a Borel equivalence relation on a Polish space X. Then either E is smooth or else E0 c E. The real theorem being proved is the following effective version of the Glimm–Effros dichotomy. Theorem 6.3.1 follows immediately by relativization. Theorem 6.3.2 Let E be a Δ11 equivalence relation on ω ω . Then either there is a Δ11 generating family for E or else E0 c E. The rest of this section is devoted to a proof of Theorem 6.3.2. We leave further remarks and corollaries to the next section. First we state an effective version of Lemma 5.4.6. The proof is left as an exercise (Exercise 6.3.1). Lemma 6.3.3 Let E be a Σ11 equivalence relation on ω ω . Let A, B be disjoint Σ11 E-invariant sets. Then there is a Δ11 E-invariant set C with A ⊆ C and B ∩ C = ∅. Next for any Σ11 equivalence relation E on ω ω we define a derived equivalence relation E ∗ by xE ∗ y ⇐⇒ for all Δ11 E-invariant sets C, x ∈ C iff y ∈ C. Then the following lemma collects some basic properties of E ∗ .

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Lemma 6.3.4 E ∗ ⊇ E is a Σ11 equivalence relation with a Δ11 generating family. Proof. It follows straightforward from the definition of E ∗ that E ⊆ E ∗ , E ∗ is an equivalence relation, and that E ∗ has a Δ11 generating family, that is, the family of all Δ11 E-invariant sets. We show that E ∗ is Σ11 . Using the coding in Theorem 1.7.4 we have xE ∗ y ⇐⇒ ∀n { ∃w∃z (w ∈ Pn− ∧ z ∈ Pn+ ∧ wEz ) ∨ [ ( x ∈ Pn+ ⇒ y ∈ Pn− ) ∧ ( y ∈ Pn+ ⇒ x ∈ Pn− ) ]} . Thus E ∗ is Σ11 . To consider further properties of E ∗ we take into account the Gandy– Harrington topologies. Let τ be the Gandy–Harrington topology on ω ω . Let τ 2 = τ ×τ be the product topology on (ω ω )2 and τ2 be the Gandy–Harrington topology on (ω ω )2 . Then τ 2 ⊆ τ2 . Lemma 6.3.5 E ∗ is Gδ in τ 2 . Proof.

Let C be the (countable) family of all Δ11 E-invariant sets. Then E∗ =



{(C × C) ∪ [(ω ω − C) × (ω ω − C)]}.

C∈C

Since (ω ω , τ ) is a second countable Baire space, it follows that E ∗ is Baire. Lemma 6.3.6 E ∗ is the closure of E in τ 2 . Proof. We first show that E is dense in E ∗ . For this we show that, for any Σ11 A, B ⊆ ω ω , if (A × B) ∩ E = ∅, then (A × B) ∩ E ∗ = ∅. Assume A, B are Σ11 sets with (A × B) ∩ E = ∅. Then [A]E ∩ [B]E = ∅. Since both [A]E and [B]E are Σ11 , by Lemma 6.3.3 there is a Δ11 E-invariant set C with A ⊆ C and B ∩ C = ∅. Now for any x ∈ A and y ∈ B, x ∈ C but y ∈ C, hence x E ∗ y. This means (A × B) ∩ E ∗ = ∅ as required. Next we show that E ∗ is closed in τ 2 . For this let (x, y) ∈ E ∗ , then there is a Δ11 E-invariant set C with x ∈ C and y ∈ C. Let A = C and B = ω ω − C. Then both A and B are Σ11 sets, and (A × B) ∩ E ∗ = ∅. This shows that (ω ω )2 − E ∗ is open in τ 2 , and therefore E ∗ is closed.

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If E ∗ = E then E has a Δ11 generating family and the first alternative of the conclusion of Theorem 6.3.2 is obtained. From now on we assume E ∗ = E. Consider the set X = {x ∈ ω ω : [x]E ∗ = [x]E }. Then X is nonempty. If in addition E is Δ11 then X is Σ11 , since x ∈ X ⇐⇒ ∃y ( yE ∗ x ∧ y E  x ). From now on we also assume that E is Δ11 . We are now ready to take category arguments into account. Lemma 6.3.7 If A, B ⊆ X are nonempty Σ11 sets such that E is comeager in (A × B) ∩ E ∗ , then (A × A) ∩ E ∗ ⊆ E. Proof. Consider Y = {(x, y, z) ∈ A × A × B : xE ∗ yE ∗ z} with the τ2 × τ topology. Similar to Lemma 6.3.5 we have that  {(C × C × C) ∪ [(ω ω − C) × (ω ω − C) × (ω ω − C)]}, Y = (A × A × B) ∩ C∈C

where C is the countable family of all Δ11 E-invariant sets. Thus Y is Gδ in τ 2 × τ as well as in τ2 × τ . Let Y1 = {(x, y, z) ∈ Y : xEz} and Y2 = {(x, y, z) ∈ Y : yEz}. Since E is comeager in (A × B) ∩ E ∗ , both Y1 and Y2 are comeager in Y . Now suppose (A × A) ∩ E ∗ ⊆ E. Consider Y3 = {(x, y, z) ∈ Y : x  E y}. Then Y3 is nonempty. Since E is Δ11 , Y3 is Σ11 and open in τ2 × τ . It follows that Y1 ∩ Y2 ∩ Y3 = ∅. But if (x, y, z) ∈ Y1 ∩ Y2 ∩ Y3 , then xEz, yEz, and xE  y, a contradiction. Lemma 6.3.8 E is dense and meager in X 2 ∩ E ∗ in τ 2 . Proof. Since X 2 is Σ11 , and thus open in τ 2 , the density is immediate from Lemma 6.3.6. Assume toward a contradiction that E is nonmeager in X 2 ∩E ∗ . Then there exist nonempty Σ11 sets A, B ⊆ X such that E is comeager in (A × B) ∩ E ∗ . By the preceding lemma we have that (A × A) ∩ E ∗ ⊆ E. Therefore in fact ([A]E × [A]E ) ∩ E ∗ ⊆ E. We claim that [A]E = [A]E ∗ . To see this, assume that [A]E = [A]E ∗ . Then the set Z = {x ∈ [A]E ∗ : ∃y ∈  y )} is nonempty, Σ11 , and (Z×[A]E )∩E ∗ = ∅. By the density [A]E ( xE ∗ y ∧ x E of E we have that (Z × [A]E ) ∩ E = ∅. But if z ∈ Z, x ∈ [A]E , and xEz then  y; since (x, y) ∈ ([A]E × [A]E ) ∩ E ∗ ⊆ E, there is y ∈ [A]E with zE ∗ y and z E we have xEy, a contradiction. Finally we claim that for any x ∈ A, [x]E = [x]E ∗ , a contradiction to the assumption that x ∈ X. So let x ∈ A and yE ∗ x. Then y ∈ [A]E ∗ = [A]E , and hence (x, y) ∈ ([A]E × [A]E ) ∩ E ∗ ⊆ E. This means that xEy and y ∈ [x]E .

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We are finally ready to embed E0 into E. Let us comment on the main ideas before giving the construction itself. As usual we will construct a sequence {Us }s∈2<ω of open sets and define the embedding from 2ω by letting f (z) to be the unique element of the singleton n Uzn . However, since we are working with the Gandy–Harrington topology, the strong Choquet property is used to guarantee the nonemptiness of the intersection n Uzn in place of complete metrizability. For this we need to play the strong Choquet game and let Us be coming from a winning strategy for Player II. Thus for each s ∈ 2<ω we need to find xs and Vs as Player I’s move. To make sure we are creating a reduction of E0 into E a scheme similar to the proof of Theorem 6.2.1 is used. To ensure that z E 0 y ⇒ f (z) E f (y) we ask that Us × Ut be contained in some dense open set avoiding the meager relation E if s, t have the same length but their last entries differ. For the other implication, namely zE0 y ⇒ f (z)Ef (y), the difficulty now is that no group action is present to provide the uniform linkage we had in the proof of Theorem 6.2.1. The alternative is also making use of the strong Choquet property, this time on the space of pairs, to guarantee that appropriate linkage exists among the elements xs . When zE0 y, the pairs (xzn , xyn ) will converge to a unique linked pair, forcing that f (z)Ef (y). To deal with the combinatorics associated with the linkage of elements xs , we introduce the following notation. For s, t ∈ 2<ω with lh(s) = lh(t) and k ∈ ω, define a relation Rk by sRk t ⇐⇒ ∀i < k s(i) = t(i) = 0 ∧ s(k) = t(k) ∧ ∀k < i < lh(s) s(i) = t(i). Note that if k = k  then Rk ∩ Rk = ∅. If sRk t then (s u)Rk (t u) for any u ∈ 2<ω . Also if sRk t and k < n < lh(s) then (s  n) Rk (t  n). Let also R = k Rk . The following lemma collects some other basic properties of the relation R. The proof is an easy induction on n ∈ ω, which we leave as an exercise (Exercise 6.3.3). Lemma 6.3.9 For each n ∈ ω the following are true:

(i) R ∩ (2n × 2n ) ⊆ k
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Proof. Fix winning strategies for Player II in the strong Choquet games played on (X, τ ) and ((X × X) ∩ E, τ2 ) respectively. Since E is meager in (X × X) ∩ E ∗ in τ 2 we can find a decreasing sequence of dense open subsets of (X × X) ∩ E ∗ in τ 2 such that E ∩ n Wn = ∅. Without loss of generality we may assume that W0 ∩ {(x, x) : x ∈ X} = ∅. We define sequences {xs }s∈2<ω of elements of X, {Vs }s∈2<ω , {Us }s∈2<ω of τ -open subsets of X, and {Fs,t }sRt , {Es,t }sRt of τ2 -open subsets of (X ×X)∩E such that, for any s, t ∈ 2<ω with lh(s) = lh(t), the following requirements are fulfilled: (i) V∅ = U∅ = X; (ii) The following is a play in the strong Choquet game on X according to Player II’s winning strategy: I

xs1 , Vs1

II

···

xs2 , Vs2 Us1

xs , Vs ···

Us2

Us

In particular, xs ∈ Us ⊆ Vs ; (iii) diam(Vs ) ≤ 2−lh(s) (in the usual topology of ω ω ); (iv) If lh(s) = lh(t) = n and s(n − 1) = t(n − 1) then Vs × Vt ⊆ Wn ; (v) If lh(s) = lh(t) = n and sRk t then the following is a play in the strong Choquet game on (X ×X)∩E according to Player II’s winning strategy: I

···

(xsk , xtk ), Fsk,tk

II

Esk,tk

(xs , xt ), Fs,t ···

Es,t

(vi) diam(Fs,t ) ≤ 2−lh(s) for sRt (in the usual topology of (ω ω )2 ). The construction is by induction on lh(s). We start with lh(s) = 1. At this stage we need Σ11 sets V0 , V1 ⊆ X of diameter ≤ 1/2 such that V0 × V1 ⊆ W1 and elements x0 ∈ V0 , x1 ∈ V1 so that (x0 , x1 ) ∈ E. Since W1 is dense open in (X × X) ∩ E ∗ in τ 2 such V0 , V1 exist; and since E is dense in (X × X) ∩ E ∗ , E ∩ (V0 × V1 ) = ∅. Let F0,1 be any Σ11 subset of (X × X) ∩ E containing (x0 , x1 ) and of diameter < 1/2. The sets U0 , U1 and E0,1 are obtained by playing the strong Choquet games. This finishes the construction for lh(s) = 1. In the inductive step assume all xs , Vs , Us , Fs,t , and Es,t have been defined for lh(s) ≤ n and sRt. We need to define xs i , Vs i , Fs i,t i for lh(s) = n, sRt, and i ∈ {0, 1}, as well as F0 0,0 1 . Once these are defined the sets Us i , Es i,t i , and E0 0,0 1 are obtained by playing the strong Choquet games.

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To satisfy the requirements we need at least to maintain, for all s, t with lh(s) = lh(t) = n and sRt, that xs 0 , xs 1 ∈ Us , xt 0 , xt 1 ∈ Ut , and (xs 0 , xt 0 ), (xs 1 , xt 1 ) ∈ Es,t . We consider the set of all such tuples Y = {((ys 0 )s∈2n , (ys 1 )s∈2n ) : ∀s, t ∈ 2n [ys i ∈ Us ∧ (ys i , yt i ) ∈ E ∗ ∧ (sRt ⇒ (ys i , yt i ) ∈ Es,t )], i = 0, 1}. n

n

As a subset of X 2 × X 2 the set Y is obviously of the form Z × Z for n n some Z ⊆ X 2 open in the Gandy–Harrington topology on X 2 . And it is nonempty since (xs )s∈2n ∈ Z and therefore ((xs )s∈2n , (xs )s∈2n ) ∈ Y . To further satisfy requirement (iv) we fix any s0 , t0 ∈ 2n . Since Wn+1 is dense open in (X × X) ∩ E ∗ , the set Ys0 ,t0 = {((ys 0 )s∈2n , (ys 1 )s∈2n ) ∈ Y : (ys 0 , yt 1 ) ∈ Wn+1 } 0

0

is dense open in Y . It follows that the finite intersection s,t∈2n Ys,t is still dense open, and thus there are Σ11 sets Vs 0 , Vt 1 such that    Vs 0 × Vt 1 ⊆ Ys,t . s

t

s,t

With possible shrinking we may require that diam(Vs 0 ), diam(Vt 1 ) ≤ 2−lh(s)−1 . Thus (iii) and (iv) are satisfied. Furthermore since E is dense in (X × X) ∩ E ∗ , the set Y  = {((ys 0 )s∈2n , (ys 1 )s∈2n ) ∈ Y : (y0 0 , y0 1 ) ∈ E} is dense in Y . Let ((xs 0 )s∈2n , (xs 1 )s∈2n ) be any element of the nonempty intersection   Vs 0 × Vt 1 . Y∩ s

t

We have defined the required elements (xu )u∈2n+1 satisfying (ii). Finally for each pair (s, t) with sRt and i ∈ {0, 1} let Fs i,t i be any Σ11 subset of (X × X) ∩ E containing (xs 0 , xt 1 ) and with diameter ≤ 2−lh(s)−1 . Let also F0 0,0 1 be any Σ11 subset of (X × X) ∩ E containing (x0 0 , x0 1 ) and with diameter ≤ 2−lh(s)−1 . We have thus finished the definition of all elements and sets involved, and conditions (i) through (vi) are all satisfied. It follows from (ii) that for any z ∈ 2ω the objects xzn , Vzn , Uzn form a play in the strong Choquet game on X according to Player II’s winning strategy. Thus n Uzn is nonempty, and by (iii), is a singleton. We define f (z) to be this unique element. This defines a continuous embedding of 2ω into X. It remains to verify that f is a reduction from E0 to E. First suppose z E  0 y. Then for infinitely many n, z(n − 1) = y(n − 1), and by (iv), for infinitely many n, Uzn ×Uyn ⊆ Wn . It follows that (f (z), f (y)) ∈ W , and hence (f (z), f (y) ∈ E. n n

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For the other direction we show that for any s, t with lh(s) = lh(t) = n and z ∈ 2ω , f (s 0 z)Ef (t 1 z). This is by induction on n. For n = 0 we note that (0 z  k)R1 (1 z  k) for any k ∈ ω. Thus the objects (x0 zk , x1 zk ), F0 zk,1 zk , E0 zk,1 zk form a play in the strong Choquet game on (X × X) ∩ E according to Player II’s winning strategy. It follows from (v) and (vi) that k E0 zk,1 zk is a singleton. By our construction this unique pair must be (f (0 z), f (1 z)). Therefore f (0 z)Ef (1 z). For the inductive step when n > 0, consider the elements f (0 0 z) and f (0 1 z). By the inductive hypothesis f (0 0 z)Ef (s 0 z) and f (0 1 z)Ef (t 1 z). Thus it suffices to show that f (0 0 z)Ef (0 1 z). However this is similar to the argument for the case n = 0. For any k ∈ ω let uk = 0 0 z  k and vk = 0 1 z  k. By our construction the objects (xuk , xvk ), Fuk ,vk , Euk ,vk form a play in the strong Choquet game on (X × X) ∩ E according to Player II’s winning strategy. It follows again from (v) and (vi) that k Euk ,vk is a singleton and its unique element must be (f (0 0 z), f (0 1 z)). Therefore f (0 0 z)Ef (0 1 z). Exercise 6.3.1 Prove Lemma 6.3.3. (Hint: Δ11 is closed under effective countable unions but not arbitrary countable unions.) Exercise 6.3.2 Show that if E is a Σ11 equivalence relation on ω ω then E ∗ is clopen in the Gandy–Harrington topology τ2 . Exercise 6.3.3 Prove Lemma 6.3.9.

6.4

Consequences of the Glimm–Effros dichotomy

In this section we derive some consequences of the Glimm–Effros dichotomy for Borel equivalence relations. The main results of this section are due to Harrington, Kechris, and Louveau [64]. We first recall the following definition. Definition 6.4.1 Let X, Y be standard Borel spaces. A subset A ⊆ X is called universally measurable if it is μ-measurable for any σ-finite Borel measure μ on X. A function f : X → Y is universally measurable if it is μ-measurable for any σ-finite Borel measure μ on X. A theorem of Luzin states that every analytic set is universally measurable (see Reference [97] Theorem 21.10). Let σ(Σ11 ) be the σ-algebra generated by the analytic sets. Then it follows that every set in σ(Σ11 ) is universally measurable. If X and Y are standard Borel spaces, then a function f : X → Y

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is σ(Σ11 )-measurable if for any Borel set B ⊆ Y , f −1 (B) is a σ(Σ11 ) subset of X. We also recall the following definition. Definition 6.4.2 Let X, Y be sets and P ⊆ X × Y . Let projX (P ) = {x ∈ X : ∃y ∈ Y (x, y) ∈ P }. A uniformization of P is a function f : projX (P ) → Y such that for all x ∈ projX (P ), (x, f (x)) ∈ P . We will use the following theorem known as the Jankov–von Neumann uniformization theorem. Theorem 6.4.3 (Jankov–von Neumann) Let X, Y be standard Borel spaces and P ⊆ X × Y an analytic set. Then there is a uniformization of P that is σ(Σ11 )-measurable. For a proof of this theorem see Reference [97] Theorem 18.1. We are now ready to derive some consequences of the Glimm–Effros dichotomy. Theorem 6.4.4 Let X be a standard Borel space and E a Borel equivalence relation on X. Then the following are equivalent: (i) E is smooth; (ii) There is a compatible Polish topology τ on X such that E is closed in (X 2 , τ 2 ); (iii) There is a compatible Polish topology τ on X such that E is Gδ in (X 2 , τ 2 ); (iv) There is a compatible Polish topology τ on X such that every Eequivalence class is Gδ ; (v) There is a countable generating family for E consisting of analytic sets; (vi) There is a countable generating family for E consisting of universally measurable sets; (vii) There is a universally measurable reduction from E to id(2ω ); (viii) E has a σ(Σ11 )-measurable selector; (ix) E has a universally measurable selector.

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Proof. The implication (i)⇒(ii) was proved in Corollary 5.4.5. The implications (ii)⇒(iii)⇒(iv), (v)⇒(vi), and (viii)⇒(ix) are obvious. The equivalence between (vi) and (vii) follows from the proof of Proposition 5.4.4. Also (ix)⇒(vi) by a similar argument. We prove the other implications. For (iv)⇒(v), let {Un }n∈ω enumerate a countable base for (X, τ ). We claim that {[Un ]E }n∈ω is a countable generating family for E consisting of analytic sets. It is clear that each [Un ]E is analytic. To see that they form a generating family for E, let x, y ∈ X be such that x ∈ [Un ]E ⇐⇒ y ∈ [Un ]E for all n ∈ ω. Since for any n ∈ ω, [x]E ∩ Un = ∅ ⇐⇒ [x]E ∩ Un = ∅ ⇐⇒ x ∈ [Un ]E , we have that for all n ∈ ω, [x]E ∩ Un = ∅ ⇐⇒ [y]E ∩ Un = ∅, which in turn implies that [x]E = [y]E . Now [x]E , [y]E are both dense Gδ subsets of [x]E , thus [x]E ∩ [y]E = ∅, and therefore in fact xEy. Next we show that (vi)⇒(i) from the Glimm–Effros dichotomy. Let {An }n∈ω be a countable generating family for E where each An is universally measurable. Assume that E is not smooth. Then there is a Borel embedding θ : 2ω → X of E0 into E. It is easy to verify that the family {θ−1 (An )}n∈ω is a generating family for E0 . Note also that each θ−1 (An ) is universally measurable. To see this let μ be a σ-finite Borel measure on 2ω . Then θμ defined by θμ (A) = μ(θ−1 (A)) is a σ-finite Borel measure on X. Since each An is universally measurable, An is θμ -measurable, and hence θ−1 (An ) is μmeasurable. To summarize, we now have a countable separating family for E0 consisting of universally measurable sets. By the proofs of Propositions 6.1.6 and 6.1.7 show that E0 does not admit such families. Finally we show that (i)⇒(viii). For this let ϕ : X → 2ω be a Borel reduction of E to id(2ω ). Consider A = {(z, x) ∈ 2ω × X : z = ϕ(x)}. Then A is Borel, in particular Σ11 , and by the Jankov–von Neumann uniformization theorem, allows a σ(Σ11 )-measurable uniformization. Let ψ : ϕ(X) → X be a σ(Σ11 )-measurable uniformization of A. Then ψ ◦ ϕ is σ(Σ11 )-measurable. But ψ ◦ ϕ is clearly a selector for E. Recall that smooth equivalence relations need not have Borel selectors. In contrast, clauses (viii) and (ix) provide characterizations for smoothness in terms of σ(Σ11 )-measurable and universally measurable selectors. Also for Borel equivalence relations the existence of Borel selectors and of Borel transversals are equivalent (see the remarks preceding Proposition 3.4.6), therefore smooth equivalence relations need not have Borel transversals either. On the flip side we also get characterizations for E0 -embeddability.

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Theorem 6.4.5 Let X be a Polish space and E a Borel equivalence relation on X. Then the following are equivalent: (a) E0 c E; (b) E0 ≤B E; (c) There is a universally measurable reduction from E0 to E; (d) There is an E-nonatomic, E-ergodic probability Borel measure on X. Proof. The implications (a)⇒(b)⇒(c) are obvious. To see that (c)⇒(a), suppose there is a universally measurable reduction θ from E0 to E. By the proofs of Propositions 6.1.6 and 6.1.7 we know that there are no countable generating families of E0 consisting of universally measurable sets. However, if (a) fails then by the Glimm–Effros dichotomy we have that E is smooth and thus admitting a countable Borel generating family {An }n∈ω . It follows that {θ−1 (An )}n∈ω is a generating family for E0 consisting of universally measurable sets, a contradiction. A similar argument shows that (d)⇒(a). We finally show that (a)⇒(d). Let μ be the product measure on 2ω and θ : 2ω → X be a continuous embedding of E0 into E. Then the measure θμ defined by θμ (A) = μ(θ−1 (A)) is a probability Borel measure on X. It is E-nonatomic because for any x ∈ X, θμ ([x]E ) = μ(θ−1 ([x]E )) = 0 since θ−1 ([x]E ) is contained in at most one E0 -equivalence class. It is also E-ergodic since for any invariant Borel set A ⊆ X, θ−1 (A) is also invariant Borel. Exercise 6.4.1 Give a direct proof that an equivalence relation E on a Polish space X with every E-equivalence class Gδ admits a σ(Σ11 )-measurable selector. Exercise 6.4.2 Show that a Borel equivalence relation with a Σ11 transversal is smooth. Exercise 6.4.3 Let G be a Polish group and X a Polish G-space. Suppose X is Fσ . Show that the following are equivalent: EG X (1) EG is smooth;

(2) Every orbit is Gδ ; X (3) EG is Gδ ; X . (4) There is a Borel selector for EG

Exercise 6.4.4 Let G be a Polish group and H a Borel subgroup of G. Let E be the coset equivalence relation on G defined by xEy ⇐⇒ xH = yH. Show that E is smooth iff H is closed.

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The Glimm–Effros Dichotomy

6.5

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Actions of cli Polish groups

In this section we prove a theorem of Becker [5] that establishes the Glimm– Effros dichotomy for orbit equivalence relations induced by Borel actions of cli Polish groups. The main theorem of this section is the following. Theorem 6.5.1 (Becker) Let G be a cli Polish group and X a Borel G-space. Then there is a Polish topology on X such that the action is continuous and every Gδ orbit is closed. By Theorem 6.2.3, the Glimm–Effros dichotomy follows immediately. Theorem 6.5.2 (Becker) X Let G be a cli Polish group and X a Borel G-space. Then either EG is smooth X . or else E0 B EG The rest of the section is devoted to a proof of Theorem 6.5.1. We fix a cli Polish group G and a compatible complete left-invariant metric dG on G. Also fix a countable dense subgroup G0 of G and a countable base U for the topology of G such that U is closed under left-translates by elements of G0 , that is, for g ∈ G0 and U ∈ U, gU ∈ U. Without loss of generality assume that G ∈ U and ∅ ∈ U. For U, V ∈ U, define G0 (U, V ) = {g ∈ G0 : gU ∩ V = ∅}. Then note that G0 (U, V ) is nonempty. Lemma 6.5.3

For any open U  ⊆ U , {gU  : g ∈ G, gU  ∩ V = ∅} ⊆ G0 (U, V )U . In particular, V ⊆ G0 (U, V )U . Proof. Let g ∈ G and h, k ∈ U  with gh ∈ V . We need to show that gk ∈ G0 (U, V )U . By the continuity of the group operations there are open sets U0 , U1 , U2 , N such that g ∈ U0 , 1G ∈ N , h ∈ U1 , k ∈ U2 , N U1 ⊆ U  , N U2 ⊆ U , and U0 N U1 ⊆ V . Note that gN −1 ∩ U0 is nonempty open. We let g0 ∈ G0 ∩ gN −1 ∩ U0 . Then g0 h ∈ U0 N U1 ⊆ V , and therefore g0 ∈ G0 (U, V ). Also gk = g0 (g0−1 g)k, where (g0−1 g)k ∈ N U2 ⊆ U . Thus gk ∈ G0 (U, V )U . This proves the first half of the lemma. The second half follows since V ⊆ {gU : g ∈ G, gU ∩ V = ∅}. Next we consider the action of G on X and find a Polish topology on X by the method of Section 4.4. Lemma 6.5.4 There is a countable collection B of Borel sets on X such that

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(i) the topology τ generated by B is Polish, (ii) the action of G on (X, τ ) is continuous, and (iii) for all A, B ∈ B and U, V ∈ U, the set [A − G0 (U, V ) · B] G is τ -open. Proof. Let B0 be a countable base for a Polish topology on X generating the given Borel structure. Let A be a countable Boolean algebra of Borel subsets of X with the following closure properties: (a) B0 ⊆ A; (b) For all A ∈ A and U, V ∈ U, G0 (U, V ) · A ∈ A; (c) For all A ∈ A and U ∈ U, A U ∈ A; (d) The topology generated by A is Polish. This can be easily done by the remarks preceding Lemma 4.4.5. Now let τ be the topology on X generated by B = A U . By the lemmas in Section 4.4 (X, τ ) becomes a Polish G-space. To check (iii) note that B ⊆ A, and for any A, B ∈ A and U, V ∈ U, [A − G0 (U, V ) · B] G ∈ A U by condition (b) and the fact that A is a Boolean algebra. We fix a countable base B and the Polish topology τ on X with the properties described above, and investigate the Gδ orbits. For this we also fix x ∈ X so that [x]G is Gδ . By Effros’ Theorem 3.2.4 the canonical map g → g · x is open from G onto [x]G . It follows that for any open U ⊆ G, U ·x is a relatively open subset of [x]G . We define for open U ⊆ G,  Ω(U ) = {A ⊆ X : A ∈ τ, A ∩ [x]G ⊆ U · x}. It is clear that Ω(U ) is the largest open subset O of X such that O∩[x]G = U ·x. It is also easy to see that U ⊆ V implies Ω(U ) ⊆ Ω(V ), and for any g ∈ G, g · Ω(U ) = Ω(gU ). Note as well that for any sequence (Un ) of open sets in G, n Ω(Un ) ⊆ Ω( n Un ). We also define, for U, V ∈ U,  Λ(U, V ) = [Ω(V ) − {Ω(hU ) : h ∈ G0 (U, V )}] G . Then Λ(U, V ) is an invariant Borel subset of X. The main issue is whether it is empty. We first show that it is empty on the orbit [x]G . Lemma 6.5.5 For any U, V ∈ U, Λ(U, V ) ∩ [x]G = ∅. Proof. Assume Λ(U, V ) ∩ [x]G =  ∅. Since Λ(U, V ) is invariant, we have x ∈ Λ(U, V ). By definition there is g ∈ G such that g · x ∈ Ω(V ) but for all

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h ∈ G0 (U, V ), g · x ∈ Ω(hU ). From g · x ∈ Ω(V ) we get that g · x ⊆ V · x. Thus we may assume without loss of generality that g ∈ V . But by Lemma 6.5.3 g ∈ G0 (U, V )U . Thus for some h ∈ G0 (U, V ), g ∈ hU , and g · x ∈ hU · x ⊆ Ω(hU ), a contradiction. The following lemma is one of two key lemmas for the proof of Theorem 6.5.1. Lemma 6.5.6 Suppose y ∈ Λ(U, V ). Then there are A, B ∈ B such that [y]G ⊆ [A − G0 (U, V ) · B] G and [x]G ∩ [A − G0 (U, V ) · B] G = ∅. Proof. Since y ∈ Λ(U, V ) there is a nonmeager set N0 ⊆ G such that for all g ∈ N0 ,  g · y ∈ Ω(V ) − {Ω(hU ) : h ∈ G0 (U, V )}. Thus for all g ∈ N0 , g ·y ∈ Ω(V ) but for all h ∈ G0 (U, V ), g ·y ∈ Ω(hU ). Since Ω(V ) is open in X and B is a countable base, Ω(V ) = {A : A ∈ B ∧ A ⊆ Ω(V )}. It follows that there is A ∈ B and a nonmeager set N1 ⊆ N0 such that for all g ∈ N1 , g · y ∈ A ⊆ Ω(V ). Let B ∈ B be such that B ⊆ Ω(U ) and B ∩ U · x = ∅. Then B ∩ [x]G ⊆ U · x. It follows that for h ∈ G0 (U, V ), h · B ∩ [x]G ⊆ h · (U · x) = (hU ) · x, and thus h · B ⊆ Ω(hU ). This implies that for all g ∈ N1 , g · y ∈ h · B. We have shown that y ∈ [A − G0 (U, V ) · B] G . It remains to show [x]G ∩ [A − G0 (U, V ) · B] G = ∅. Assume this is not the case. Then there is k ∈ G such that k · x ∈ A − G0 (U, V ) · B. Since A ⊆ Ω(V ) we may assume without loss of generality that k ∈ V . Since B ⊆ Ω(U ) is open there is an open U  ⊆ U such that B ∩ [x]G = U  · x. U  = ∅ since B ∩ U · x = ∅. Note that G0 (U  , V ) ⊆ G0 (U, V ), and hence k · x ∈ G0 (U  , V ) · B. However, by Lemma 6.5.3 V ⊆ G0 (U  , V )U  . Thus k · x ∈ G0 (U  , V ) · (U  · x) ⊆ G0 (U  , V ) · B, a contradiction. Note that so far we have not used the assumption that G is a cli Polish group. The next key lemma is the only place where we essentially use this assumption. Lemma 6.5.7 If y ∈ [x]G − [x]G , then there are U, V ∈ U such that y ∈ Λ(U, V ). Proof. Suppose y ∈ [x]G but assume there are no U, V ∈ U such that y ∈ Λ(U, V ). We construct sequences (Un )n∈ω of elements of U and (gn )n∈ω of elements of G such that, for all n ∈ ω, (i) diam(Un ) < 2−n , (ii) Un ∩ Un+1 = ∅,

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(iii) dG (gn , gn+1 ) < 2−n , (iv) gn · y ∈ Ω(Un ). The construction is by induction on n. To begin with, let U0 = G and g0 = 1G . Since Ω(G) = X we have that g0 · y ∈ Ω(U0 ). In general, suppose Un and gn have been defined to satisfy the requirements. We now define Un+1 ∈ U and gn+1 ∈ G. Let V = Un and U ∈ U such that diam(U ) < 2−(n+1) . Since y ∈ Λ(U, V ) and Λ(U, V ) is invariant, we have that gn · y ∈ Λ(U, V ). Thus there is a comeager set C ⊆ G such that for all g ∈ C,  g · (gn · y) ∈ Ω(Un ) − {Ω(hU ) : h ∈ G0 (U, Un )}. Let

W = {g ∈ G : g · (gn · y) ∈ Ω(Un ) ∧ dG (ggn , gn ) < 2−n }.

By (iv) gn · y ∈ Ω(Un ), and since Ω(Un ) is open in X, W is nonempty open. Let g ∈ C ∩ W and gn+1 = ggn . Then dG (gn , gn+1 ) = dG (gn , ggn ) < 2−n . Now gn+1 · y = g · (gn · y) ∈ Ω(Un ) since g ∈ W , so by g ∈ C we obtain an h ∈ G0 (U, Un ) such that gn+1 · y = g · (gn · y) ∈ Ω(hU ). Let Un+1 = hU . Since diam(U ) < 2−(n+1) , the left invariance gives that diam(Un+1 ) = diam(hU ) < 2−(n+1) . Also Un ∩ Un+1 = Un ∩ hU = ∅ since h ∈ G0 (U, Un ). Lastly, gn+1 · y = g · (gn · y) ∈ Ω(hU ) = Ω(Un+1 ). Thus the construction is finished with all the requirements fulfilled. By the construction we have that the sequence (gn ) is dG -Cauchy, and therefore by the completeness of dG there is g∞ ∈ G such that gn → g∞ as n → ∞. Let y∞ = g∞ · y. Let dX be a complete metric on X compatible with τ . Let (Mn )n∈ω be a countable open nbhd base of y∞ with gn · y ∈ Mn . We claim that Un · x ∩ Mn = ∅ for all n ∈ ω. To see this fix an n and let A = Ω(Un ) ∩ Mn . Then A is open and gn · y ∈ A by (iv). Since y ∈ [x]G , we get that gn · y ∈ [x]G and thus A ∩ [x]G = ∅. It follows that A ∩ [x]G = ∅, and therefore Un · x ∩ Mn = ∅. For each n ∈ ω let xn ∈ Un · x ∩ Mn and hn ∈ Un be such that hn · x = xn . By the above construction (hn ) is a dG -Cauchy sequence, and therefore there is h∞ ∈ G such that hn → h∞ as n → ∞. By the continuity of the action we thus have that xn = hn · x → h∞ · x as n → ∞. On the other hand, xn → y∞ as (Mn ) is a nbhd base for y∞ . Thus y∞ = h∞ · x, and y ∈ [x]G . We are now ready to give the proof of Theorem 6.5.1. Proof. Following the notation of this section let x ∈ X be such that [x]G is Gδ . We show that [x]G is closed. Assume that y ∈ [x]G −[x]G . By Lemma 6.5.7 there are U, V ∈ U such that y ∈ Λ(U, V ). And so by Lemma 6.5.6 there are A, B ∈ B such that, letting D = [A − G0 (U, V ) · B] G , we have [y]G ⊆ D and [x]G ∩ D = ∅. By Lemma 6.5.4 D is open, and so y ∈ [x]G , contradicting our assumption.

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Becker’s theorem implies earlier results of Sami (for abelian Polish groups [134]; see Exercises 9.5.3 through 9.5.5), Hjorth–Solecki (for nilpotent Polish groups [83]), and Solecki (for Polish groups with two-sided invariant metrics [83]). The following exercise problems follow the notation of this section, where G is an arbitrary Polish group. Exercise 6.5.1 Show that for any g ∈ G and open U ⊆ G, g ·Ω(U ) = Ω(gU ). Exercise 6.5.2 Show that for any A ∈ B, [A]G is clopen. Exercise 6.5.3 Define an equivalence relation ∼ on X by x ∼ y iff [x]G = [y]G . Show that every ∼-equivalence class is closed.

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Chapter 7 Countable Borel Equivalence Relations

The topic of Borel equivalence relations is so vast that it certainly cannot fit into two chapters of any book. In this and the next chapters we just give an introduction of the topic and leave further explorations of the literature to the interested reader. There are many theoretical results about Borel equivalence relations, and at the same time we have also identified many classification problems and equivalence relations from other fields of mathematics that turn out to be Borel equivalence relations on standard Borel spaces. In this chapter we will discuss countable Borel equivalence relations. For reasons that will become clear the study of countable Borel equivalence relations have become intertwined with the study of countable group theory and ergodic theory. Thus most of the important results about countable Borel equivalence relations cannot be proved using set theoretic or general topological tools alone. Our objective in this chapter is to provide a self-contained inroad into the subject.

7.1

Generalities of countable Borel equivalence relations

Definition 7.1.1 Let X be a standard Borel space. An equivalence relation E on X is called finite if every E-equivalence class is finite. E is called countable if every E-equivalence class is countable. Any finite Borel equivalence relation is smooth (see Exercise 7.1.1). The following theorem of Luzin–Novikov on Borel uniformizations will be useful in proving theorems about countable Borel equivalence relations. The proof can be found in Reference [97] Theorem 18.10. Theorem 7.1.2 (Luzin–Novikov) Let X, Y be standard Borel spaces and let P ⊆ X × Y be Borel. Suppose every section Px = {y ∈ Y : (x, y) ∈ P } is countable. Then projX (P ) = {x ∈ X : ∃y (x, y) ∈ P } is Borel, and P has a Borel uniformization, that is, there is a Borel function f : projX (P ) → Y such that (x, f (x)) ∈ P for any x ∈ projX (P ).

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Moreover, P can be written as n Pn where each Pn is the graph of some Borel function fn : An → Y for An Borel, that is, Pn = {(x, fn (x)) : x ∈ An }. This uniformization theorem greatly simplifies the study of reductions among countable Borel equivalence relations, as the following lemma shows. Lemma 7.1.3 Let E be a countable Borel equivalence relation on a standard Borel space X, Y a standard Borel space, and f : X → Y a Borel function such that f (x) = f (y) implies xEy. Then f (X) is Borel in Y and there is a Borel function g : f (X) → X such that f ◦ g = id. Proof. Consider P = {(z, x) ∈ Y ×X : f (x) = z}. Then P is a Borel subset of Y × X with each Pz countable. By the Luzin–Novikov uniformization theorem f (X) = projY (P ) is Borel. Let g : f (X) → X be a Borel uniformization function. Then f ◦ g = id. The following is another application of the Luzin–Novikov uniformization theorem. The theorem states that all countable Borel equivalence relations are orbit equivalence relations of countable group actions. Theorem 7.1.4 (Feldman–Moore) Let E be a countable Borel equivalence relation on a standard Borel space X. Then there is a countable group G and a Borel action of G on X such that X . E = EG Proof. Consider E as a Borel subset of X × X. Since E is countable, each section Ex is

countable. By the Luzin–Novikov uniformization theorem we can write E as n Pn where each Pn is the graph of a Borel function fn : An → X for An Borel. Without loss of generality we may assume that Pn ∩ Pm = ∅ for n = m. Let Δ = {(x, x) : x ∈ X}. First note that X − Δ can be partitioned into countably many sets of the form A × B, where A, B are disjoint Borel subsets of X (see Exercise 7.1.3). We write X − Δ = k Rk , where {Rk }k∈ω is such a partition. Next for n, m, k ∈ ω we let En,m,k = {(x, y) ∈ E : (x, y) ∈ Pn , (y, x) ∈ Pm , x = y, (x, y) ∈ Rk }. Then {En,m,k } is a partition of E − Δ. And since each Pn is the graph of a Borel function, so is each En,m,k . We let hn,m,k : An,m,k → Bn,m,k be a Borel isomorphism between Borel subsets An,m,k and Bn,m,k such that En,m,k = {(x, hn,m,k (x)) : x ∈ An,m,k }. Define a Borel automorphism gn,m,k of X by ⎧ ⎨ hn,m,k (x), if x ∈ An,m,k , (x), if x ∈ Bn,m,k , gn,m,k (x) = h−1 ⎩ n,m,k x, otherwise.

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Let G be the group of Borel automorphisms generated by gn,m,k for n, m, k ∈ ω. Then xEy ⇐⇒ ∃n, m, k (gn,m,k (x) = y) ∨ (x = y) ⇐⇒ ∃g ∈ G(g · x = y). X Thus E = EG .

The following lemma is another basic and useful result about countable Borel equivalence relations. It is often referred to as the marker lemma. Lemma 7.1.5 (Slaman–Steel) Let E be a countable Borel equivalence relation on a standard Borel space X. Let A = {x ∈ X : [x]E is finite} and B = X − A. Then there are Borel sets B ⊇ S0 ⊇ S1 ⊇ S2 ⊇ . . . such that [Sn ]E = B for all n, and n Sn = ∅. Proof. Without loss of generality we may assume that X = 2ω . Let G be X . Enumerate a countable group acting on X in a Borel manner with E = EG G as {gn }n∈ω . For any E-equivalence class C, let C be the closure of C and xC be the lexicographically least element of C. Then map y → x[y]E is Borel, since for any s ∈ 2<ω , x[y]E ∈ Ns ⇐⇒ ∃n ( gn · y ∈ Ns ) ∧ ∀s
(i) ω ω − Δ = {Ns × Nt : sRt}. (ii) If sRt and s Rt but (s, t) = (s , t ), then (Ns × Nt ) ∩ (Ns × Nt ) = ∅.

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Exercise 7.1.4 Show that the sets Sn defined in the proof of Lemma 7.1.5 are as required. Exercise 7.1.5 Show that every countable Borel equivalence relation is idealistic. Exercise 7.1.6 Show that the following are equivalent for a countable Borel equivalence relation E: (i) E is smooth; (ii) E has a Borel transversal; (iii) E has a Borel selector. Exercise 7.1.7 Show that for every countable Borel equivalence relation E

there exist finite equivalence relations En such that E = n En . Exercise 7.1.8 Let X, Y be standard Borel spaces and E, F countable Borel equivalence relations on X, Y , respectively. Show that if E ≤B F then there is an F -invariant Borel subset B of Y such that E ∼B F  B.

7.2

Hyperfinite equivalence relations

Definition 7.2.1 Let X be a standard Borel space. A Borel equivalence relation E on X is hyperfinite if there are finite Borel equivalence relations En , n ∈ ω, with En ⊆ En+1 for all n, such that E = n En . By definition every hyperfinite equivalence relation is a countable Borel equivalence relation. Note that the equivalence relation E0 is hyperfinite. In fact, we can define, for n ≥ 1, equivalence relations En on 2ω by xEn y ⇐⇒ ∀m ≥ n x(m) = y(m).

Then each En is apparently Borel and finite, En ⊆ En+1 , and E0 = n≥1 En . The theorems proved in this section will show that E0 is a typical hyperfinite equivalence relation in a strong sense. The following proposition collects some basic closure properties of hyperfinite equivalence relations. Proposition 7.2.2 Let E and F be countable Borel equivalence relations on standard Borel spaces X and Y , respectively. Then the following hold:

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(1) If X = Y , F is hyperfinite and E ⊆ F , then E is hyperfinite. (2) If E is hyperfinite and A ⊆ X is Borel, then E  A is hyperfinite. (3) If A ⊆ X is Borel, [A]E = X, and E  A is hyperfinite, then E is hyperfinite. (4) If E ≤B F and F is hyperfinite, then E is hyperfinite. (5) If both E and F are hyperfinite, then E × F is hyperfinite. Proof. Properties (1), (2) and (5) follow easily

from the definition. We only show (3) and (4). For (3) suppose E  A = n Fn , where Fn are finite Borel X equivalence relations and Fn ⊆ Fn+1 for all n. By Theorem 7.1.4 E = EG for some countable group G acting in a Borel manner on X. Enumerate G by {gm }m∈ω . For each x ∈ X, define m(x) to be the least m such that gm ·x ∈ A. By the assumption [A]E = X, m(x) is well defined for all x ∈ X. We then define xEn y ⇐⇒ ( m(x), m(y) < n ∧ gm(x) · x Fn gm(y) · y ) ∨ x = y.

Then En are finite Borel equivalence relations, En ⊆ En+1 , and E = n En . Thus E is hyperfinite. For (4) we let f : X → Y be a Borel reduction of E to F . Then f (x) = f (y) implies xEy. By Lemma 7.1.3 there is a Borel function g : f (X) → X so that f ◦g = id. Now g is a Borel injection, thus g(f (X)) is Borel. Let A = g(f (X)). Then [A]E = X. Moreover E  A is Borel isomorphic (via g −1 ) to F  f (X). Therefore by (2) F  f (X) is hyperfinite. It follows that E  A is hyperfinite and by (3) E is hyperfinite. The following theorem gives various useful characterizations of hyperfiniteness. Especially of interest is the characterization in terms of Borel reducibility to E0 . The theorem was based on earlier work by Weiss and by Slaman–Steel, and appeared in this form in Reference [30]. Theorem 7.2.3 (Dougherty–Jackson–Kechris) Let X be a standard Borel space and E a countable Borel equivalence relation. Then the following are equivalent: (i) E ≤B E0 . (ii) E is hyperfinite.

(iii) E = n En , where En are finite Borel equivalence relations, En ⊆ En+1 , and every En -equivalence class has at most n elements. (iv) There is a Borel assignment C →
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(v) There is a Borel automorphism T of X such that xEy Z T n (x) = y.

⇐⇒ ∃n ∈

(vi) There is a Borel action of the additive group Z on X such that E = EZX . Proof. The implication (i)⇒(ii) follows immediately from Proposition 7.2.2 (4). The implication (iii)⇒(ii) and the equivalence of (iv), (v), and (vi) are obvious. We show the other implications. (ii)⇒(iii): Let E = n Fn , where Fn are finite Borel equivalence relations and Fn ⊆ Fn+1 . Without loss of generality we may assume F1 = id(X). We define a sequence of finite Borel equivalence relations En , n ≥ 1, such that En ⊆ En+1 and each En -equivalence class has at most n elements. Define E1 = id(X) = F1 . For n > 1, inductively define the following sequence of sets: Xn = {x ∈ X : [x]Fn has at most n elements}, Xn−1 = {x ∈ Xn : [x]Fn−1 has at most n elements}, · · · · · · Xi = {x ∈ i<j≤n Xj : [x]Fi has at most n elements}, ······ X2 = {x ∈ Xn ∪ · · · ∪ X3 : [x]F2 has at most n elements}, X1 = X −

1<j≤n

Xj .

Note that the definition of X1 can be formulated in the same manner as those of the other sets. Since F1 ⊆ F2 ⊆ · · · ⊆ Fn , each Xi is Fj -invariant if j ≤ i. Let En = (Fn  Xn ) ∪ (Fn−1  Xn−1 ) ∪ · · · ∪ (F2  X2 ) ∪ (F1  X1 ). Then En is a finite Borel equivalence relation with each En -equivalence class containing at most n elements. To see that En ⊆ En+1 let Yn+1 , Yn , . . . , Y2 , Y1 be the sequence of sets defined in the definition of En+1 . Suppose xEn y. Then for some i ≤ n, x, y ∈ Xi and xFi y. Since Xi ⊆ Yn+1 ∪ Yn ∪ · · · ∪ Yi and each of Yn+1 , Yn , . . . , Yi is Fi -invariant, there is j ≥ i such that x, y ∈ Yj and xFj y, and thus xEn+1 y. This shows that En ⊆ En+1 . It is easy to see that E = n En .

(ii)⇒(iv): Let E = n En where En are finite Borel equivalence relations and En ⊆ En+1 . Fix a Borel linear order < on X. We define
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It follows from this construction that if C is an En -equivalence class and D is an En+1 -equivalence class with C ⊆ D, then
from the property that Sn ⊇ Sn+1 , and E  X2 = n En since n Sn = ∅. (vi)⇒(i): This would make the above proof of (iv)⇒(ii) appear redundant, but in fact we are going to give a proof based on the above one. First by Theorem 3.3.4 the orbit equivalence relation induced by the Z-action on 2ω×Z is a universal equivalence relation for all Z-orbit equivalence relations. Denote this equivalence relation by EZ . It is thus sufficient to show that EZ ≤B E0 . For convenience we use the variation of E0 on the Baire space ω ω and continue to denote EZ by E. By Exercise 7.2.1 it again suffices to find a Borel partition {Xn }n∈ω so that E  Xn ≤B E0 for all n ∈ ω. We repeat the above proof of (iv)⇒(ii) to define the invariant Borel sets X0 , X1 , and X2 . It suffices to show that E  X2 ≤B E0 . And we also have the finite equivalence relations En defined. Now for each x ∈ X2 and n ∈ ω we let πn (x) = {g ∈ Z : g · xEn x}. We note that πn (x) is a finite set of consecutive integers and for xEn y, πn (x) · x = πn (y) · y. Let m = |πn (x)| and gn (x) = inf πn (x). Then define an n × m matrix Mn (x) = (ai,j )i
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have that Mn (x) = Mn (y). Moreover, Mn−1 (x) is a submatrix of Mn (x), and its relative position in Mn (x) can be determined by gn−1 (x) − gn (x). It is also true that if xEn−1 y then Mn−1 (x) = Mn−1 (y), Mn (x) = Mn (y), and gn−1 (x) − gn (x) = gn−1 (y) − gn (y). Let cn (x) be an integer coding the matrix Mn,m (x) and the integer gn−1 (x) − gn (x). Then we have that if xEn−1 y, then rn (x) = rn (y). We claim that xEy iff (cn (x))E0 (cn (y)). If xEy then for some n we have that xEn y for all n > n. By the construction we have that πn (x) · x = πn (y) · y and eventually cn (x) = cn (y). Thus (cn (x))E0 (cn (y)). Conversely, from the sequence (cn (x)) we can determine x ∈ 2ω×Z up to a shift by an element of Z. This is because, from the information of the sequence Mn (x) and gn−1 (x) − gn (x) we may recover x if g0 (x) is given. Thus if (cn (x)) = (cn (y)) then x and y are in the same orbit of the Z-action, and so xEy. Example 7.2.4 Let X be a standard Borel space and let Z act on X Z by (n · x)(m) = x(m − n). Then the orbit equivalence relation is hyperfinite. In view of the Glimm–Effros dichotomy, we have the following immediate corollary of the theorem. Corollary 7.2.5 Let X be a standard Borel space and E a hyperfinite equivalence relation on X. Then either E is smooth or else E ∼B E0 . Thus E0 is a typical nonsmooth hyperfinite equivalence relation and in fact the unique one up to Borel bireducibility. The theorem also motivates the following definitions. Definition 7.2.6 Let X be a standard Borel space. An equivalence relation E is essentially hyperfinite if E ≤B E0 . E is essentially countable if there is a countable Borel equivalence relation F such that E ≤B F . However, no good characterizations of essential hyperfiniteness are known. Exercise 7.2.1 Let E be a countable Borel equivalence relation on a standard Borel space X. Suppose {Xn }n∈ω is a partition of X into disjoint Borel sets and E  Xn ≤B E0 for all n ∈ ω. Show that E ≤B E0 . Exercise 7.2.2 (Jackson) Let E, F be countable Borel equivalence relations on a standard Borel space X. Suppose E ⊆ F , E is hyperfinite, and every F -equivalence class contains only finitely many E-equivalence classes. Show that F is hyperfinite.

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Countable Borel Equivalence Relations

7.3

165

Universal countable Borel equivalence relations

In this section we investigate countable Borel equivalence relations that are most complicated in the Borel reducibility hierarchy. Such equivalence relations are called universal countable Borel equivalence relations. All results in this section are due to Dougherty, Jackson, and Kechris [30]. By the Feldman–Moore theorem (Theorem 7.1.4) every countable Borel equivalence relation is the orbit equivalence relation of a Borel action of a countable group. If G is a fixed countable group, then among all Borel actions of G one is identified as universal by Theorem 3.3.4. The action is the shift action of G on the power of the Effros Borel space F (G)ω . Here, however, G has the discrete topology, and hence every subset of G is closed. It is thus equivalent to consider the shift action of G on the space (2G )ω or 2G×ω defined by (g · f )(h, n) = f (g −1 h, n) for g ∈ G, f ∈ 2G×ω , h ∈ G, and n ∈ ω. Note that this action is continuous. We have the following immediate corollary. Proposition 7.3.1 Let G be a countable group and Let X be the Polish G-space 2G×ω . Then X is a universal G-orbit equivalence X is a universal Borel G-space and EG relation. Also by Theorem 3.5.2 and a trivial analog of it for quotient groups we have the following extension result. Proposition 7.3.2 Xi for i = 1, 2. If Let G1 , G2 be countable groups, Xi = 2Gi ×ω , and Ei = EG i G1 ≤w G2 , then E1 B E2 . Thus the Borel complexity of the orbit equivalence relations of the shift actions reflect the complexity of the structures of the acting groups. It turns out that this connection between the group structures and the dynamics they produce is much stronger than we would suspect. This strong bondage is now refered to as the rigidity or superrigidity of the groups. It is therefore not surprising that ergodic theory, especially the superrigidity theory, is playing a central role in the study of countable Borel equivalence relations nowadays. Here we only provide an inroad to the study by focusing on the universal countable Borel equivalence relations. For 1 < n < ω let Fn be the free group with n generators. Let Fω be the free group with countably infinitely many generators. It is an easy fact of group theory that Fω is a surjectively universal countable group, that is, every countable group is a quotient group of Fω . Also Fω can be embedded as a subgroup of F2 . This implies that each of Fn , for 1 < n ≤ ω, is a

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weakly universal countable group. Therefore we have the following examples of universal countable Borel equivalence relations. Proposition 7.3.3 Let 1 < n ≤ ω, X = 2Fn ×ω , and E = EFXn . Then E is a universal countable Borel equivalence relation. In search for a simpler representation of the universal countable Borel equivalence relation, we prove the following sharper results. Proposition 7.3.4 Let G1 , G2 be countable groups and Y a Polish space. For i = 1, 2 let Xi = Xi . If G1 ≤w G2 then E1 B E2 . Y Gi with the shift action of Gi and Ei = EG i Proof. X2 by

First suppose G1 ≤ G2 . Let y0 ∈ Y . Define an embedding ϕ : X1 →  ϕ(f )(g) =

f (g), if g ∈ G1 , if g ∈ G1 . y0 ,

Then for g1 ∈ G1 , ϕ(g1 · f ) = g1 · ϕ(f ). Thus f E1 f  implies ϕ(f )E2 ϕ(f  ). Conversely, suppose g ·ϕ(f ) = ϕ(f  ) for some g ∈ G2 . If g ∈ G1 then g ·f = f  . Otherwise it must be the case that both f and f  take constant value of y0 , and thus f = f  . Next suppose G1 is a quotient of G2 , and let π : G2 → G1 be an onto homomorphism. Then define ψ : X1 → X2 by ψ(f )(g) = f (π(g)). Then for any g2 ∈ G2 , g2 ·ψ(f ) = ψ(π(g2 )·f ). It follows easily that E1 B E2 . The general case G1 ≤w G2 is a composition of the above cases considered, and we are done by the transitivity of B . In several steps we will build up a reduction of the shift action of F2 on 2F2 ×ω to simply its shift action on 2F2 . Proposition 7.3.5 X Let G be a countable group, X = 2G×ω , and E = EG . Let Y = 3G×Z with the shift action of G × Z and let F be the orbit equivalence relation. Then E B F . Proof. Fix a bijection of ω with Z−{0}. We can then view X as 2G×(Z−{0}) since they are Borel isomorphic as Borel G-spaces. Define an embedding ϕ : 2G×(Z−{0}) → Y by  f (h, n), if n = 0, ϕ(f )(h, n) = 2, if n = 0.

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Then for any g ∈ G, ϕ(g · f ) = (g, 0) · ϕ(f ). Conversely, if ϕ(f  ) = (g, m) · ϕ(f ), then m = 0 and f  = g · f . This is because, suppose m = 0, then ϕ(f )(h, 0) = 2 for all h ∈ G and therefore ϕ(f  )(h, m) = ((g, m)·ϕ(f ))(h, m) = ϕ(f )(g −1 h, 0) = 2 for all h ∈ G, but this is a contradiction since ϕ(f  )(h, m) = f  (h, m) ∈ {0, 1} by definition. Proposition 7.3.6 X Let G be a countable group, X = 3G , and E = EG . Let F be the orbit equivalence relation of the shift action of G × Z2 on 2G×Z2 . Then E B F . Proof. by

View 0 as encoded by 00, 1 by 01, and 2 by 11. Define ϕ : X → 2G×Z2  ϕ(f )(h, i) =

0, if f (h) = 0 or if f (h) = 1 and i = 0, 1, if f (h) = 1 and i = 1 or if f (h) = 2.

Then for any g ∈ G, ϕ(g · f ) = (g, 0) · ϕ(f ). Conversely, if ϕ(f  ) = (g, i)ϕ(f ), then we claim that ϕ(f  ) = (g, 0) · ϕ(f ) and f  = g · f . Suppose ϕ(f  ) = (g, 1) · ϕ(f ), then ϕ(f  ) = (1G , 1) · ϕ(g · f ), and thus without loss of generality we may assume ϕ(f  ) = (1G , 1) · ϕ(f ). It is a property of the coding used that if ϕ(f  )(h, 0) = 1 then ϕ(f  )(h, 1) = 1 as well. Thus by our assumption if ϕ(f )(h, 1) = ϕ(f  )(h, 0) = 1 then ϕ(f )(h, 0) = ϕ(f  )(h, 1) = 1. Similarly we also have that if ϕ(f )(h, 1) = 0 then ϕ(f )(h, 0) = 0. It follows that for any h ∈ G, ϕ(f )(h, 0) = ϕ(f )(h, 1) = ϕ(f  )(h, 0) = ϕ(f  )(h, 1). So f = f  . Notation 7.3.7 Let E∞ denote the orbit equivalence relation induced by the shift action of F2 on 2F2 . Theorem 7.3.8 E∞ is a universal countable Borel equivalence relation. Proof. By Proposition 7.3.3 the orbit equivalence relation of F2 shift action on 2F2 ×ω is already a universal countable Borel equivalence relation. By Proposition 7.3.5 it is Borel embeddable into orbit equivalence relation of the shift action of F2 × Z on 3F2 ×Z , and by Proposition 7.3.6 the latter orbit equivalence relation is in turn Borel embeddable into that of the shift action of F2 × Z × Z2 on 2F2 ×Z×Z2 . The group F2 × Z × Z2 is a quotient of Fω , and thus by Proposition 7.3.4, the last equivalence relation is Borel embeddable into that of the shift action of Fω on 2Fω . This shows that the shift action of Fω on 2Fω is a universal countable Borel equivalence relation. Now since Fω is a subgroup of F2 , by Proposition 7.3.4 again E∞ is also universal. In the next section we will show that E0
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complicated (see Reference [1] by Adams and Kechris). Exercise 7.3.1 Give a direct proof of Proposition 7.3.1. Exercise 7.3.2 Prove Proposition 7.3.2. Exercise 7.3.3 Show that for any n ≥ 2 the shift action of Fn on 2Fn gives rise to a universal countable Borel equivalence relation. Consider the shift action of Fn on 2Fn . Let Fn be the free part of this action, that is, Fn = {x ∈ 2Fn : ∀g ∈ Fn ( g = 1G ⇒ g · x = x ) }. Exercise 7.3.4 Show that Fn is an invariant dense Gδ subset of 2Fn . Exercise 7.3.5 Let μn be the usual product measure on 2Fn . Show that μn (Fn ) = 1. Exercise 7.3.6 Let En be the orbit equivalence relation on Fn of the induced Fn action. Show that En B E2 for all n ≥ 2.

7.4

Amenable groups and amenable equivalence relations

In this section we show that the universal countable Borel equivalence relation E∞ defined in the preceding section is not hyperfinite. To do this we discuss the concept of amenability for countable groups and their orbit equivalence relations. The study of amenable groups and amenable equivalence relations is interesting in its own right and is a well-established subject in ergodic theory. We first give the definitions. Definition 7.4.1 Let X be a set. A finitely additive probability measure on X is a map μ : P (X) → [0, 1], where P (X) = {A : A ⊆ X} is the power set of X, such that μ(∅) = 0, μ(X) = 1, and μ(A ∪ B) = μ(A) + μ(B) if A ∩ B = ∅. If G is a group, a finitely additive probability measure μ on G is leftinvariant if for all g ∈ G and A ⊆ G, μ(gA) = μ(A). A countable group G is amenable if there is a left-invariant finitely additive probability measure μ on G.

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Definition 7.4.2 A countable group G satisfies the Reiter condition if for all  > 0 and  g1 , . . . , gn ∈ G there is f : G → R≥0 such that g∈G f (g) = 1 and for any  1 ≤ i ≤ n, g∈G |f (g) − f (gi−1 g)| < . G satisfies the Følner condition if for all  > 0 and g1 , . . . , gn ∈ G there is a finite F ⊆ G such that for any 1 ≤ i ≤ n, |gi F F | < |F |. It is known that the Reiter condition and the Følner condition are equivalent to amenability for countable groups. Here, to keep our exposition selfcontained, we only prove the following weaker theorem. Theorem 7.4.3 Let G be a countable group. If G satisfies the Reiter condition then it also satisfies the Følner condition. If G satisfies the Følner condition then it is amenable. Proof. First we assume that G satisfies the Reitercondition. Let  > 0 and g1 , . . . , gn ∈ G. Let f : G → R≥0 be such that g∈G f (g) = 1 and for  any 1 ≤ i ≤ n, g∈G |f (g) − f (gi−1 g)| < /n. For each 0 < r ≤ 1 we let Fr = {g ∈ G : f (g) ≥ r} and χr be the characteristic function of Fr . Since  1 g∈G f (g) = 1 the set Fr is finite for any r > 0. Note that f (g) = 0 χr (g)dr,   |Fr | = g∈G χr (g), and |Fr gi Fr | = |Fr gi−1 Fr | = g∈G |χr (g)− χr (gi−1 g)| for any 1 ≤ i ≤ n. Then for any 1 ≤ i ≤ n, ! 1 ! 1 |Fr gi Fr |dr = |χr (g) − χr (gi−1 g)|dr 0

=

0 g∈G

! g∈G

<

1

0

|χr (g) − χr (gi−1 g)|dr =

|f (g) − f (gi−1 g)|

g∈G

! !     1  1  = f (g) = χr (g)dr = χr (g)dr n n n n 0 0 g∈G

=



 n

!

It follows that

1

g∈G

g∈G

|Fr |dr.

0

! 0

n 1 i=1

! |Fr gi Fr |dr <  0

1

|Fr |dr,

n and thus there is 0 < r ≤ 1 such that i=1 |Fr gi Fr | < |Fr |. This Fr witnesses the Følner condition for G. Now for the second part of the theorem assume that G satisfies the Følner condition. Enumerate G as g0 , g1 , . . . , gn , . . . and for each n ∈ ω let Fn ⊆ G be

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a finite subset of G such that |Fn gi Fn | < 2−n |Fn |. Let U be a nonprincipal ultrafilter on ω. For A ⊆ G, define μ(A) = lim

n∈U

|A ∩ Fn | . |Fn |

Then μ is a finitely additive probability measure on G. It remains to check that μ is left-invariant. For this let g ∈ G and A ⊆ G. Then | |gA ∩ Fn | − |A ∩ Fn | | |Fn | " " " |A ∩ g −1 Fn | − |A ∩ Fn | " = lim n∈U |Fn | " −1 " "g Fn Fn " ≤ lim = 0. n∈U |Fn |

|μ(gA) − μ(A)| = lim

n∈U

Thus G is amenable. It is easy to see that Z satisfies the Reiter condition. In fact for any integer N > 1 let fN : Z → R≥0 be defined as  fN (n) =

1/N, if 0 < n ≤ N , 0, otherwise.

Then for any finite collection of elements of Z one of the functions fN witnesses the Reiter condition. It follows that Z is amenable. Proposition 7.4.4 F2 is not amenable. Proof. Assume that μ is a left-invariant finitely additive probability measure on F2 . Let a and b be the generators of F2 . For each s ∈ {a, b, a−1 , b−1 } let A(s) be the set of all elements of F2 whose reduced form starts with s. Then A(a), A(b), A(a−1 ), A(b−1 ) form a partition of F2 . Also we have that A(a) = a(A(b) ∪ A(b−1 ) ∪ A(a)). Since μ is left-invariant, we have that μ(A(a)) = μ(A(b) ∪ A(b−1 ) ∪ A(a)) = μ(A(b)) + μ(A(b−1 )) + μ(A(a)). It follows that μ(A(b)) = μ(A(b−1 )) = 0. Similarly we also obtain that μ(A(a)) = μ(A(a−1 )) = 0. But then μ(F2 ) = 0, a contradiction. We now turn to equivalence relations. The following definition of amenable equivalence relations is inspired by the Reiter condition.

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Definition 7.4.5 Let X be a standard Borel space, μ a probability Borel measure on X, and E a countable equivalence relation. We say that (X, E, μ) is amenable, or E is μ-amenable, if there is a sequence of Borel functions ϕn : E → R≥0 such that (a) for any x ∈ X,

 yEx

ϕn (x, y) = 1, and

(b) there is a Borel E-invariant set A ⊆ X with μ(A) = 1 such that for all (x, x ) ∈ E ∩ (A × A), 

|ϕn (x, y) − ϕn (x , y)| → 0 as n → ∞.

yEx

Proposition 7.4.6 Let G be a countable group, X a Borel G-space, and μ a Borel probability X measure on X. If G satisfies the Reiter condition, then EG is μ-amenable. Proof. Enumerate . . . and for each n ≥ 1 let fn : G → G as g0 , g1 , . . . , gn , R≥0 be such that g∈G f (g) = 1 and g∈G |f (g) − f (gi−1 g)| < 1/n for all i < n. Then for x, y ∈ X let  ϕn (x, y) = fn (g). g·y=x



  Then for every x ∈ X, yEx ϕn (x, y) = g∈G fn (g) = 1. If xEx and  x = h · x , then for any yEx,    fn (g) − fn (g  ) = fn (g) − fn (h−1 g), ϕn (x, y) − ϕn (x , y) = g·y=x

and



g ·y=x

|ϕn (x, y) − ϕn (x , y)| ≤

yEx



g·y=x

|fn (g) − fn (h−1 g)| → 0

g∈G

X is μ-amenable. as n → ∞. Thus EG

In particular, every hyperfinite equivalence relation is μ-amenable for any Borel probability measure μ. We will show below that the amenability of an orbit equivalence relation can be pulled back to conclude the amenability of the group. For this we need the notion of G-invariance for measures, as follows.

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Definition 7.4.7 Let G be a countable group, X a Borel G-space, and μ a Borel probability measure on X. We say that μ is G-invariant if for all g ∈ G and Borel S ⊆ X, μ(S) = μ(g · S). Theorem 7.4.8 Let G be a countable group, X a Borel G-space, and μ a Borel probability X is measure on X. Suppose the action is free and μ is G-invariant. If EG μ-amenable, then G is amenable. Proof. It suffices to show that G satisfies the Reiter condition. Let ϕn be X . Define Borel functions witnessing the μ-amenability of EG ! fn (g) = ϕn (x, g · x)dμ(x).   Then g∈G fn (g) = g∈G ϕn (x, g · x)dμ(x) = 1. In addition, for every h ∈ G, by the assumptions we have  |fn (g) − fn (h−1 g)| g∈G ! !  = | ϕn (x, g · x)dμ(x) − ϕn (x, h−1 g · x)dμ(x)| g∈G ! !  = | ϕn (x, g · x)dμ(x) − ϕn (h · x, g · x)dμ(x)| g∈G !  ≤ |ϕn (x, g · x) − ϕ(h · x, g · x)|dμ(x) g∈G !  = |ϕn (x, g · x) − ϕn (h · x, g · x)|dμ(x) ! g∈G  = |ϕn (x, y) − ϕn (h · x, y)|dμ(x) → 0 yEx

as > 0 and g1 , . . . , gm ∈ G, then for large enough n we have  n → ∞. Now if  −1 |f (g) − f (g g)| <  for all 1 ≤ i ≤ m. This shows that G satisfies n n i g∈G the Reiter condition, and is therefore amenable. The usual product measure on 2G is a canonical example of a G-invariant measure (Exercise 7.4.5). To apply the above theorem we check below that its free part has full measure. Lemma 7.4.9 Let G be a countably infinite group, X = 2G with the shift action of G, and μ the usual product measure on X. Let F be the free part of X. Then μ(F ) = 1.

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Proof. We show that for any g ∈ G, the set {x ∈ 2G : g · x = x} has measure 0. Consider two cases. Case 1: g is of infinite order, that is, for any n = m ∈ Z, g n = g m . In this case note that if g · x = x then for all n ∈ Z, we have x(g 2n ) = x(g 2n+1 ). Thus # $ μ ({x : g · x = x}) ≤ μ {x : ∀n ∈ Z x(g 2n ) = x(g 2n+1 )} =

 n∈Z

# $ 1 = 0. μ {x : x(g 2n ) = x(g 2n+1 )} = 2 n∈Z

Case 2: g has finite order, that is, for some n ∈ Z, g n = 1G . In this case let H = g ≤ G. Then H is a finite subgroup of G. Since G is infinite, there are infinitely many right-cosets of H in G. Let g0 , g1 , . . . be chosen from infinitely many distinct cosets of H in G. Note that if g · x = x then for all n ∈ ω, we have x(gn ) = x(ggn ). Moreover, for distinct n, m ∈ ω, {gn , ggn } ∩ {gm , ggm } = ∅. Thus similar to Case 1 we have μ ({x : g · x = x}) ≤ μ ({x : ∀n ∈ ω x(gn ) = x(ggn )}) =



μ ({x : x(gn ) = x(ggn )}) =

n∈ω

1 = 0. 2

n∈Z

We are finally ready to prove the main theorem of this section. Theorem 7.4.10 E∞ is not hyperfinite. In particular, E0
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Exercise 7.4.4 Prove that if G and H both satisfy the Reiter condition then so does G × H. Exercise 7.4.5 Let G be a countable group, X = 2G with the shift action of G, and μ the usual product measure on X. Show that μ is G-invariant. Exercise 7.4.6 Let F2 be the free part of 2F2 with the shift action of F2 . Let E∞T denote the equivalence relation E∞ restricted to F2 . Show that E∞T is not hyperfinite.

7.5

Actions of locally compact Polish groups

In this section we prove a theorem of Kechris [92] that any orbit equivalence relation of an action of a locally compact Polish group is essentially countable, that is, Borel reducible to a countable Borel equivalence relation. We will use the following theorem, which is a generalization of the Luzin–Novikov uniformization theorem (Theorem 7.1.2) to sets with Kσ sections. Recall that a subset of topological space is Kσ if it is a union of countably many compact sets. Theorem 7.5.1 (Arsenin–Kunugui) Let Y be a Polish space, X a standard Borel space, and P ⊆ X × Y Borel. Suppose each section Px is Kσ . Then projX (P ) is Borel and P has a Borel uniformization. A proof can be found in Reference [97] Theorem 35.46. Also recall that a complete section of an equivalence relation E on X is a subset Y of X which meets every E-equivalence class. We say that a complete section Y is countable if Y ∩ [x]E is countable for every x ∈ X. Theorem 7.5.2 (Kechris) Let G be a locally compact Polish group and X a Borel G-space. Then there X . is a Borel countable complete section of EG Proof. In view of Theorem 4.4.6 we may assume X is a Polish G-space without loss of generality. Let d be a compatible metric on X. Let K0 be a compact nbhd of 1G and K a compact symmetric nbhd (that is, K = K −1 ) of 1G with K 2 ⊆ K0 . Let N = Int(K). Consider the following relation defined on X: R(x, y) ⇐⇒ ∃g ∈ K (g · x = y).

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R is symmetric and reflexive, but not necessarily an equivalence relation on X. However, we will construct a sequence of Borel sets Xn ⊆ X with X = n Xn such that R  Xn is an equivalence relation. For  > 0 put A = {x ∈ X : ∀g ∈ K0 ( d(x, g · x) ≤  ⇒ g ∈ N Gx ) }. Recall that Gx is the stabilizer of x. We check that each A is Borel. Note that x ∈ A ⇐⇒ ∃g P (x, g), where P (x, g) ⇐⇒ g ∈ K0 − N Gx and d(x, g · x) ≤ . By the Arsenin–Kunugui uniformization theorem it suffices to show that P is Borel and that each section Px is Kσ . It is clear that every section Px is Kσ since G as a locally compact Polish group is Kσ and each Px is closed in G. To see that P is Borel, note that the conditions g ∈ K0 and d(x, g · x) ≤  are closed. Hence it is enough to see that g ∈ N Gx is a Borel condition. For this we note that g ∈ N Gx ⇐⇒ ∃h ∈ Gx ∃k ∈ N ( g = hk ) ⇐⇒ ∃h∃k Q(g, x, h, k), where Q(g, x, h, k) ⇐⇒ h · x = x ∧ k ∈ N ∧ g = hk. Q is obviously Borel, with sections Qg,x Fσ in G × G and hence Kσ . Thus again by the Arsenin–Kunugui theorem we have that g ∈ N Gx is Borel, which in turn implies that A is Borel.

Next we claim that X = n≥1 A1/n . Toward a contradiction assume x ∈ A1/n for any n ≥ 1. We obtain a sequence gn ∈ K0 such that d(x, gn ·x) ≤ 1/n but gn ∈ N Gx . By compactness of K0 there is an accumulation point g∞ ∈ K0 of gn . Then g∞ · x = x but g∞ ∈ N Gx since N Gx is open. This is a contradiction since 1G ∈ N . Next we note that if B ⊆ A has diameter ≤  then R  B is an equivalence relation. If suffices to check that R  B is transitive. For this let x, y, z ∈ B with R(x, y) and R(y, z). So there are g, h ∈ K with g · x = y and h · y = z. Since K 2 ⊆ K0 , hg ∈ K0 and d(x, hg · x) = d(x, z) ≤ diam(B) ≤ . Thus hg ∈ N Gx . This means that there is k ∈ N ⊆ K with z = k · x, and hence R(x, z). For each n ≥ 1 let {Bn,m : m ∈ ω} be a countable open cover of X with diam(Bn,m ) ≤ 1/n. Then the collection {A1/n ∩ Bn,m : n ≥ 1, m ∈ ω} is a countable Borel cover of X so that the restriction of R on each set is an equivalence relation. Let Xn enumerate this collection of Borel sets.

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We claim that R  Xn is smooth. To see this note that R is closed in X ×X. Let τ be a Polish topology on X extending the topology induced by d so that Xn is clopen in (X, τ ). Then R is still closed in τ × τ . Now R  Xn is a closed equivalence relation on the Polish space (Xn , τ  Xn ), therefore it is smooth. X By Proposition 5.4.10 EG is idealistic, that is, there is a Borel map C → IC X assigning to each E -equivalence class C a σ-ideal IC so that C ∈ IC G

. Now X since X = n Xn , for each EG equivalence class C we also have C = n (C ∩ Xn ). It follows that there is an n ∈ ω such that C ∩ Xn ∈ IC . Let n(C) be the least such n for the class C. Put  Y0 = [x]G ∩ Xn([x]G ) . x∈X

Then Y0 is Borel since for C = [x]G , x ∈ Y0 ⇐⇒ ∃n [ ∀m < n ( C ∩ Xm ∈ IC ) ∧ C ∩ Xn ∈ IC ∧ x ∈ Xn ]. X It is clear that Y0 is a complete section for EG . Let Y0,n = Y0 ∩ Xn . Then {Y0,n : n ∈ ω} is a partition of Y0 . Since R  Xn is a smooth equivalence relation, it follows that R  Y0,n is also a smooth equivalence relation, and X therefore so is R  Y0 . It is clear that each EG  Y0 equivalence class in Y0 contains only countably many R  Y0 -classes. Define Y ⊆ Y0 by

x ∈ Y ⇐⇒ x ∈ Y0 ∧ [x]RY0 ∈ I[x]G . X . We still have that R  Y is Then again Y is a Borel complete section of EG smooth. Now R  Y is idealistic, witnessed by the assignment

[x]RY −→ I[x]G  [x]RY . By Theorem 5.4.11 R  Y has a Borel selector as well as a Borel transversal. X , A is a complete Let A be a Borel transversal of R  Y on Y . Since R ⊆ EG X X section of EG . Each EG  Y class in Y contains only countably many R  Y X , as required. classes. Thus A is a Borel countable complete section for EG Corollary 7.5.3 X Let G be a locally compact Polish group and X a Borel G-space. Then EG X is essentially countable, that is, EG ≤B E∞ . X Proof. Let A ⊆ X be a Borel countable complete section of EG . Let X F = EG  A. Then F is a countable Borel equivalence relation on the standard Borel space A. By Theorem 7.1.2 there is a Borel function f : X → A X X x for all x ∈ X. Then f witnesses that EG ≤B F . so that f (x)EG

Exercise 7.5.1 Show that the following are equivalent for a Borel equivalence relation E:

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(a) E is essentially countable; (b) there is a Borel countable complete section of E;

(c) E = n En where each En is smooth and idealistic;

(d) E = n En where each En has a Borel selector. Exercise 7.5.2 Let n ≥ 1. Consider the space F (Rn ) of all closed subsets of Rn . If A, B ∈ F (Rn ) we say that A is congruent to B, denoted A ∼ B, if there is an isometry from A onto B. Show that the congruence relation on F (Rn ) is essentially countable.

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Chapter 8 Borel Equivalence Relations

In this chapter we continue our introduction of Borel equivalence relations. The primary classification here is orbit equivalence relations versus equivalence relations not reducible to orbit equivalence relations. Again we only give a self-contained account of the basic facts. The more interested reader should go beyond this book and find a vast literature about the subject. Along with many theorems there are also a large number of open problems. Many of the problems are fundamental, the answers of which might potentially redefine the subject.

8.1

Hypersmooth equivalence relations

The concept of a hypersmooth equivalence relation is a natural generalization of hyperfiniteness. Definition 8.1.1 Let X be a standard Borel space. An equivalence relation E on X is hypersmooth if E = n En , where En are smooth equivalence relations and En ⊆ En+1 for all n. Similar to the equivalence relation E0 we define a canonical hypersmooth equivalence relation E1 . Definition 8.1.2 The equivalence relation E1 is defined on 2ω×ω as xE1 y ⇐⇒ ∃m ∀n ≥ m ∀k x(n, k) = y(n, k). It is easy to check that E1 is hypersmooth. We can also consider the eventual agreement for sequences of actual real numbers. Let E1 (R) denote the equivalence relation on Rω defined by (xn )E1 (yn ) ⇐⇒ ∃m ∀n ≥ m xn = yn . It is easy to see that E1 (R) is also hypersmooth and that E1 ∼B E1 (R). For notational simplicity we also write x = (xn ) for x ∈ 2ω×ω and xn ∈ 2ω , where x(n, k) = xn (k) for all n, k ∈ ω.

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We have the following closure properties for hypersmooth equivalence relations. The proof is left to the reader (Exercise 8.1.1). Proposition 8.1.3 Let X, Y be standard Borel spaces and E, F equivalence relations on X, Y , respectively. Then the following are true: (1) If F is hypersmooth and E ≤B F , then E is hypersmooth. (2) If E is hypersmooth and A ⊆ X is Borel, then E  A is hypersmooth. (3) If E and F are hypersmooth, then so is E × F . The following characterization of hypersmoothness is analogous to the one on hyperfiniteness. Proposition 8.1.4 Let X be a standard Borel space and E an equivalence relation on X. Then the following are equivalent: (i) E is hypersmooth. (ii) E ≤B E1 . (iii) E B E1 . Proof. By Proposition 8.1.3 (1) we have that (iii)⇒(ii)⇒(i). It suffices to

show (i)⇒(iii). For this let E be hypersmooth. Let E = n Fn where Fn is smooth and Fn ⊆ Fn+1 for all n ∈ ω. Without loss of generality assume that F0 = id(X). Let fn : X → 2ω witness that Fn ≤B id(2ω ). Then in particular f0 is a Borel injection from X into 2ω . Define f : X → 2ω×ω by f (x)(n, k) = fn (x)(k). Then f is a Borel injection and xEy ⇐⇒ f (x)E1 f (y), as required. Hyperfinite equivalence relations are of course hypersmooth. Conversely, we have the following theorem. Theorem 8.1.5 (Dougherty–Jackson–Kechris) Let X be a standard Borel space and E a countable Borel equivalence relation on X. If E is hypersmooth, then it is hyperfinite.

Proof. Let E = n Fn , where each Fn is smooth, Fn ⊆ Fn+1 for all n ∈ ω. Without loss of generality assume F0 = id(X). Each Fn is a countable equivalence relation. Hence by the Feldman–Moore Theorem 7.1.4 there are countX . By Proposiable groups Gn with Borel actions on X so that Fn = EG n tion 5.4.10 and Theorem 5.4.11 there are Borel selectors sn for Fn , that is,

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sn (x)Fn x and sn (x) = sn (y) for all x, y ∈ X with xEn y. Fix an enumeration for Gn as {gn,k }k∈ω . We define a sequence of relations Rn as follows: xRn y ⇐⇒ ∃m ≤ n ( xFm y∧ ∃k0 , . . . , km ≤ n [x = g0,k0 s0 g1,k1 s1 . . . gm,km sm (x)]∧ ∃l0 , . . . , lm ≤ n [y = g0,l0 s0 g1,l1 s1 . . . gm,lm sm (y)]). Then Rn ⊆ Fn and Rn ⊆ Rn+1 . We claim that E = n Rn . Suppose xEy. Then there is m ∈ ω such that xFm y. For every j ≤ m, there is gj,kj ∈ G with x = gj,kj sj (x). Similarly, for each j ≤ m there is gj,lj ∈ G with y = gj,lj sj (y). Then we have that x = g0,j0 s0 g1,j1 s1 . . . gm,km sm (x) and y = g0,l0 s0 g1,l1 s1 . . . gm,lm sm (y). Let n = max{m, k0 , k1 , . . . , km , l0 , . . . , lm }. Then xRn y. Next we check that Rn is an equivalence relation. Clearly Rn is reflexive and symmetric. We only show its transitivity. For this let xRn yRn z. Let m, k0 , . . . , km , l0 , . . . , lm ≤ n witness that xRn y and, without loss of generality, let p ≤ m and h0 , . . . , hp ≤ n be such that yEp z and g0,h0 s0 g1,h1 s1 . . . gp,hp sp (z) = z. If m = p then there is nothing to prove. We assume p < m. Then we have that xFm yFp z and thus xFm z. Note that sm (y) = sm (z) since yFp z and p < m. Let v = gp+1,lp+1 sp+1 . . . gm,lm sm (y) = gp+1,lp+1 sp+1 . . . gm,lm sm (z). Then y = g0,l0 s0 g1,l1 s1 . . . gp,lp sp (v) and hence vFp y. Since yFp z it follows that vFp z, and thus sp (v) = sp (z). Therefore z = g0,h0 s0 g1,h1 s1 . . . gp,hp sp (z) = g0,h0 s0 g1,h1 s1 . . . gp,hp sp (v) = g0,h0 s0 g1,h1 s1 . . . gp,hp sp gp+1,lp+1 sp+1 . . . gm,lm sm (y) = g0,h0 s0 g1,h1 s1 . . . gp,hp sp gp+1,lp+1 sp+1 . . . gm,lm sm (z). This shows that xRn z, and thus Rn is an equivalence relation. Finally note that Rn is finite, since sm (x) = sm (y) for any yFm x. We have thus obtained

an increasing sequence of finite Borel equivalence relations Rn with E = n Rn , and therefore E is hyperfinite. In the rest of this section we analyze the equivalence relation E1 and show that it is not essentially countable. This implies in particular that it is not essentially hyperfinite, and hence E0
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Theorem 8.1.6 (Kechris–Louveau) Let X be a standard Borel space, F a countable Borel equivalence relation on X, and f : 2ω×ω → X a Borel map such that xE1 y ⇒ f (x)F f (y). Then there are x, y ∈ 2ω×ω such that (a) for n = m, xn = xm and yn = ym , (b) for n, m ∈ ω, xn = ym , and (c) f (x) = f (y). In particular, E1 ≤B F . Proof. Let C = n Un ⊆ 2ω×ω where Un ⊇ Un+1 are dense open such that f  C is continuous. We construct sequences (ln ), (kn ), (pn ), (qn ), (xn ), (yn ) such that the following conditions are met: (i) ln , kn ∈ ω, ln < ln+1 , kn < kn+1 for all n ∈ ω; (ii) pn , qn ∈ 2ln ×kn ; (iii) Npn , Nqn ⊆ Un , where Ns = {x ∈ 2ω×ω : s ⊆ x} for any s ∈ 2l×k with l, k ∈ ω; (iv) xn , yn ∈ 2ln ×ω with pn ⊆ xn ⊆ xn+1 , qn ⊆ yn ⊆ yn+1 ; for i = j < ln , (xn )i = (xn )j , (yn )i = (yn )j ; for any i, j < ln , (xn )i = (xn )j ; (v) ∀∗ z ∈ 2ω×ω (xn  z ∈ C ∧ yn  z ∈ C); (vi) ∃∗ z ∈ 2ω×ω f (xn  z) = f (yn  z).

Granting this construction, we can let x = n xn and y = n yn . Then x and y satisfy conditions (a) and (b) of the theorem, and by (iii) x, y ∈ C. Let zn ∈ 2ω×ω be such that xn  zn , yn  zn ∈ C and f (xn  zn ) = f (yn  zn ). Such zn exists by (v) and (vi). Then xn  zn → x and yn  zn → y as n → ∞, and by the continuity of f on C we get that f (x) = f (y). To show that the construction is possible, we need to use the following claim. Claim. Let p ∈ 2l×k and ϕ : D → X, where D ⊆ Np is comeager in Np , such that for x, y ∈ D, xE1 y ⇒ ϕ(x)F ϕ(y). Then there is p ≥ p and r both in 2<ω×<ω such that ∀∗ u ∈ Np ∀∗ v ∈ Np ∀∗ z ∈ Nr ϕ(u z) = ϕ(v  z). To prove the claim, consider a family of functions defined as follows. For x ∈ 2ω×ω define ψx : 2l×ω → X by letting ψx (u) = ϕ(u x) wherever ϕ is defined. For a comeager set of x in 2ω×ω , ψx is well defined on a comeager set of u in Np . Since for u, v ∈ 2l×ω , ux E1 v x for any x ∈ 2ω×ω , we also get that ψx (u) = ϕ(u x)F ϕ(v  x) = ψx (v).

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Therefore, whenever ψx is defined, its range is a countable set. Thus there are px ⊇ p in 2l×<ω such that ψx is constant on a comeager subset of Npx in 2l×ω . A similar argument shows that there is r ∈ 2<ω×<ω such that for a comeager set of x in Nr the map x → px is constant. Let p be this constant value of px for comeager many x ∈ Nr . Thus ∀∗ x ∈ Nr ∀∗ u ∈ Np ∀∗ v ∈ Np ϕ(u x) = ϕ(v  x). The claim follows by the Kuratowski–Ulam theorem. We now turn to the construction by induction on n. Assume that ln , kn , pn , qn , xn , and yn have been constructed to satisfy (i) through (vi). By (v) and (vi) there is r ∈ 2<ω×<ω such that $ # ∀∗ z ∈ Nr xn  z, yn  z ∈ C ∧ f (xn  z) = f (yn  z) . 



Without loss of generality assume r ∈ 2l ×k for some l > 0 and k  > kn .  Also by extending we may assume that if we let s = xn  (ln × k  ) r then ∗ l ×ω ∗ ω×ω ∀ z∈2 Ns ⊆ Un+1 . Then it follows that ∀ u ∈ Nr ∩ 2 f (xn  u z) = f (yn  u z). Now define ϕ(x) = f (xn  x) = f (yn  x) for x = u z where u ∈ Nr and z ∈ 2ω×ω . Then on a comeager set of Nr ϕ is well defined. It certainly satisfies that xE1 y ⇒ ϕ(x)F ϕ(y) since f satisfies it. Thus by the above claim we can find extensions r ⊇ r such that ∀∗ u ∈ Nr ∀∗ v ∈ Nr ∀∗ z ∈ 2ω×ω : f (xn  u z) = f (yn  v  z). Now it is easy to extend r to pn+1 and qn+1 so that for any u ⊇ pn+1 and v ⊇ qn+1 , condition (iv) holds for u, v. Let xn+1 ⊇ pn+1 and yn+1 ⊇ qn+1 be witnesses in the appropriate comeager set. Then (v) and (vi) are satisfied. This completes the construction as well as the proof of the theorem. Kechris and Louveau [102] have shown a dichotomy theorem that any hypersmooth equivalence relation is either essentially hyperfinite or else Borel bireducible with E1 . The proof uses the Gandy–Harrington topology as in the proof of the Glimm–Effros dichotomy theorem. This penetrating theorem completely determines the Borel reducibility structure of hypersmooth equivalence relations. A by-product of their proof is that E1 is not Borel reducible to any idealistic Borel equivalence relation. A further result of theirs states X (even if it that E1 is not Borel reducible to any orbit equivalence relation EG is non-Borel). These results suggest the possibility that idealistic equivalence relations coincide with the orbit equivalence relations. We will give a proof of this last result, that E1 is not Borel reducible to any orbit equivalence relation, later in Section 10.6 (Theorem 10.6.1) as an application of the techniques developed in Chapter 10.

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Exercise 8.1.1 Prove Proposition 8.1.3. Exercise 8.1.2 Recall that the tail equivalence relation Et on 2ω is defined by xEt y ⇐⇒ ∃n∃m∀k ( x(n + k) = y(m + k) ). Show that Et is hyperfinite but not smooth. Thus E ∼B E0 . Exercise 8.1.3 Show that a hypersmooth equivalence relation is essentially countable iff it is idealistic. (Hint: Use Exercise 7.5.1.)

8.2

Borel orbit equivalence relations

In Section 3.4 we already analyzed some conditions guaranteeing Borelness of orbit equivalence relations. For instance, if a Borel action of a Polish group is free then the orbit equivalence relation is Borel. We now give some more characterizations of Borelness of orbit equivalence relations. Recall the following notation about group actions. Let G be a Polish group and X a Borel G-space. For x ∈ X the stabilizer of x is the closed subgroup Gx = {g ∈ G : g · x = x}. For x, y ∈ X we denote Gx,y = {g ∈ G : g · x = y}. Then Gx,y is a left coset of Gx since for any g, h ∈ Gx,y , h−1 g ∈ Gx . We view Gx and Gx,y as elements of the Effros Borel space F (G) of all closed subsets of G. Theorem 8.2.1 (Becker–Kechris) Let G be a Polish group and X a Borel G-space. Then the following are equivalent: X (i) EG is Borel.

(ii) The map x → Gx from X into F (G) is Borel. (iii) The map (x, y) → Gx,y from X × X into F (G) is Borel.

X y Proof. We show that (i)⇒(ii)⇒(iii)⇒(i). (iii)⇒(i) is obvious since xEG iff Gx,y = ∅. We next show (ii)⇒(iii). For this let U ⊆ G be nonempty open. Then

Gx,y ∩ U = ∅ ⇐⇒ ∃g ∈ G (g · x = y ∧ Gx ∩ g −1 U = ∅) ⇐⇒ Gx,y = ∅ ∧ ∀g ∈ G (g · x = y ⇒ Gx ∩ g −1 U = ∅)

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The computation shows that the condition is both Σ11 and Π11 , and is therefore Borel. X It remains to show (i)⇒(ii). For this we assume EG is Borel. We use the X ⊆ X × X, to obtain uniform version of Theorem 4.4.6, Exercise 4.4.4, for EG a Borel set B ⊆ X ×X ×ω with the following properties. For y ∈ X and n ∈ ω let By,n = {x ∈ X : (x, y, n) ∈ B} and let σy be the topology generated by the countable collection {By,n : n ∈ ω}. Then for each y ∈ X, By,0 = [y]G , and (X, σy ) is a Polish G-space. We may assume that {By,n : n ∈ ω} is a countable base for σy . We are now ready to show that the map x → Gx from X into F (G) is Borel. For this we fix a countable base U = {Um } of nonempty open sets for −1 Uk : the topology of G. It is easy to see that the countable collection {Um m, k ∈ ω} is still a base for the topology of G. Moreover, −1 Uk = ∅ ⇐⇒ Um Gx ∩ Uk = ∅. Gx ∩ Um

Thus, in order to finish the proof, it suffices to show that for any nonempty open U, V ⊆ G, the set {x ∈ X : U Gx ∩ V = ∅} is Borel. For this we claim that Um V → x ∈ Bx,n ). U Gx ∩ V = ∅ ⇐⇒ ∃m ∀n (Um ⊆ U ∧ x ∈ Bx,n

Suppose U Gx ∩ V = ∅ and let g0 ∈ Gx and h0 ∈ U with h0 g0 ∈ V . Let Um ⊆ U with h0 ∈ Um and Um g0 ⊆ V . Now for any A ⊆ X, x ∈ A Um ⇐⇒ ∃∗ h ∈ Um (h · x ∈ A) ⇐⇒ ∃∗ h ∈ Um g0 ⊆ V (h · x ∈ A) =⇒ x ∈ A V . Thus the direction (⇒) follows with A = Bn,x for any n ∈ ω. For the direction (⇐) assume U Gx ∩V = ∅. Then U ·x∩V ·x = ∅. Consider the topology σx on X. (X, σx ) is a Polish G-space and [x]G is σx -open. By Effros’ Theorem 3.2.4 the map gGx → g · x is a homeomorphism from G/Gx onto [x]G . It follows that U · x = U Gx · x is σx -open. Since {Bx,n : n ∈ ω} is a countable base for V σx , there exists n ∈ ω such that x ∈ Bx,n ⊆ U · x. It follows that x ∈ Bn,x . This completes the proof of the claim. The following is another useful characterization of Borelness of orbit equivalence relations. Theorem 8.2.2 (Sami) X Let G be a Polish group and X a Polish G-space. Then EG is Borel iff there 0 is α < ω1 such that every G-orbit is Πα . X X Proof. If EG is Borel then there is α < ω1 such that EG is a Π0α subset of X X × X. Then every G-orbit is a section of EG , and hence is Π0α . Conversely,

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consider for any α < ω1 the equivalence relation Eα defined by xEα y ⇐⇒ for all Π0α invariant subsets A ⊆ X, x ∈ A iff y ∈ A. X If every G-orbit is Π0α then Eα = EG . Hence it suffices to show that Eα is 1 Π1 for all α < ω1 . For this we use the fact that there is a Π0α subset U of ω ω × X which is universal for Π0α subsets of X (see Exercise 1.5.6). Consider the action of G on ω ω × X defined by

g · (z, x) = (z, g · x). Then the action is still continuous. Now let T = U ∗G ⊆ ω ω × X. Then for any z ∈ ω ω and x ∈ X, x ∈ Tz ⇐⇒ ∀∗ g ∈ G g ·(z, x) ∈ U ⇐⇒ ∀∗ g ∈ G g ·x ∈ Uz ⇐⇒ x ∈ (Uz )∗G . We claim that {Tz : z ∈ ω ω } consists of all Π0α invariant subsets of X. Since Tz = (Uz )∗G and Uz ∈ Π0α , Tz is Π0α invariant. Conversely, if A is Π0α invariant, then there is z ∈ ω ω with Uz = A, and furthermore, Tz = (Uz )∗G = A∗G = A. This finishes the proof of the claim. We thus have that xEα y ⇐⇒ ∀z ∈ ω ω (x ∈ Tz ↔ y ∈ Tz ), and hence Eα is Π11 . The assumption of a Polish group action is crucial in the theorem. It is easy to construct non-Borel equivalence relations with all equivalence classes closed (see Exercise 7.1.2). We give some examples of Borel orbit equivalence relations. By Corollary 7.5.3 any locally compact Polish group action generates an essentially countable equivalence relation, and thus a Borel one. In particular, the following scheme will yield many examples of Borel orbit equivalence relations with potentially distinct complexity in the Borel reducibility hierarchy. Consider an arbitrary uncountable Polish group H. Let G be an arbitrary countable subgroup of H. Then the coset equivalence relation H/G is a countable equivalence relation. Note that G is regarded to have the discrete topology, rather than the subspace topology inherited from H. However, the Borel structures on G generated by these topologies are the same. The following concept is an abstraction of this circumstance. Definition 8.2.3 Let H be a Polish group. A Polishable subgroup G of H is a Borel subgroup of H which admits a Polish topology τ generating the Borel structure on G inherited from that of H, and such that (G, τ ) is a topological group. If H is a Polish group and G is a Polishable subgroup of H, then the translation action of G on H is continuous and the coset equivalence relation

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H/G is obviously a Borel orbit equivalence relation. The seemingly technical requirement of Polishability is actually fulfilled by many examples other than countable subgroups. For example, the classical Banach spaces p (1 ≤ p < ∞) and c0 are Polishable subgroups of the additive group Rω . The Borel reducibility relation among these examples of Borel orbit equivalence relations are extremely complicated. In fact many questions about them remain open. We will discuss them again in the next few sections and the rest of the book. Exercise 8.2.1 Let G be a Polish group. Let S(G) be the set of all closed subgroups of G and C(G) be the set of all (left) cosets of closed subgroups of G. Show that both S(G) and C(G) are Borel subsets of F (G), and hence are standard Borel spaces. Exercise 8.2.2 (Dixmier) Show that for any Polish group G there is a Borel set T ⊆ S(G) × G such that for any H ∈ S(G), TH is a transversal for the left cosets of H. Exercise 8.2.3 Let G be a Polish group and X a Borel G-space. Show that X EG is Borel iff there is a Borel function g : X × X → G such that for all X xEG y, g(x, y) · x = y. Exercise 8.2.4 Show that p (1 ≤ p < ∞) and c0 are Polishable subgroups of Rω .

8.3

A jump operator for Borel equivalence relations

In this section we discuss an operation that will increase the Borel complexity of any nontrivial Borel equivalence relation. Definition 8.3.1 Let E be a Borel equivalence relation on a standard Borel space X. The Friedman–Stanley jump (or simply jump) of E, denoted by E + , is the equivalence relation on X ω defined by (xn )E + (yn ) ⇐⇒ {[xn ]E : n ∈ ω} = {[yn ]E : n ∈ ω}. E + is obviously an equivalence relation. To see that it is Borel, just note that (xn )E + (yn ) ⇐⇒ ∀n∃m(xn Eym ) ∧ ∀m∃n(xn Eym ). Thus in general if E is Σ0α for α < ω1 , then E + is Π0α+1 . It is easy to see that E ≤B E + for any equivalence relation E. We will show later in this

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section that E
Let C ⊆ 2ω × ω ω be Π01 such that x ∈ A ⇐⇒ ∃y ∈ ω ω (x, y) ∈ C.

Let π : 2ω × ω ω → 2ω be the projection to the first coordinate. Then for any z ∈ C, π(z) ∈ A, and in particular, π(z) ∈ LO. Assume toward a contradiction that the described Δ11 set B exists. Then for any u, v ∈ C, if π(u) ∈ WO and π(v) ∈ WO then (π(u), π(v)) ∈ B, and if π(u) ∈ WO and π(v) ∈ WO then (π(u), π(v)) ∈ B. For each Π11 set P we consider a game GP as follows. First let C P ⊆ ω ω ×ω ω be Π01 such that x ∈ P ⇐⇒ ∃y ∈ ω ω (x, y) ∈ C P .

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Then let T P be a tree on ω × ω such that [T P ] = C P . For each x ∈ ω ω , let TxP be the tree on ω defined by s ∈ TxP ⇐⇒ (x  lh(s), s) ∈ T P . Then we have that x ∈ P ⇐⇒ ∀y ∈ ω ω (x, y) ∈ [T P ] ⇐⇒ TxP is well-founded. The game GP is now played as follows: I n II



a1 , u1 , ϕ1 a2 , u2 , ϕ2

where (i) n ∈ ω,  ∈ {0, 1}, a1 , a2 ∈ 2ω ; (ii) if  = 0 then n ∈ a1 but n ∈ a2 ; if  = 1 then n ∈ a2 but n ∈ a1 ; (iii) u1 , u2 ∈ C (thus π(u1 ), π(u2 ) ∈ A ⊆ LO); (iv) for i = 1, 2, ϕi : TaPi → ω is strictly (reverse) order-preserving: if s, t ∈ TaPi and s  t then ϕi (t) <π(ui ) ϕi (s). Then elements a1 , u1 , ϕ1 , a2 , u2 , ϕ2 are regarded as coded by subsets of ω and are understood to be played alternatively by the two players just as in the standard game. Then Player I wins the play iff (π(u1 ), π(u2 )) ∈ B. For each fixed Π11 set P the above game can be equivalently played on ω as a standard game where the payoff set is Borel. Thus by Borel determinacy the game GP is determined. We make some observations about the game GP where P is a singleton. For this we fix an arbitrary Π11 singleton a and replace all notation with superscript P by superscript a. We claim that Player II has a winning strategy in the game Ga . Since Ga is determined, it is enough to show that Player I does not have a winning strategy in Ga . Toward a contradiction assume he does. Consider the following play of the game. At the beginning Player I chooses n ∈ ω. Let Player II play  according to whether n ∈ a. If n ∈ a let  = 1; otherwise let  = 0. Then let Player II play a2 = a. Now Taa is well-founded, hence there exist well-ordering x of ω and strictly order-preserving map ϕ : Taa → ω with ϕ(t) <x ϕ(s) for all s  t in Taa . Let u2 ∈ C be such that π(u2 ) = x and ϕ2 = ϕ. Then the rules (i) through (iv) of the game are obeyed. Let Player I play according to his winning strategy. Then (i) through (iv) hold and (π(u1 ), π(u2 )) ∈ B. Since π(u2 ) ∈ WO the separation property of B requires that π(u1 ) ∈ WO. Thus we have that ϕ1 : Taa1 → ω is a strictly order-preserving map from Taa1 into a well-order coded by π(u1 ). This implies that Taa1 is well-founded, and thus a1 ∈ {a} and in fact a1 = a = a2 . However, by (ii) n ∈ a2 iff n ∈ a1 , and thus a1 = a2 , a contradiction.

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Let σ be a winning strategy of Player II in Ga . σ is coded by a real in ω . Let σ ∗ : ω → {0, 1} be such that σ ∗ (n) =  iff Player II responses with  according to σ when Player I’s first move is n. Apparently σ ∗ is recursive in σ. We note that σ ∗ (n) = 1 iff n ∈ a. To see this suppose n ∈ a but σ ∗ (n) = 0. Then let Player I plays a1 = a (note that (ii) is satisfied), u1 ∈ C with π(u1 ) ∈ WO, ϕ1 : Taa → ω strictly order-preserving into <π(u1 ) , where the choice of u1 and ϕ1 are similar to the argument in the preceding paragraph. Then since Player II wins we have (π(u1 ), π(u2 )) ∈ B, and by the separation property of B this requires that π(u2 ) ∈ WO. By the above argument again we have that a2 = a, a contradiction to (ii). The argument for the situation n ∈ a but σ ∗ (n) = 1 is similar. Finally note that for all Π11 sets P the payoff set for the game GP is Δ11 , hence Σ0α for some α < ω1 . Since there are only countably many Π11 sets there exists β < ω1 such that all payoff sets are Σ0β . Let U ⊆ ω × ω ω be universal for all Σ0β subsets of ω ω . Consider the set Q of all τ ∈ (ω ω )ω such that for any k ∈ ω, τ (k) ∈ ω ω is a winning strategy for either Player I or Player II in the game GUk . Then Q is nonempty Π11 . By Theorem 1.7.7 Q contains a Π11 singleton τ0 . We arrive at a final contradiction to Corollary 1.7.8 by showing that every Π11 singleton is recursive in τ0 . Let a be an arbitrary Π11 singleton. Let k ∈ ω be such that Uk is the payoff set of the game Ga . Then τ0 (k) is a winning strategy for Player II in Ga . Therefore n ∈ a iff τ0 (k)∗ (n) = 1. This shows that a is recursive in τ0 (k)∗ , which is in turn recursive in τ0 , and hence a is recursive in τ0 . ω

Recall that D ⊆ ω ω is Π11 -complete if D is Π11 and for all Π11 sets P ⊆ ω ω there is a continuous function f : ω ω → ω ω such that f −1 (D) = P . The set WO is Π11 -complete, and it is easy to see that a set D is Π11 -complete iff there is a continuous function f : LO → ω ω such that f −1 (D) = WO. With these concepts the above theorem can be generalized and relativized. Theorem 8.3.4 Let D ⊆ ω ω be Π11 -complete and A ⊆ ω ω be Σ11 with D ⊆ A. Then there is no Borel set B ⊆ ω ω ×ω ω such that D×(A−D) ⊆ B and (A−D)×D∩B = ∅. Proof. First note that the proof of the previous theorem can be relativized. Let f : LO → ω ω be continuous such that f −1 (D) = WO. Let A = f −1 (A). Then A ⊆ LO is Σ11 and WO ⊆ A since D ⊆ A. Assume that there is a Borel set B with the separation property. Let B  = {(u, v) ∈ LO × LO : (f (u), f (v)) ∈ B}. Then B  is Borel in LO × LO, and WO × (A − WO) ⊆ B  and (A − WO) × WO ∩ B  = ∅. Let x ∈ ω ω be such that A ∈ Σ11 (x) and B  ∈ Δ11 (x). This contradicts the previous theorem relativized to x. We are now ready to give Harrington’s proof of Friedman’s Borel diagonalization theorem for Borel equivalence relations.

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Theorem 8.3.5 (Friedman) Let E be a Borel equivalence relation on a standard Borel space X. Then there is no Borel function F : X ω → X such that, for all x = (xn ) and y = (yn ) in X ω , (a) if xE + y then F (x)EF (y), and (b) (F (x), xn ) ∈ E for all n ∈ ω.

Proof. Assume that the Borel function F exists with properties (a) and (b). Without loss of generality we may assume that X = ω ω . Let z0 ∈ ω ω be an arbitrarily fixed element. Let A ⊆ 2ω × (ω ω )ω be the set of all elements u = (x, f ) where x ∈ LO and f : ω → ω ω such that, for all n ∈ ω, (i) if n is <x -least then f (n) = z0 ; and (ii) if n is not <x -least, then letting m0 , m1 , . . . enumerate the set {m ∈ ω : m <x n} in the increasing order of natural numbers (possibly with repetitions), we have that f (n) = F (f (m0 ), f (m1 ), . . . ). Because of property (a) we can write f (n) = F (f  {m : m <x n}) without specifying the order in which the elements are enumerated. Note that A is Borel. Let π : 2ω × (ω ω )ω → 2ω be the projection to the first coordinate. Then WO ⊆ π(A) since for any x ∈ WO a function f satisfying (i) and (ii) can be defined by a transfinite induction on the ordinal length of <x . Define D ⊆ A by (x, f ) ∈ D ⇐⇒ (x, f ) ∈ A ∧ x ∈ WO. Then it is clear that D is a Π11 -complete subset of 2ω × (ω ω )ω . We are going to apply Theorem 8.3.4 to the sets D and A, regarding the underlying space as a homeomorphic copy of ω ω . For any u = (x, f ) and v = (y, g) in A, we define a binary relation Ru,v ⊆ ω × ω by nRu,v m ⇐⇒ f (n)Eg(m). Because of property (b) we have that if nRu,v m1 and nRu,v m2 then m1 = m2 . Similarly, if n1 Ru,v m and n2 Ru,v m then n1 = n2 . Thus Ru,v is in fact a partial bijection between two subsets of ω. Let ϕu,v denote the partial bijection given by Ru,v . Define Iu,v ⊆ dom(ϕu,v ) and Ju,v ⊆ range(ϕu,v ) by letting n ∈ Iu,v ⇐⇒ {k : k ≤x n} ⊆ dom(ϕu,v ) ∧ ϕu,v ({k : k ≤x n}) = {l : l ≤y ϕu,v (n)} ∧ ∀k, k  ≤x n (k <x k  → ϕu,v (k)
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Ju,v = ω). Note that B is Borel. We will arrive at a contradiction by checking that D × (A − D) ⊆ B and (A − D) × D ∩ B = ∅. For this let u = (x, f ) and v = (y, g) be in A. First suppose x ∈ WO and y ∈ WO. If Iu,v = ω, then since (Iu,v , <x ) is a well-order and ϕu,v is order-preserving, we get that (Ju,v ,
Proof. It suffices to show that E + ≤B E. Assume toward a contradiction that f : X ω → X is a Borel reduction from E + to E. We will consider the space (X ω )ω and the equivalence relation (E + )+ on it. For notational clarity we denote a typical element of X ω by x = (xn ), where each xn ∈ X, and a typical element of (X ω )ω by x = (xk ), where xk ∈ X ω . We will also omit the subscripts when we denote the equivalence class of an element. This will not cause any confusion. Let C = {x = (xn ) ∈ X ω : ∀n, m xn = xm }. Thus elements of C are constant sequences in X ω . We note that there is x ∈ C such that f (x) = x0 . Otherwise we would have that f (C) = X. This implies that, for every y ∈ X ω , there is some x ∈ C with f (x) = f (y), and thus xE + y since f is a reduction, and in fact yn Ex0 for all n. It follows that there is only one E-equivalence class on X, contradicting our assumption. We fix a z ∈ C with f (z) = z0 . For each x = (xk ) ∈ (X ω )ω let F (x) ∈ X ω be a canonical enumeration (possibly with repetitions) of the set {f (z), f (xk ) : ∀n [f (xk )] = [xkn ]}. Note that F : (X ω )ω → X ω is Borel. We claim that F is invariant, that is, if x(E + )+ y then F (x)E + F (y). For this suppose x(E + )+ y. Let k be such that [f (xk )] = [xkn ] for all n. Let l be such that xk E + y l . Since f is a reduction [f (y l )] = [f (xk )] whereas {[xkn ] : n ∈ ω} = {[ynl ] : n ∈ ω}. Hence [f (y l )] = [ynl ] for all n. By symmetry we have that F (x)E + F (y). Next we claim that for any x, (F (x), xk ) ∈ E + for all k. Assume that for some x and k, F (x)E + xk . Let A = {z, xp : ∀n [f (xp )] = [xpn ]}. We consider two cases. Case 1: there is y ∈ A with f (xk )Ef (y). Then in fact xk E + y,

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[f (xk )] = [f (y)], and {[yn ] : n ∈ ω} = {[xkn ] : n ∈ ω}. This implies that [f (xk )] = [xkn ] for all n, and therefore xk ∈ A. However since F (x)E + xk , [f (xk )] ∈ {[f (y)] : y ∈ A} = {[xkn ] : n ∈ ω}, a contradiction. Case 2: there is no y ∈ A with f (xk )Ef (y). In particular xk ∈ A. Thus for some n, [f (xk )] = [xkn ]. Again by F (x)E + xk it follows that [f (xk )] ∈ {[f (y)] : y ∈ A}, contradicting our case assumption. We have thus constructed a Borel diagonalizer of E + , contradicting Theorem 8.3.5. Exercise 8.3.1 Show that E ≤B E + for any equivalence relation E on a standard Borel space X. Exercise 8.3.2 Show that if E ≤B F then E + ≤B F + . Exercise 8.3.3 Let =+ denote id(ω ω )+ . Show that E∞ ≤B =+ and in fact + ∼B =+ . E∞ Let E be an equivalence relation on a standard Borel space X. Define an equivalence relation E ∗ on X ω by xE ∗ y ⇐⇒ xE + y ∧ ∀z ∈ X |{n ∈ ω : xn = z}| = |{n ∈ ω : yn = z}|. Exercise 8.3.4 Show that E ∗ ∼B E + . Exercise 8.3.5 Let G be a Polish group and X a Polish G-space. Show that X ∗ ) is the orbit equivalence relation induced by an action of G × S∞ on (EG ω X , where S∞ is the infinite permutation group.

8.4

Examples of Fσ equivalence relations

Recall from Theorem 6.4.4 that every Gδ equivalence relation is smooth. Thus from the point of view of descriptive complexity the simplest kind of Borel equivalence relations are Fσ ones. However, this is not to say that the Borel complexity of these equivalence relations in the Borel reducibility hierarchy is easy to describe. In this section we give an overview of some canonical examples of Fσ equivalence relations. We have introduced at least three concrete nonsmooth Fσ equivalence relations: E0 , E∞ , and E1 . Of course, any countable Borel equivalence relation is Fσ . Consider the following additional examples arising from classical Banach spaces.

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Recall that for a real number 1 ≤ p < ∞,   ∞  ω p p = x = (xn ) ∈ R : |xn | < ∞ , n=0

and

 ∞ =

x = (xn ) ∈ R

ω

: sup |xn | < ∞ . n∈ω

All of them are Fσ subgroups of the additive group Rω . Thus for 1 ≤ p ≤ ∞ the coset equivalence relation Rω /p defined by x Rω/p y ⇐⇒ x − y ∈ p is an Fσ equivalence relation on the Polish space Rω . For notational simplicity we also denote the equivalence relation by p . We first note that the equivalence relations p all have nontrivial Borel complexity. Lemma 8.4.1 For any 1 ≤ p ≤ ∞, E0 ≤B p . Proof. We use Theorem 6.2.1. For this note that each p is a group of homeomorphisms on Rω . It is also easy to see that p is dense, and thus the equivalence relation has a dense orbit. It remains to show that the equivalence relation is meager. By Kuratowski–Ulam it suffices to show that every orbit is meager. Since each orbit is a homeomorphic copy of p , it suffices to show that p is meager in Rω . We do this for p = ∞ first. For each N ∈ ω let AN = {x ∈ Rω : supn |xn | ≤ N }. Then AN is closed in Rω and ∞ = N ∈ω AN . We claim that AN is nowhere dense. Otherwise there is a basic open U ⊆ Rω with U ⊆ AN . Without loss of generality assume U = (a0 , b0 ) × (a1 , b1 ) × . . . (an , bn ) × Rω . Consider V ⊆ U defined by V = (a0 , b0 ) × (a1 , b1 ) × . . . (an , bn ) × (N, N + 1) × Rω . Then V ⊆ AN . However, for any x ∈ V we have that |xn+1 | > N and therefore supn |xn | > N , contradicting x ∈ AN . A similar argument shows that p is also meager for 1 ≤ p < ∞. Our next goal is to show that ∞ is the most complex among these equivalence relations. In fact, we show that ∞ is universal among all Kσ equivalence relations on Polish spaces. Recall that a subset of a topological space is Kσ if it is the union of countably many compact sets. If X is a Polish space, E a Kσ equivalence relation on X, and Δ = {(x, x) : x ∈ X}, then Δ is a closed subset of E which is homeomorphic to X, and it follows that X is Kσ .

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Theorem 8.4.2 (Rosendal) Let E be a Kσ equivalence relation on a Polish space X. Then E ≤B ∞ .

Proof.

Let (Xn ) be an increasing sequence of compact subsets of X with X = n Xn . Let Δ = {(x, x) : x ∈ X}. Let (Fn ) be an increasing sequence of compact subsets of X × X with E = n Fn . Without loss of generality we may assume Δ ∩ Xn2 ⊆ Fn . Define an increasing sequence (Rn ) of compact relations by induction as follows. Let R0 = F0 ∩ X02 . Assuming Rn has been defined, let Rn+1 = Rn ∪ (Rn ◦ Rn ) ∪ (Fn ∩ Xn2 ).

It is easy to see that each Rn is compact and that E = n Rn . Let {Um }m∈ω be a countable base for the topology of X. Fix a bijection ·, · : ω × ω → ω such that for all n, m ∈ ω, n, m ≥ n. We define a function ϕ : X → Rω by  k, if k ≤ n is the least such that ∃y ∈ Um (x, y) ∈ Rk , ϕ(x)(n, m) = n, if ∀y ∈ Um (x, y) ∈ Rn . Note that ϕ is a Borel map. To see this it suffices to check that for any m, n ∈ ω, the set {x ∈ X : ∃y ∈ Um (x, y) ∈ Rn } is Borel. We have that ∃y ∈ Um (x, y) ∈ Rn ⇐⇒ ∃l(Ul ⊆ Um ∧ ∃y ∈ Ul ∩ Xn (x, y) ∈ Rn ). The set {x : ∃y ∈ Ul ∩ Xn (x, y) ∈ Rn } is compact, and thus the desired set is in fact Fσ . We claim that ϕ is Borel reduction from E to ∞ . Suppose x, y ∈ X and xEy. Fix the least n0 ∈ ω with (x, y) ∈ Rn0 and (y, x) ∈ Rn0 . Then for any z ∈ X and k ∈ ω, if (x, z) ∈ Rk then (y, z) ∈ Rn0 ◦ Rk ⊆ Rmax{n0 ,k}+1 , and by symmetry, if (y, z) ∈ Rk then (y, z) ∈ Rmax{n0 ,k}+1 . It follows that if ϕ(x)(n, m) = k < n, then ϕ(y)(n, m) ≤ max{n0 , k} + 1, and therefore |ϕ(x) − ϕ(y)| ≤ n0 + 1. By symmetry if ϕ(x)(n, m) < n then |ϕ(x)(n, m) − ϕ(y)(n, m)| ≤ n0 +1. If both ϕ(x)(n, m) = n = ϕ(y)(n, m), then trivially |ϕ(x)(n, m)−ϕ(y)(n, m)| ≤ n0 +1. We have shown that ϕ(x)−ϕ(y) ∈ ∞ . Conversely, suppose ϕ(x) − ϕ(y) ∈ ∞ . Let k be large enough such that y ∈ Xk and |ϕ(x)(N ) − ϕ(y)(N )| ≤ k for all N . Fix an arbitrary m with y ∈ Um . Since Δ ∩ Xk2 ⊆ Fk , (y, y) ∈ Rk+1 and then ϕ(y)(2k + 2, m) ≤ k + 1. It follows that ϕ(x)(2k + 2, m) ≤ 2k + 1, and thus there is z ∈ Um with (x, z) ∈ Rk+1 . We have shown that, for any m with y ∈ Um there is z ∈ Um with (x, z) ∈ Rk+1 . Thus we have a sequence (zm ) with zm → y and (x, zm ) ∈ Rk+1 . Since Rk+1 is compact, we have that (x, y) ∈ Rk+1 , and therefore (x, y) ∈ E. It is easy to see that all of the equivalence relations E0 , E∞ , E1 , and p (1 ≤ p ≤ ∞) are Kσ (see Exercise 8.4.3). From this we immediately get that

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all of them are Borel reducible to ∞ . It is not known if ∞ is universal among all Fσ equivalence relations. Before closing this section we mention some results without proofs. Dougherty and Hjorth [29] have shown that the Borel reducibility order
∞ 

1/p |xn |

p

.

n=0

This makes the equivalence relation p an orbit equivalence relation of a Polish group action. In particular, all p (1 ≤ p < ∞) are idealistic. Since E1 is not Borel reducible to any orbit equivalence relations (Theorem 10.6.1), E1 ≤B p for 1 ≤ p < ∞. On the other hand we just showed that E1 ≤B ∞ , and so the equivalence relation ∞ is not an orbit equivalence relation of a Polish group action. This implies that topological group ∞ is not a Polishable subgroup of Rω , which can also be seen directly (see Lemma 9.3.3). For now we note that, with the usual norm x∞ = sup |xn |, n∈ω

the space ∞ is not separable. Exercise 8.4.1 Show that for any 1 ≤ p ≤ ∞, p is an Fσ subset of Rω . Exercise 8.4.2 Show that p is meager in Rω for 1 ≤ p < ∞. Exercise 8.4.3 Show that all equivalence relations E0 , E∞ , E1 , and p (1 ≤ p ≤ ∞) are Kσ .

8.5

Examples of Π03 equivalence relations

We define the power operator on equivalence relations. Definition 8.5.1 Let E be an equivalence relation on a standard Borel space X. The power of E is the equivalence relation E ω on X ω defined by xE ω y ⇐⇒ ∀n ∈ ω xn Eyn .

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Apparently E ≤B E ω and E ω ≤B F ω if E ≤B F . However, (E ω )ω ∼B E ω . Thus the power operator is not a jump operator. If E is Π0α for α < ω1 , then E ω is also Π0α . Thus the powers of all Fσ equivalence relations we considered in the last section are examples of Π03 equivalence relations, among which E0ω has the simplest Borel complexity. It is easy to see that if E is an orbit equivalence relation induced by an action of a Polish group G, then E ω is induced by an action of Gω . We show that E0ω is not Borel reducible to any Σ03 equivalence relation. It follows that the powers of all other equivalence relations have the same property. Theorem 8.5.2 Let E be a Σ03 equivalence relation on a Polish space X. Then E0ω ≤B E. Proof. Assume toward a contradiction that f : (2ω )ω → X is a Borel reduction of E0ω to E. Since f is Borel, and hence Baire measurable, there is a comeager set C ⊆ (2ω )ω such that f  C is continuous. Without loss of generality we may assume C is Gδ . Then E0ω ∩ C 2 is Σ03 , since (x, y) ∈ E0ω ∩ C 2 ⇐⇒ x, y ∈ C ∧ (f (x), f (y)) ∈ E. To arrive at a contradiction it suffices to establish the following claim: For every dense Gδ set C ⊆ (2ω )ω , E0ω ∩ C 2 is Π03 -complete. Fix any bijection ·, · : ω × ω → ω. Let B = {x ∈ 2ω : ∀i ∃j ∀k ≥ j x(i, k) = 1}. Then B is Π03 -complete. To see this, let A ⊆ 2ω be Π03 , say  A= Ai,j,k , i

j

k

where each Ai,j,k is clopen. Define ρ : 2ω → 2ω as follows. Given x ∈ 2ω , for each i, k ∈ ω let ax,i,k ∈ ω be the least integer j ≤ k, if one exists, such that ∀k  ≤ k (x ∈ Ai,j,k ). Let  1, if ax,i,k and ax,i,k−1 are both defined and are equal, ρ(x)(i, k) = 0, otherwise. It is easy to see that ρ is continuous and x ∈ A ⇐⇒ ρ(x) ∈ B. This shows that B is Π03 -complete. Write C = n Dn where each Dn is open dense in (2ω )ω . We next define two continuous functions π1 , π2 : 2ω → 2ω×ω so that x → (π1 (x), π2 (x)) reduces the set B to E0ω  C 2 . For this we define two sequences (us )s∈2<ω , (vs )s∈2<ω and a function p : ω → ω so that the following hold: (i) for any n ∈ ω, if n = i, k then i, p(n) + k < p(n + 1);

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(ii) for any n ∈ ω and lh(s) = n, us , vs ∈ 2p(n)×p(n) ; (iii) for s ⊆ t, us ⊆ ut and vs ⊆ vt ; (iv) if lh(s) = n then Nus , Nvs ⊆ Dn ; (v) for any n ∈ ω, if s(n) = 1 then us (j, l) = vs (j, l) for all (j, l) ∈ p(n + 1) × p(n + 1) − p(n) × p(n); (vi) for n = i, k, if s(n) = 0 then us (i, p(n) + k) = vs (i, p(n) + k) and us (j, l) = vs (j, l) for (j, l) ∈ p(n + 1) × p(n + 1) − p(n) × p(n) − {(i, p(n) + k)}. It follows easily from

the density of Dn that

such sequences can be constructed. We let π1 (x) = n uxn and π2 (x) = n vxn . Clearly π1 , π2 are continuous. By (iv) π1 (x), π2 (x) ∈ C for any x ∈ 2ω . If x ∈ B, then for each i let k(i) be such that ∀k ≥ k(i) x(i, k) = 1. Fix i ∈ ω. For any n ≥ i, k(i), if x(n) = 1 then by (v) for any p(n) ≤ j < p(n + 1) we have π1 (x)(i, j) = π2 (x)(i, j); if x(n) = 0 then n = i , k for some i = i, since otherwise k ≥ k(i) and we have x(i, k) = 1 by assumption. In this latter case by (vi) we still have that π1 (x)(i, j) = π2 (x)(i, j) if i < p(n + 1) and p(n) ≤ j < p(n + 1). Thus π1 (x)(i, k) = π2 (x)(i, k) for all k ≥ k0 . This shows that π1 (x) E0ω π2 (x). If x ∈ / B, then for some i we have that x(i, k) = 0 for infinitely many k. Fix such an i. Then π1 (x)(i, p(i, k) + k) = π2 (x)(i, p(i, k) + k) for infinitely many k. This shows that (π1 (x), π2 (x)) ∈ E0ω . We introduce some other Π03 equivalence relations. First consider the classical Banach space c0 = {x ∈ Rω : lim |xn | = 0} n

with the norm  · ∞ . c0 is a subgroup of Rω . We consider the coset ω equivalence relation R /c0 defined by Π03

x Rω/c0 y ⇐⇒ x − y ∈ c0 and still denote it by c0 . Then the equivalence relation c0 is Π03 . Lemma 8.5.3 E0ω ≤B c0 . Proof. Let ·, · : ω × ω → ω be a bijection such that n, m ≤ n , m  if n ≤ n and m ≤ m . Define ϕ : 2ω×ω → Rω by ϕ(x)(n, m) = 2−n x(n, m). Clearly ϕ is a Borel function. We check that it is a reduction from E0ω to c0 . Let xE0ω y, that is, ∀n∃m(n)∀m ≥ m(n) x(n, m) = y(n, m). Let  > 0. Fix

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n0 large enough such that 2−n0 < . Let m0 = max{m(i) : i < n0 }. We show that for all N ≥ n0 , m0 , |ϕ(x)(N ) − ϕ(y)(N )| < . Let N ≥ n0 , m0 . Say N = n, m. If n ≥ n0 then |ϕ(x)(N ) − ϕ(y)(N )| ≤ 2−n ≤ 2−n0 < . If n < n0 then m ≥ m(n) since N = n, m ≥ n0 , m0  ≥ n, m0  ≥ n, m(n). Thus x(n, m) = y(n, m) and ϕ(x)(N ) − ϕ(y)(N ) = 0. This shows that limN ϕ(x)(N ) − ϕ(y)(N ) = 0 and so ϕ(x) − ϕ(y) ∈ c0 . Conversely let (x, y) ∈ E0ω . Fix n such that (xn , yn ) ∈ E0 , that is, x(n, m) = y(n, m) for infinitely many m. Then |ϕ(x)(n, m) − ϕ(y)(n, m)| = 2−n for infinitely many m, and thus limN [ϕ(x)(N ) − ϕ(y)(N )] = 0. This shows that ϕ(x) − ϕ(y) ∈ c0 . Another class of examples of Π03 equivalence relations are the Friedman– Stanley jumps of Fσ equivalence relations. Among these the simplest is the jump of the identity relation on a Polish space, for instance, id(2ω )+ . For notational simplicity we denote this equivalence relation by =+ . Lemma 8.5.4 E0ω ≤B =+ . Proof. It is easy to see that E0+ ∼B =+ (see Exercise 8.3.3). Thus it suffices to show that E0ω ≤ E0+ . Fix a bijection ·, · : ω × ω → ω. Let τ : 2ω → 2ω be defined by  z(j), if i = 0, τ (z)(i, j) = 1, otherwise. For each n ∈ ω define a function ϕn : 2ω → 2ω by  τ (z)(j), if i = n, ϕn (z)(i, j) = 0, otherwise. Then for each n ∈ ω and z1 , z2 ∈ 2ω , we have that z1 E0 z2 iff ϕn (z1 )E0 ϕn (z2 ). Moreover, for n1 = n2 and any z1 , z2 ∈ 2ω , (ϕn1 (z1 ), ϕn2 (z2 )) ∈ E0 since for infinitely many m, ϕn1 (z1 )(n1 , m) = τ (z1 )(m) = 1, but for all m, ϕn2 (z2 )(n1 , m) = 0. We then define ρ : (2ω )ω → (2ω )ω by letting (ρ(x))n = ϕn (xn ). Suppose xE0ω y. Then for all n ∈ ω, xn Eyn , and therefore ϕn (xn )E0 ϕn (yn ). It follows that {[ϕn (xn )]E0 : n ∈ ω} = {[ϕn (yn )]E0 : n ∈ ω}, and thus ρ(x)E0+ ρ(y). Conversely, suppose (x, y) ∈ E0ω . Let n be such that (xn , yn ) ∈ E0 . Then (ϕn (xn ), ϕm (y)) ∈ E0 for all m ∈ ω. Thus [ϕn (xn )]E0 ∈ {[ϕn (yn )]E0 : n ∈ ω} and (ρ(x), ρ(y)) ∈ E0+ . The last equivalence relation we consider in this section is measure equivalence. Let M be the space of Borel probability measures on 2ω . Every μ ∈ M

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is completely determined by the sequence (μ(Ns ))s∈2<ω . Hence M can be viewed as a subspace of [0, 1]ω , and therefore is a Polish space. For μ, ν ∈ M , we say that μ is absolutely continuous with respect to ν, denoted μ  ν, if for any Borel set A ⊆ 2ω , ν(A) = 0 ⇒ μ(A) = 0. Two measures μ and ν are equivalent if both μ  ν and ν  μ. We use ≡m to denote the measure equivalence relation. It is easy to see that μ  ν ⇐⇒ ∀ > 0∃δ > 0 ∀ basic open U (ν(U ) < δ ⇒ μ(U ) < ). It follows that  is a Π03 relation and ≡m is a Π03 equivalence relation. Lemma 8.5.5 =+ ≤B ≡m . Proof. We give an informal sketch of the proof. The exact details of the definitions are left to the reader. Intuitively we will assign to each countable subset S of 2ω a Borel probability measure μS so that S = S  iff μS ≡m μS  . This is easy to do: let μS be any Borel probability measure supported on S. Then for any Borel set A ⊆ 2ω , μ(A) = 0 iff A ∩ S = ∅ iff ν(A) = 0. There are many important facts omitted in our discussion so far. Before closing this section we mention some known facts without proof. The Spectral Theorem in functional analysis essentially established that the measure equivalence relation is Borel bireducible to the orbit equivalence relation of the conjugacy action of the unitary group on itself. In particular this establishes that the measure equivalence is Borel bireducible with an orbit equivalence relation of a Polish group action. This implies (by Theorem 10.6.1) that E1 is not reducible to any of the equivalence relations we discussed in this section. Kakutani studied the measure equivalence relation on product measures. He essentially showed that 2 ≤B ≡m (see Reference [104]). It follows from Theorem 8.4.3 that p ≤B ≡m if 1 ≤ p ≤ 2. It is not known if p ≤B ≡m for any p > 2. Hjorth and Kechris [78] proved a dichotomy theorem that for any E ≤ E0ω , either E ∼B E0ω or E is hyperfinite. Exercise 8.5.1 Let C ⊆ Rω be comeager. Show that c0 ∩C 2 is Π03 -complete. Exercise 8.5.2 Show that (c0 )ω ∼B c0 . Exercise 8.5.3 Give an example of an equivalence relation E such that E +
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Chapter 9 Analytic Equivalence Relations

The study of general analytic equivalence relations is a very important and yet challenging subject. In this chapter we will discuss general dichotomy theorems for analytic equivalence relations, universal analytic equivalence relations, and other examples of analytic but non-Borel equivalence relations. Orbit equivalence relations of Polish group actions are in general analytic, and often non-Borel, and therefore provide many interesting examples and opportunities of applications of general theorems. The Borel reducibility hierarchy for analytic equivalence relations is more complicated than the one for Borel equivalence relations.

9.1

The Burgess trichotomy theorem

The Silver dichotomy theorem for Π11 equivalence relations fails for Σ11 equivalence relations. For an example, consider the equivalence relation Eω1 defined on the standard Borel space LO by xEω1 y ⇐⇒ (x ∈ WO ∧ y ∈ WO) ∨ ∃ϕ : (ω, <x ) ∼ = (ω,
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countably many equivalence classes it is in fact Borel and smooth. Thus having ω1 many equivalence classes is typical of all counterexamples of the Silver dichotomy for Σ11 equivalence relations. To prove the Burgess trichotomy theorem we need to show another theorem of Burguess representing every Σ11 equivalence relation as a intersection of ω1 many Borel equivalence relations. We need also recall some basic notions about ordinals. The notation ω1 is used to represent the least uncountable ordinal as well as the set of all countable ordinals. We also use it to denote the least uncountable cardinal number. The space ω1 of all countable ordinals is endowed with the order topology, where subbasic open sets are of the form {α < ω1 : α < α0 } and {α < ω1 : α > α0 } for α0 < ω1 . A subset C of ω1 is closed if C is closed in the order topology of ω1 , and C is a club if it is closed and unbounded above. The intersection of countably many clubs in ω1 is still a club in ω1 . If f : ω1 → ω1 is a continuous function satisfying f (α) ≥ α for all α < ω1 , then there is a club C ⊆ ω1 such that f (α) = α for all α ∈ C. The following lemma is an immediate corollary of the boundedness principle Theorem 1.6.10. Lemma 9.1.1 Let X be a standard Borel space and f1 , . . . , fn , g1 , . . . , gm : X → LO be Borel −1 functions. Suppose f1−1 (WO) ∩ · · · ∩ fn−1 (WO) ⊆ g1−1 (WO) ∪ · · · ∪ gm (WO). Then there is a club C ⊆ ω1 such that for all α ∈ C, −1 (WOα ). f1−1 (WOα ) ∩ · · · ∩ fn−1 (WOα ) ⊆ g1−1 (WOα ) ∪ · · · ∪ gm

Proof. We define a continuous function α → β(α) as follows. For each α < ω1 , let β(α) be the least β ≥ α such that −1 (WOβ ). f1−1 (WOα ) ∩ · · · ∩ fn−1 (WOα ) ⊆ g1−1 (WOβ ) ∪ · · · ∪ gm

To see that this β(α) is well defined, note that by our assumption −1 (WO). f1−1 (WOα ) ∩ · · · ∩ fn−1 (WOα ) ⊆ g1−1 (WO) ∪ · · · ∪ gm

Let A denote the set on the left-hand side. Then A is a Borel subset of X. It −1 (WO)) is Σ11 . Since follows that A − (g2−1 (WO) ∪ · · · ∪ gm −1 (WO)) ⊆ g1−1 (WO), A − (g2−1 (WO) ∪ · · · ∪ gm

we have that −1 (WO))) ⊆ WO. g1 (A − (g2−1 (WO) ∪ · · · ∪ gm

The set on the left-hand side is Σ11 . Hence by the boundedness principle there is some β1 ≥ α such that −1 (WO)) ⊆ g1−1 (WOβ1 ). A − (g2−1 (WO) ∪ · · · ∪ gm

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Repeating the argument for g2 , . . . , gm , we obtain β2 , . . . , βm ≥ α similarly. Let β = max{β1 , . . . , βm }. Then we have that A ⊆ g1−1 (WOβ ) ∪ · · · ∪ −1 gm (WOβ ). This shows that β(α) is well defined. It is easy to see that α → β(α) is continuous. We can thus get a club C ⊆ ω1 such that for all α ∈ C, β(α) = α. Theorem 9.1.2 (Burgess) Let E be a Σ11 equivalence relation on a standard Borel X. Then there are Borel equivalence relations Eα on X for α < ω1 such that E = α<ω1 Eα .

Proof. Let f : X 2 → LO be a Borel function such that X 2 − E = f −1 (WO). 2 −1 For α < ω1 let A α = X −f (WOα ). Then each Aα is a Borel binary relation on X and E = α<ω1 Aα . We claim that there is a club C ⊆ ω1 such that for all α ∈ C, Aα is an equivalence relation. Clearly the theorem follows from this claim. It is easy to see that every Aα is reflexive. To prove the claim we first show that there is a club C ⊆ ω1 such that for all α ∈ C, Aα is symmetric. For this let f1 (x, y) = f (x, y) and g1 (x, y) = f (y, x) for all (x, y) ∈ X 2 . Then since E is symmetric, f1−1 (WO) = X 2 − E = g1−1 (WO). By applying Lemma 9.1.1 for both f1−1 (WO) ⊆ g1−1 (WO) and g1−1 (WO) ⊆ f1−1 (WO), we get a club C ⊆ ω1 such that for all α ∈ C, f1−1 (WOα ) = g1−1 (WOα ). It follows that Aα is symmetric for all α ∈ C. Next we show that there is a club C ⊆ ω1 such that for all α ∈ C, Aα is transitive. The claim then follows since the intersection of two clubs is still a club. For this define f1 , g1 , g2 : X 3 → LO by f1 (x, y, z) = f (x, z), g1 (x, y, z) = f (x, y), and g2 (x, y, z) = f (y, z). Then for any (x, y, z) ∈ X 3 , (x, y, z) ∈ f1−1 (WO) ⇐⇒ (x, z) ∈ E =⇒ (x, y) ∈ E ∨ (y, z) ∈ E ⇐⇒ (x, y, z) ∈ g1−1 (WO) ∪ g2−1 (WO). This means that f1−1 (WO) ⊆ g1−1 (WO) ∪ g2−1 (WO). Thus by Lemma 9.1.1 there is a club C ⊆ ω1 such that for all α ∈ C, f1−1 (WOα ) ⊆ g1−1 (WOα ) ∪ g2−1 (WOα ). It follows that for α ∈ C, Aα is transitive, since if (x, y), (y, z) ∈ Aα then (x, y, z) ∈ g1−1 (WOα )∪g2−1 (WOα ) and therefore (x, y, z) ∈ f1−1 (WOα ) and so (x, z) ∈ Aα . Recall from Definition 5.4.3 that a collection S of subsets of a standard Borel space X generates the equivalence relation xEy ⇐⇒ ∀S ∈ S (x ∈ S ↔ y ∈ S).

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Definition 9.1.3 An equivalence relation E on a standard Borel space X is ω1 -smooth if E is generated by a set S of ω1 many Borel subsets of X. Theorem 9.1.4 Let E be an ω1 -smooth equivalence relation on a Polish space X. Then either E has ≤ ω1 many equivalence classes or else E has perfectly many classes. Proof. We fix a complete metric for X for the entire proof. Let Bα , α < ω1 , witness that E is ω1 -smooth. Since both Bα and X − Bα are Suslin, we have sequences (Csα,0 )s∈ω<ω , (Csα,1 )s∈ω<ω of closed sets with diam(Csα,i ) < 2−lh(s) for i = 0, 1 and all α < ω1 , such that   α,0   α,1 Bα = Cxn , and X − Bα = Cxn . x∈ω ω n∈ω

x∈ω ω n∈ω

As usual we may assume that Ctα,i ⊆ Csα,i if s ⊆ t. For s ∈ ω <ω we let  α,i  Bsα,i = Cxn . x∈ω ω ,s⊆x n∈ω

Then Bsα,i ⊆ Csα,i . Assume that E has more than ω1 many equivalence classes. Let Z ⊆ X be a set of ω2 pairwise E-inequivalent elements. By induction on n ∈ ω, and for each s ∈ 2n , we construct a set Ds , an ordinal αs < ω1 , and σ(s, m) ∈ ω n for m < n so that the following conditions are satisfied: (i) σ(s, m) ⊆ σ(t, m) if s ⊆ t and m < lh(s); αsm ,s(m) for n > 0; (ii) Ds = m
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which is a finite union of sets of the form  α ,s(m) sm Bσ(s,m) ∩ Btαs ,0 k m m
for km ∈ ω and t ∈ ω n+1 . At least one of these sets contains ω2 many elements of Z. Thus there exists km for m < n and t ∈ ω n+1 such that if we define σ(s 0, m) = σ(s, m) km for m < n and σ(s 0, n) = t, then the corresponding Ds 0 contains ω2 many elements of Z. It is clear that all of (i) through (iii) are satisfied. The definitions of σ(s 1, m) for m ≤ n is similar by considering the set (Z ∩ Ds ) − Bαs . This finishes our construction. Now the sequence (Ds )s∈2ω gives rise to a Suslin set   A= Dxn x∈2ω n∈ω

via the Suslin operation. Since diam(Ds ) < 2−lh(s) we get that for any x ∈ 2ω , the set n∈ω Dxn is a singleton, say with element zx . Moreover, if x(m) = 0, then by our construction, for any n > m, α

,0

x(m+1) . Dxn ⊆ Cσ(xn,m)

It follows from (i) that   αx(m+1) ,0 zx ∈ Dxn ⊆ Cσ(xn,m) ⊆ Bαx(m+1) . n>m

n>m

Thus zx ∈ Bαx(m+1) . Similarly, if y(m) = 1, then zy ∈ Bαy(m+1) . These imply that if x, y ∈ 2ω with x = y, then there is α < ω1 such that either x ∈ Bα and y ∈ Bα , or x ∈ Bα and y ∈ Bα . In either case we have that x and y are E-inequivalent. Thus we have produced a Suslin set A of E-inequivalent elements containing uncountably many elements. By the perfect set theorem (Theorem 1.6.6) A contains a perfect set of E-inequivalent elements. Theorem 9.1.5 (Burgess) Let E be a Σ11 equivalence relation on a standard Borel space X. Then E has either perfectly many, countably many, or ω1 many equivalence classes.

Proof. By Theorem 9.1.2 E can be written as the intersection of ω1 many Borel equivalence relations. Let Eα , α < ω1 , be Borel equivalence relations such that E = α<ω1 Eα . Each Eα has either countably many or perfectly many equivalence classes by the Silver dichotomy theorem. If any of Eα has perfectly many equivalence classes, then so does E. So assume each Eα has only countably many equivalence classes, and let them be enumerated as Cα,n for n ∈ ω. Then we have that xEy ⇐⇒ ∀α∀n (x ∈ Cα,n ↔ y ∈ Cα,n ).

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This means that E is ω1 -smooth, and hence by Theorem 9.1.4, E has ≤ ω1 many equivalence classes. It follows that E has either perfectly many, countably many, or ω1 many equivalence classes. Exercise 9.1.1 Write Eω1 as an intersection of ω1 many Borel equivalence relations. Exercise 9.1.2 Recall that a quasiorder is a reflexive and transitive relation. Show that any Σ11 quasiorder on a standard Borel space X is the intersection of ω1 many Borel quasiorders. Exercise 9.1.3 Let X be a standard Borel space and E ⊆ F equivalence relations on X, where E is Σ11 and F is Π11 . Show that there is a Borel equivalence relation R such that E ⊆ R ⊆ F . Exercise 9.1.4 Let X be a standard Borel space and E = α<ω1 Eα , where Eα are Σ11 equivalence relations on X. Show that E has either perfectly many, countably many, or ω1 many equivalence classes. Exercise 9.1.5 Let X be a standard Borel space and E = α<ω1 Eα , where Eα are Π11 equivalence relations on X. Show that E has either perfectly many, countably many, or ω1 many equivalence classes. Exercise 9.1.6 For a standard Borel space X and a collection S of subsets of X, let E(S) denote the equivalence relation generated by S. Show the following corollary of the proof of Theorem 9.1.4: If S is a set of ω1 many Borel subsets of a standard Borel space X, then either E(S) has ≤ ω1 many equivalence classes or there is a countable T ⊆ S such that E(T ) has perfectly many equivalence classes.

9.2

Definable reductions among analytic equivalence relations

In this section we discuss definable reductions among Σ11 equivalence relations. We start with a discussion of Borel reducibility and show that it is sometimes inadequate to capture the intuitive reductions among general Σ11 equivalence relations. We will then introduce a more general notion of definable reducibility. The downside, however, of considering this broader notion is that a significant amount of metamathematics is needed to prove statements about it. Since these metamathematical techniques are beyond the scope of this book, we will only mention some results without proving them. On the

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other hand, we will introduce the very useful notion of Ulm classifiability and also briefly discuss universal analytic equivalence relations. Recall that the conclusion of the Silver dichotomy can be reformulated in terms of Borel reducibility as either E ≤B id(ω) or id(2ω ) ≤B E. For the Burgess trichotomy theorem, these Borel reducibility statements still correspond to the cases when E has either countably many or perfectly many equivalence classes. However, when E has ω1 many equivalence classes, there is no obvious way to characterize E in the Borel reducibility hierarchy. We elaborate some more below. Consider the equivalence relation Fω1 defined on 2ω by CK(x)

xFω1 y ⇐⇒ ω1

CK(y)

= ω1

.

CK(x)

Since {ω1 : x ∈ 2ω } is unbounded in ω1 , clearly Fω1 has ω1 many equivalence classes. Lemma 9.2.1 Fω1 is Σ11 equivalence relation with every equivalence class Borel. Proof.

CK(x)

Note that ω1

CK(y)

≤ ω1

iff

∀m∃n ({m}x ∈ WO → ∃ϕ : (ω, {m}x ) ∼ = (ω, {n}y ) ) , which implies that Fω1 is Σ11 . Now fix x ∈ 2ω and m, n ∈ ω with {m}x ∈ WO. Then we also have that (ω, {m}x ) ∼ = (ω, {n}y ) iff (a) {n}y ∈ WO, (b) for all strictly order-preserving ϕ from (ω, {n}y ) into (ω, {m}x ), ϕ is onto, and (c) for all strictly order-preserving ψ from (ω, {m}x) into (ω, {n}y ), ψ is onto. Thus if we let Bm,n = {y : (ω, {m}x ) ∼ = (ω, {n}y )}, then Bm,n is Δ11 , in x particular Borel, when {m} ∈ WO. Now 

  CK(x) CK(y) = y : ω1 = ω1 Bm,n , {m}x ∈WO n

hence the Fω1 -equivalence class of x is Borel. Now Fω1 is indeed a Σ11 equivalence relation and apparently has ω1 many equivalence classes. However, if we compare it to Eω1 defined in the preceding section via Borel reducibility, they turn out to be incomparable.

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Lemma 9.2.2 Eω1 ≤B Fω1 and Fω1 ≤B Eω1 . Proof. Note that if x ∈ WO then the Eω1 -equivalence class of x is not Borel. Now every Fω1 -equivalence class is Borel, which implies that any equivalence relation Borel reducible to Fω1 has the same property. Thus Eω1 ≤B Fω1 . On the other hand, assume f : 2ω → LO is a Borel reduction from Fω1 to Eω1 . There is at most one Fω1 -equivalence class C such that f (C) ⊆ LO − WO. Since C is Borel, it follows that f (2ω −C) is Σ11 in LO. Since f (2ω −C) ⊆ WO, by the boundedness principle there is α < ω1 such that f (2ω − C) ⊆ WOα . This is a contradiction to the assumption that f is a reduction since Eω1  WOα has only countably many equivalence classes. Nevertheless on an intuitive level we should be able to define a reduction from Fω1 to Eω1 , that is, given x ∈ 2ω to define Φ(x) ∈ WO so that its order CK(x) type is ω1 . A straightforward strategy to achieve this is the following. Consider the countable set {e ∈ ω : {e}x ∈ WO} and enumerate it as {en : n ∈ ω}. Then define ⎧ if n < m, ⎨ 1, if m < n, Φ(x)(n, k, m, l) = 0, ⎩ {en }(k, l), if m = n. It is easy to see that Φ is as required. From the discussions above Φ is not a Borel function. To allow reductions like Φ we need to consider classes of functions larger than that of Borel functions. This takes us to the realm of the projective hierarchy, and one natural class of functions to use is that of Δ12 functions. One can check that Φ is Δ12 . Two basic issues arise with the consideration of Δ12 reductions. The first is that the usual axioms of set theory no longer seem sufficient to determine the answers of many questions. Thus for many results axioms beyond ZFC are assumed and it is not clear that they can be removed from the assumptions. The second, a more subtle, issue is that, even for results which turn out to be provable within ZFC, the proof techniques go beyond the classical or effective descriptive set theory we have developed and been using in this book. The most powerful method employed in these proofs is that of forcing. Because of these issues we will not prove any result involving Δ12 reductions, even if some of the proofs are classical. Instead, we will mention some results without proof so as to provide the reader a brief overview of the subject. One of the common axioms beyond ZFC used in the results about Δ12 reductions is Σ11 determinacy. This is the statement that every game GA is determined if the payoff set A ⊆ ω ω is Σ11 . Burgess has shown under Σ11 determinacy that if an analytic equivalence relation E has ≤ ω1 -many equivalence classes, then E is Δ12 reducible to Eω1 . For this reason it is customary to denote Eω1 by id(ω1 ), so that the Burgess trichotomy theorem

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becomes a full analog to the Silver dichotomy. Hjorth and Kechris [76] have considered a generalization of the Glimm– Effros dichotomy for Σ11 equivalence relations. Here the analog of the smooth clause, or the concrete classifiability, is the following notion of Ulm classifiability, inspired by Ulm’s classification of countable abelian p-groups (which we will review more in Section 13.4). For any α < ω let 2α be the space of all transfinite sequences (xβ )β<α where xβ ∈ {0, 1} for all β < α. Elements of 2α code subsets of α. If α < β then we view 2α as a subset of 2β in the natural sense: for any x ∈ 2ω , regard x(ξ) = 0 for all α ≤ ξ < β. Let 2<ω1 be the increasing union of all 2α for α < ω1 . Elements of 2<ω1 are called Ulm invariants, since Ulm used such elements to classify countable abelian p-groups up to isomorphism (see Section 13.4). We provide a way to code Ulm invariants by reals, as follows. Given any (x, y) ∈ WO × 2ω , let α = ot(<x ). For each n ∈ ω with y(n) = 1, let <x,n be the restriction of <x on the set {m ∈ ω : m <x n}, and then let βn = ot(<x,n ). We then associate U (x, y) ∈ 2α by letting it code the set {βn : y(n) = 1}. Consider the equivalence relation EUlm on LO × 2ω defined by (x, y)EUlm (x , y  ) if U (x, y) and U (x , y  ) code the same set of ordinals. A somewhat tedious but straightforward calculation shows that EUlm is a Σ11 equivalence relation (Exercise 9.2.2). Definition 9.2.3 An equivalence relation E on a standard Borel space X is Ulm classifiable if it is Δ12 reducible to EUlm . The equivalence relation EUlm is customarily denoted by id(2<ω1 ) for the obvious reason that it can be viewed as the identity relation on 2<ω1 . In this notation an equivalence relation E is Ulm classifiable iff E is Δ12 reducible to id(2<ω1 ). Hjorth and Kechris [76] proved under Σ11 determinacy that if E is a Σ11 equivalence relation, then either E is Ulm classifiable or E0 ≤B E. Finally we note that there are equivalence relations universal for all analytic equivalence relations with respect to Borel reducibility. Proposition 9.2.4 There exists a universal Σ11 equivalence relation, that is, a Σ11 equivalence relation E such that for any Σ11 equivalence relation F , F ≤B E. Proof. Let U ⊆ ω ω × (ω ω )2 be a universal set for all Σ11 subsets of (ω ω )2 , that is, for any Σ11 A ⊆ (ω ω )2 there is x ∈ ω ω such that A = Ux . Define U ∗ by letting (x, y, z) ∈ U ∗ ⇐⇒ (x, y, z) ∈ U ∨ (x, z, y) ∈ U. Then U ∗ is still Σ11 . Let ·, · : ω ω × ω ω → ω ω be a homeomorphism. Then define an equivalence relation E on ω ω by x, yEx , y   ⇐⇒ x = x ∧ (y = y  ∨ (x, y, y  ) ∈ U ∗ ∨ ∃y0 = y, y1 , . . . , yn = y  ∀i < n (x, yi , yi+1 ) ∈ U ∗ ).

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Informally E is a direct sum of transitive closures of Ux∗ for all x ∈ ω ω . It is clear that E is a Σ11 equivalence relation. Suppose F is any Σ11 equivalence relation on ω ω . Then F ⊆ (ω ω )2 and so there is x ∈ ω ω such that F = Ux . We have that Ux∗ = Ux = F is a transitive relation. Define f : ω ω → ω ω by letting f (y) = x, y. Then for any y, y  ∈ ω ω , yF y  iff (y, y  ) ∈ Ux∗ = F iff (x, y, y  ) ∈ U ∗ iff x, yEx, y  . This shows that, in fact, F c E. Exercise 9.2.1 Show that the reduction Φ defined in this section is σ(Σ11 )measurable. Exercise 9.2.2 Show that EUlm is a Σ11 equivalence relation. Exercise 9.2.3 Show that there is a universal Σ11 equivalence relation.

9.3

Actions of standard Borel groups

Orbit equivalence relations of Polish group actions form an important class of analytic equivalence relations. More generally, one can consider actions in which the acting group is standard Borel and still get that the resulting orbit equivalence relation is analytic. Definition 9.3.1 A standard Borel group is a group G with a standard Borel structure B on G such that the group operations are Borel with respect to B. Borel subgroups of Polish groups are standard Borel groups with their induced Borel structure. In particular, all Polish groups are standard Borel groups with their Borel structure. The converse, as we will see below, is not true. Definition 9.3.2 A standard Borel group G is Polishable if there is a Polish topology τ inducing the Borel structure on G so that (G, τ ) is a topological group, that is, the group operations are continuous with respect to τ . Recall that in Definition 8.2.3 we defined Polishable subgroups of Polish groups. The above definition is a generalization of Definition 8.2.3. By Theorem 2.3.3, and more explicitly by Exercise 2.3.7, the topological group topology witnessing the Polishability of a standard Borel group is unique. Lemma 9.3.3 ∞ is a standard Borel group that is not Polishable.

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Proof. ∞ is an Fσ subgroup of the Polish group Rω , hence is a standard Borel group. Assume τ is a Polish group topology on ∞ witnessing its Polishability. For each N ∈ ω define  AN = (xn ) ∈ Rω : (xn )∞ = sup |xn | ≤ N . n

Then AN is a compact, hence closed subset of Rω and ∞ = N AN . It follows that AN is a Borel subset of (∞ , τ ), and in particular has the Baire property. By the Baire category theorem there is N ∈ ω such that AN is nonmeager in (∞ , τ ). By Theorem 2.3.2 AN − AN contains a τ -open nbhd of the identity element of ∞ . Note that AN − AN ⊆ A2N , and from the separability of τ there is a countable set C ⊆ ∞ such that  (x + A2N ). ∞ = x∈C

Let C = {xn : n ∈ ω}. We construct a y ∈ A5N such that y ∈ xn + A2N for any n ∈ ω, arriving at a contradiction. The construction of y = (yn ) is by diagonalization:  5N, if |(xn )n | ≤ 2N , yn = 0, if |(xn )n | > 2N . It is easy to see that |yn − (xn )n | > 2N , hence y ∈ xn + A2N for any n ∈ ω. There are many examples of non-Polishable standard Borel groups. Another such example can be found in Exercise 9.3.1. Here we mention a third one which is also useful in our discussion later this section. Recall from Section 2.6 that for any metric space (X, d) the Graev metric δ can be defined on the free group F(X) making it a topological group. Moreover if (X, d) is separable then F(X) is separable and its completion F(X) becomes a Polish group. If in addition (X, d) is compact, then by Exercise 2.6.3 F(X) becomes a Kσ subset of F(X). We summarize these simple observations in the following lemma. Lemma 9.3.4 If X is a compact Polish space then there is a Borel structure on F(X) making it a standard Borel group. Proof. Let d be any compatible metric on X and let δ be the Graev metric on F(X). Then F(X) becomes a Borel subgroup of the Polish group F(X) and therefore a standard Borel group. If X is not finite then it has an accumulation point, and from the definition of the Graev metric we can see that δ on F(X) is not complete. In fact by a similar argument as in the proof of Lemma 9.3.3 one can show that F(X) is

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not Polishable if X is any uncountable compact Polish space. We leave the details of these proofs to the reader (Exercises 9.3.2 and 9.3.3). We now turn to actions of standard Borel groups. Definition 9.3.5 If G is a standard Borel group, X a standard Borel space and a : G × X → X a Borel action, then we call X a standard Borel G-space. If G is a standard Borel group and X a standard Borel G-space, then the X is obviously Σ11 . The following result of Shelah orbit equivalence relation EG 1 shows that every Σ1 equivalence relation arises this way. Theorem 9.3.6 (Shelah) Let X be a standard Borel space and E a Σ11 equivalence relation on X. Then there is a standard Borel group G and a Borel action of G on X such that X . E = EG Proof. Let τ be a Polish topology on X giving rise to the standard Borel structure. Since E is Σ11 as a subset of (X, τ )2 , there is a closed set F ⊆ (X, τ )2 × ω ω such that xEy ⇐⇒ ∃z ∈ ω ω (x, y, z) ∈ F . As a closed subset of (X, τ )2 × ω ω , F is a Polish space. For each (x, y, z) ∈ F we define a Borel isomorphism π(x,y,z) : X → X by ⎧ ⎨ y, if w = x, π(x,y,z) (w) = x, if w = y, ⎩ u, otherwise. Now let Y be a compact Polish space with the same cardinality as F . For example, if F is uncountable then let Y = 2ω ; if F is countable then let Y be a closed countable subset of 2ω with |F | = |Y |. Then there is a Borel isomorphism ϕ between Y and F . We can therefore associate for any a ∈ Y a Borel isomorphism πϕ(a) . Let G = F(Y ) be the standard Borel group as given in Lemma 9.3.4. Then there is a natural action of G on X induced by a · w = πϕ(a) (w) for a ∈ Y and w ∈ X. It is straightforward to check that this action is Borel X and that E = EG . Exercise 9.3.1 Let T be the multiplicative group of the unit circle {eiθ : 0 ≤ θ < 2π}. Let H = Tω and G = {(xn ) ∈ H : ∃m ∀n ≥ m ( xn = 1 ) }. Show that

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(i) T and H are compact Polish groups; (ii) G is an Fσ subgroup of the Polish group H, hence is a standard Borel group; (iii) G is not Polishable; (iv) E1 is Borel reducible to the coset equivalence relation H/G. Exercise 9.3.2 Let X be a compact Polish space, d a compatible metric on X, and x an accumulation point in X. Show that there is a sequence (yn ) in X with yn → x as n → ∞ such that the following sequence (un ) in F(X) is Cauchy in the Graev metric δ but does not converge in F(X): u0 = e, un+1 = un yn x−1 . Thus δ is not complete on F(X). Exercise 9.3.3 Show that for any uncountable compact Polish space X the standard Borel group F(X) is not Polishable.

9.4

Wild Polish groups

For a general Σ11 equivalence relation the very first attempt to classify it is to decide whether it is Borel or non-Borel. In practice a good place to start is to examine its equivalence classes. Of course, if there is one non-Borel class then we conclude that the equivalence relation is non-Borel. However, for an orbit equivalence relation induced by a Polish group action this strategy will not work since every one of its orbits is Borel. Thus it is somewhat more challenging to decide if an orbit equivalence relation, being Σ11 in general, is Borel or not. In this section we investigate some non-Borel orbit equivalence relations. Theorem 8.2.2 provides a criterion: an orbit equivalence relation on a Polish space is non-Borel iff it has orbits of arbitrarily high complexity. This is useful in some context. For instance, if a locally compact Polish group acts continuously on a Polish space, then every orbit is Fσ and hence the equivalence relation is Borel (in fact Fσ ). However, it is not clear how to directly apply this theorem when the acting group is not locally compact. For instance, one of the simplest nonlocally compact Polish groups is Zω . More work is needed to decide if Zω can induce a non-Borel orbit equivalence relation. In fact, it was a documented question asked by Sami [134] whether any abelian group can induce a non-Borel orbit equivalence relation. This was answered positively by Solecki [140]. We introduce some terminology.

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Definition 9.4.1 A Polish group G is tame if every G-orbit equivalence relation is Borel; it is wild if it is not tame. In this section we present a proof by  Solecki that Zω is wild. In fact, Solecki has considered groups of the form n Gn , where Gn are abelian, and has completely determined which of them are tame. Let X be the set of all trees on Z, that is, T ∈ X iff T ⊆ Z<ω is closed <ω under taking initial segments. X is a Borel subset of 2Z , hence is itself a ω standard Borel space. We define an action of Z on X as follows. For T ∈ X and σ ∈ Zω , let  σ·T = (σ  n) + (T ∩ Zn ). n∈ω

Clearly σ · T ∈ X and the action is Borel. Throughout this section we denote the orbit equivalence relation by E(Zω ), or E for simplicity. Our objective is to show that E is non-Borel. Recall that a tree is well-founded if it does not have any infinite branch, and is ill-founded otherwise. If T1 , T2 ∈ X, we let Φ(T1 , T2 ) = {t ∈ Z<ω : ∀m ≤ lh(t) (t  m) + (T1 ∩ Zm ) = T2 ∩ Zm }. Then it is easy to see that Φ(T1 , T2 ) ∈ X, and T1 ET2 iff [Φ(T1 , T2 )] = ∅ iff Φ(T1 , T2 ) is ill-founded. Note that for any n ∈ ω, Φ(T1 , T2 ) ∩ Zn is a coset of a subgroup of Zn . This motivates the following concepts. Definition 9.4.2 Let T ⊆ Z<ω be a tree on Z. T is a coset tree if for all n ∈ ω, T ∩ Zn is a coset of a subgroup of Zn . T is a group tree if for all n ∈ ω, T ∩ Zn is a subgroup of Zn . Note that in general if G is a group and C ⊆ G is nonempty, then C is a left-coset of a subgroup of G iff for all g1 , g2 , g3 ∈ C, g1 g2−1 g3 ∈ C. The same condition also characterizes the right-cosets in G. Thus even without commutativity we may address them simply as cosets. If C is a coset in G and g ∈ C, then g −1 C is a subgroup of G. In fact g −1 C is independent of the choice of g ∈ C, and it is the unique subgroup of G for which C is a left-coset. For an arbitrary C ⊆ G, we define ⎧ −1 ⎨ g C, if C is a nonempty coset in G and g ∈ C, γC = ⎩ ∅, otherwise. Then for any C ⊆ G, γC is a subgroup of G. Now if T is a coset tree on Z, we let  γ(T ∩ Zn ). γT = n∈ω

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Then γT is a group tree on Z. Moreover, Φ(γT, T ) = T . When a tree T on Z is well-founded we may speak of its rank, denoted by rk(T ), which is a countable ordinal. Here in order to study ill-founded trees, we define the following more general notion. For any tree T on Z, let define T  = {t ∈ T : ∃s ∈ T t  s}. Then T  ⊆ T . By transfinite induction T α for α < ω1 as follows: T 0 = T, T α+1 = (T α ) , and T λ = α<λ T α , if λ is limit. Since T is countable there is a countable ordinal β < ω1 such that T β = T β+1. Define the height of T , denoted ht(T ), to be the least ordinal β such that T β = T β+1 . For a well-founded tree T the height coincides with the rank of T , and if β = ht(T ) = rk(T ), then T β = ∅. Putting all these concepts together we can now reduce the problem to the construction of well-founded coset trees on Z of arbitrarily high rank. Lemma 9.4.3 If there are well-founded coset trees on Z of arbitrarily high rank then E(Zω ) is non-Borel. Proof. Assume toward a contradiction that E is Borel. Note that Φ is a Borel map from X 2 into X, and (T1 , T2 ) ∈ E ⇐⇒ Φ(T1 , T2 ) is well-founded. Thus Φ(X − E) is a Σ11 set of well-founded trees, and by the boundedness principle for Σ11 sets of well-founded trees (Theorem 1.6.11) there is α < ω1 such that rk(Φ(T1 , T2 )) < α for all (T1 , T2 ) ∈ E. Now if T is any well-founded coset tree with rk(T ) ≥ ω, γT is ill-founded, and since Φ(γT, T ) = T is wellfounded, (γT, T ) ∈ E. It follows that rk(T ) = rk(Φ(γT, T )) < α. This shows that all well-founded coset trees on Z have rank < max{ω, α}, contradicting our assumption. 2

Next we further reduce the problem to the construction of group trees on Z of arbitrarily high height. Note that there are no well-founded group trees of rank ≥ ω, since the element (0, 0, 0, . . . ) is always a branch of such a tree. So we do need to consider the concept of height when dealing with complex group trees. We also need to examine closely the tree nodes in the transfinite pruning process. For this define for any t ∈ T the rank of t in T by  the least β such that t ∈ T β , if such β exists, rkT (t) = ω1 , otherwise. For any T ∈ X, if α ≤ ht(T ) is a successor ordinal, then there is t ∈ T with rkT (t) = α. Lemma 9.4.4 Let T be a group tree with ht(T ) > ω. tn ∈ Zn such that for

Then there exist n all n ∈ ω, (tn+1  n) − tn ∈ T , and n tn + (T ∩ Z ) is a well-founded coset tree.

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Proof. First we note a general fact about countable groups. If G is a countable group and (Gn ) is a strictly decreasing sequence of subgroups of G, then there exist gn ∈ G such that for all n ∈ ω, gn+1 ∈ gn Gn , and n gn Gn = ∅. To see this, enumerate G as {hn : n ∈ ω} and inductively define gn+1 ∈ G such that gn+1 ∈ gn Gn and hn ∈ gn+1 Gn+1 . Now let T be a group tree with ht(T ) > ω. Let s0 ∈ T be such that rkT (s0 ) = ω + 1. Let k0 = lh(s0 ) + 1. Then there exists a sequence (un )n∈ω such that for all n ∈ ω, lh(un ) = k0 , s ⊆ un , and rkT (un ) < rkT (un+1 ) < ω. Let ln = k0 + rkT (un ) + 1 and πn be the projection of Zln onto the first k0 coordinates. Then πn is a group homomorphism from Zln to Zk0 , and therefore πn (T ∩ Zln ) is a subgroup of T ∩ Zk0 . Note that un ∈ πn (T ∩ Zln ) but un ∈ πm (T ∩ Zlm ) for all m > n. Thus if we let Gn = πn (T ∩ Zln ), then (Gn ) is a strictly decreasing sequence of subgroups of T ∩ Zk0 . By the general fact above we may let vn ∈ T ∩ Zk0 be such that vn+1 ∈ vn + Gn and n (vn + Gn ) = ∅. Next we inductively define a sequence (wn ) such that wn ∈ Zln , πn (wn ) = vn , and (wn+1  ln ) − wn ∈ T . To begin with, let w0 ∈ Zl0 be any extension of v0 . In general, assume wn has been defined to satisfy the inductive hypothesis. Since vn+1 ∈ vn + πn (T ∩ Zln ), there is w ∈ T ∩ Zln such that πn (w) = vn+1 − vn = vn+1 − πn (wn ). Thus vn+1 = πn (w + wn ). Let wn+1 ∈ Zln+1 be any extension of w + wn . Then πn+1 (wn+1 ) = πn (w + wn ) = vn+1 and (wn+1  ln ) − wn = w ∈ T . Finally we define tn ∈ Zn . For any n ∈ ω let m ∈ ω be the least such that n ≤ lm . Then let tn = wm  n. We verify that the tn have the required properties. To see that (tn+1  n) − tn ∈ T , it is without loss of generality to assume n = lm for some m ∈ ω. Then tn = wm but tn+1 = wm+1  (n + 1). Therefore, (tn+1  n) − tn = (wm+1  lm ) − wm ∈ T . Now let S = n tn + (T ∩ Zω ). It follows that S is a coset tree. It remains to verify that S is well-founded. We claim that for any s ∈ S ∩ Zk0 , rkS (s) < ω. For this let s ∈ S ∩ Zk0 . Since n vn + πn (T ∩ Zln ) = ∅, we have that s ∈ vn + πn (T ∩ Zln ) for some n ∈ ω. Assume s ∈ S ∩ Zln with s ⊆ s . Then s = πn (s ) ∈ πn (tln )+ πn (T ∩Zln ) = πn (wn )+ πn (T ∩Zln ) = vn + πn (T ∩Zln ), a contradiction. This shows that rkS (s) < ln , and therefore rkS (s) < ω. It follows that rk(S) < ω + k0 , and hence S is well-founded. Lemma 9.4.5 Let T be a group tree, α < ω1 , and tn ∈ Zn satisfy (tn+1  n) − tn ∈ T α . Let S = n tn + (T ∩ Zn ). Then S is a coset tree, and for all β ≤ α, S β = n tn + (T α ∩ Zn ). Proof. It is easy to see that if T is a coset (group) tree then T  is a coset (respectively group) tree. It follows that for any β < ω1 , T β is a coset (group)

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tree if T is. We only check that S  = n tn + (T  ∩ Zn ). The lemma then follows by an easy induction. Let s ∈ S  ∩Zn . Then there are t ∈ T ∩Zn , s ∈ S ∩Zn+1 , and t ∈ T ∩Zn+1 such that s = tn + t, s ⊆ s , and s = tn+1 + t . Since (tn+1  n) − tn ∈ T  there is also u ∈ T ∩ Zn+1 with (tn+1  n) ⊆ tn ⊆ u. Then t = s−tn = (s  n)−tn = (tn+1  n)+(t  n)−tn = u  n+t  n = (u+t )  n.

Since T is a group tree u + t ∈ T , and so S  ⊆ n tn + (T  ∩Zn ). The converse is obvious. Lemma 9.4.6 For any α < ω1 if there is a group tree of height > α + ω then there is a well-founded coset tree S of rank ≥ α. Proof. Let T be a group tree of height > α + ω. Then T α = ∅. Apply α n that (tn+1  n) − tn ∈ T α Lemma

9.4.4 to αobtainω tn ∈ T ∩ Z for n ∈ ω such and n tn + (T ∩ Z ) is well-founded. Let S = n tn + (T ∩ Zn ). Then by Lemma 9.4.5 S is a coset tree with S α = ∅ well-founded. Thus S is a well-founded coset tree with rk(S) ≥ α. Theorem 9.4.7 (Makkai–Solecki) There are group trees on Z of arbitrarily high height. Proof. Let φ : Z2 → Z be the homomorphism φ(a, b) = a+b. (G 0n ), (G1n ) Let 0 be strictly decreasing sequences of subgroups of Z such that n Gn = n G1n = {0} and φ(G0n × G1n ) = Z for all n ∈ ω. For example, we may take G0n = 2n Z and G1n = 3n Z. By transfinite induction on α < ω1 we define a group tree Tα such that (a) if α = β + 1 is a successor, then (Tα )β ∩ Z = Z and (Tα )a is well-founded if a = 0; (b) if α is a limit, then for any β < α there is n ∈ ω such that G0n × G1n ⊆ (Tα )β ∩ Z2 , and if t ∈ Z2 and t = 0 0, then (Tα )t is well-founded. Granting this construction it is clear that ht(Tα ) ≥ α. Let T0 = {0n : n ∈ ω} and T1 = T0 ∪ Z. If α = β + 1 and β is a successor, let Tα = {∅} ∪ Z ∪ {t(0) t : t ∈ Tβ , lh(t) ≥ 1}. If α = β + 1 and β is a limit, let Tα = the tree generated by {φ(t(0), t(1)) t : t ∈ Tβ , lh(t) ≥ 2}.

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With these definitions (a) is satisfied. Suppose α is a limit. To satisfy (b) we construct two group trees S0 and S1 and an increasing sequence of successor ordinals βn → α such that (i) G0n ⊆ (S0 )βn and G1n ⊆ (S1 )βn for all n ∈ ω; (ii) (S0 )a and (S1 )a are well-founded if a = 0. If S0 and S1 are defined as above, then let Tα = {t : t  {2k : k ∈ ω} ∈ S0 and t  {2k + 1 : k ∈ ω} ∈ S1 } . Then it is easy to see that (b) is satisfied. Fix any increasing sequence of successor ordinals βn → α. Apparently the constructions of S0 and S1 are parallel. We only construct S0 and then the construction of S1 will be similar. Note that Tβn have already been constructed to satisfy (a). S0 will be an amalgamation of the Tβn as follows. Let {In }n∈ω be a partition of ω − {0} into infinitely many infinite subsets. For each n ∈ ω define a tree Un by s ∈ Un ⇐⇒ s = ∅ ∨ [ s(0) ∈ G0n ∧ s  In ∈ Tβn ∧ ∀i < lh(s)(i = min In ⇒ s(i) = s(0) ∧ i ∈ In ⇒ s(i) = 0) ]. Then Un is a group tree. Let S0 =

   Un ∩ Zk . k

n

Then S0 is a group tree by definition. To see that G0n ⊆ (S0 )βn let a ∈ G0n . Let s ∈ Zmin In +1 be such that s(0) = s(min In ) = a and s(i) = 0 for all 0 < i < min In . Then s ∈ Un ⊆ S0 . Since (Tβn )βn −1 ∩ Z = Z, s ∈ (Un )βn −1 ⊆ (S0 )βn −1 . Thus a ∈ (S0 )βn and (i) is satisfied. To show (ii) let a = 0 and toward a contradiction let σ ∈ Zω be an infinite branch of (S0 )a . Then σ(0) = a. Let n ∈ ω be large enough such that a ∈ G0n . We have that σ(min Ii ) = 0 for all i < n. Otherwise if σ(min Ii ) = 0 for i < n, then by our construction of S0 , σ  Ii is an infinite branch of Tβi with (σ  Ii )(0) = σ(min Ii ) = 0, violating our inductive assumption (a) for Tβi . Let k > max{min Ii : i < n}

and s = σ  k ∈ S0 . Then by the definition of S0 we have that s ∈  m Um ∩ Zk . Since Um are group trees we may assume s = sm1 + sm2 + · · · + sml + sml+1 + · · · + smp with m1 < m2 < ml < n ≤ ml+1 < · · · < mp , where smj ∈ Umj for j = 1, . . . , p. It follows that a = sm1 (min Im1 )+· · ·+sml (min Iml )+sml+1 (min Iml+1 )+· · ·+smp (min Imp ). From the above argument we get that smj (min Imj ) = 0 for j = 1, . . . , l. However, smj (min Imj ) ∈ G0n for j = l + 1, . . . , p. Hence a ∈ G0n , a contradiction. This shows that S0 has the required properties.

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Corollary 9.4.8 (Solecki) E(Zω ) is non-Borel and Zω is wild. Exercise 9.4.1 Let G be a group and C ⊆ G is nonempty. Show that the following are equivalent: (i) C is a left-coset of some subgroup of G. (ii) C is a right-coset of some subgroup of G. (iii) For all g1 , g2 , g3 ∈ C, g1 g2−1 g3 ∈ C. Exercise 9.4.2 Show that if T is a coset tree then Φ(γT, T ) = T . Give a counterexample where T is not a coset tree. Exercise 9.4.3 Show that if T is a coset tree then (γT ) = γ(T  ). Exercise 9.4.4 Let T be a group tree and tn ∈ Zn such that (tn+1  n)−tn ∈ T ∩ Zn . Show that n tn + (T ∩ Zn ) is a coset tree.

9.5

The topological Vaught conjecture

It is not known if the Silver dichotomy holds for orbit equivalence relations of Polish group actions. This is one of the central open problems in the entire invariant descriptive set theory. Proposition 9.5.1 Let G be a Polish group. Then the following statements are equivalent: (a) For any Polish G-space X, either there are countably many orbits or there are perfectly many orbits. (b) For any Polish G-space X and invariant Borel set B ⊆ X, either B contains countably many orbits or B contains perfectly many orbits. (c) For any Borel G-space X, either there are countably many orbits or there are perfectly many orbits.

Proof. The implications (c)⇒(b)⇒(a) are obvious. (a)⇒(c) follows from Theorem 4.4.6. For any Polish group G, the topological Vaught conjecture for G refers to any of the above equivalent statements. We denote it as TVC(G). Then the topological Vaught conjecture is the statement

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The topological Vaught conjecture was first formulated by Miller. But its counterpart in countable model theory goes back to Vaught. We will elaborate on the connections in Chapter 11. The topological Vaught conjecture for any tame Polish group is true by the Silver dichotomy. In particular, TVC(G) holds for any locally compact Polish group G. Sami [134] proved the topological Vaught conjecture for all abelian Polish groups. Hjorth and Solecki [83] proved it for all nilpotent Polish groups. Solecki [83] proved it for all Polish groups with two-sided invariant metrics. The strongest result up to date is the following theorem of Becker. Theorem 9.5.2 (Becker) TVC(G) holds for all cli Polish groups, that is, Polish groups with complete left-invariant metrics. Proof.

This is a corollary of Theorem 6.5.1.

By Theorem 2.2.11 the class of cli Polish groups is closed under closed subgroups, quotient groups, and group extensions. Also it is easy to see that it is closed under taking countable products. We next consider the abstract closure properties of the class of all Polish groups satisfying the topological Vaught conjecture. Theorem 9.5.3 Let C be the class of Polish groups G for which TVC(G) holds. Then (i) if G ∈ C and H is a closed subgroup of G, then H ∈ C; (ii) if G ∈ C and H is a closed normal subgroup of G, then G/H ∈ C; (iii) if G ∈ C and G is a closed subgroup of H where H/G is countable, then H ∈ C. Proof. For (i) we let X be a Polish H-space. By Theorem 3.5.2 there is a Polish G-space Y such that X is a closed subset of Y and that every G-orbit contains a unique H-orbit. It follows that if Y has only countably many G-orbits then X has only countably many G-orbits. Assume Y has Y perfectly many G-orbits, and let Z ⊆ Y be a perfect set of pairwise EG Y 1 inequivalent elements. Then the relation R = EG  (Z × X) is Σ1 and for any Y y ∈ Z there is x ∈ X such that yEG x by Theorem 3.5.2. By the Jankov–von Neumann uniformization theorem R has a σ(Σ11 )-measurable uniformization, that is, there is a σ(Σ11 )-measurable function f : Z → X such that for any y ∈ Z, yRf (y). Since σ(Σ11 )-measurable functions are Baire measurable (by the remarks preceding Proposition 2.3.1), it follows that there is a comeager set C ⊆ Z such that f  C is continuous. Without loss of generality assume

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C is Gδ , hence an uncountable Polish space. Then f : C → X is a continuous X Y reduction from id(C) to EH , since for any y1 = y2 ∈ C, (y1 , y2 ) ∈ EG , and Y X thus (f (y1 ), f (y2 )) ∈ EG and therefore (f (y1 ), f (y2 )) ∈ EH . This shows that X has perfectly many G-orbits. (ii) is obvious since every Polish G/H-space is also a G-space. For (iii) we consider a Polish H-space X. X is also a Polish G-space with the restricted action of G, and so either there are only countably many G-orbits or there are perfectly many G-orbits. The assumption implies that every H-orbit contains countably many G-orbits. Thus in the case there are only countably many G-orbits there are only countably many H-orbits as well. In the second X -inequivalent elements. Then case, assume Y ⊆ X is a perfect subset of EG X 1 E = EG  Y is a Σ1 equivalence relation on Y with every equivalence class countable. Thus E has the Baire property. By Theorem 5.3.1, and more explicitly Exercise 5.3.1, there are perfectly many E-equivalence classes in Y , X -inequivalent elements in X. which gives a perfect set of EH It follows that the topological Vaught conjecture is equivalent to TVC(G) for any universal Polish group G. By the results of Section 2.5 it is equivalent to TVC(Iso(U)) or to TVC(H(IN )). In the rest of this section we consider a more general notion. Definition 9.5.4 Let X be a standard Borel space and E an equivalence relation on X. The topological Vaught conjecture for (X, E), denoted TVC(X, E), is the statement every invariant Borel subset of X either contains only countably many E-equivalence classes or perfectly many E-equivalence classes. Recall from Section 5.2 that a faithful Borel reduction from an equivalence relation E on X to F on Y is a Borel reduction f : X → Y witnessing E ≤B F and such that for any invariant Borel subset A ⊆ X, [f (A)]F is Borel. The following proposition shows that TVC is closed under faithful Borel reductions. Theorem 9.5.5 Let X, Y be standard Borel spaces and E, F be Σ11 equivalence relations on X, Y , respectively. If E ≤f B F , then TVC(Y, F ) implies TVC(X, E). Proof. Without loss of generality assume that X and Y are Polish spaces. Let f be a faithful Borel reduction from E to F . Let A ⊆ X be invariant Borel. Then [f (A)]F is invariant Borel. Suppose TVC(Y, F ). If [f (A)]F contains only countably many F -classes, then A contains only countably many E-classes. Otherwise, suppose [f (A)]F contains perfectly many F -classes, and let P ⊆ [f (A)]F be a perfect subset of F -inequivalent elements. Consider the

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set D = {(y, x) ∈ P × X : f (x)F y}. and for all y ∈ P there is x ∈ X with (y, x) ∈ D. By the Then D is Jankov–von Neumann uniformization theorem D has a σ(Σ11 )-measurable uniformization g. Thus g : P → X is Baire measurable and for y1 = y2 ∈ P , (g(y1 ), g(y2 )) ∈ E. Let now C ⊆ P be a dense Gδ subset such that g  C is continuous. Then g  C : C → X and g(C) is a perfect subset of X with E-inequivalent elements. Σ11

Thus to prove TVC(G) for a Polish group it suffices to consider a Polish G-space to which every other Polish G-space is faithfully Borel reducible. Exercise 9.5.1 Show that if H is a countable group then TVC(G) holds iff TVC(G × H) holds. For any standard Borel space X and equivalence relation E on X the Glimm–Effros dichotomy for (X, E), denoted GED(X, E), is the statement for any invariant Borel subset A ⊆ X, either E  A is smooth or E0 c E  A. Exercise 9.5.2 Let X, Y be standard Borel spaces and E, F be Σ11 equivalence relations on X, Y , respectively. Show that if E ≤f B F , then GED(Y, F ) implies GED(X, E). The following exercise problems outline a proof of the topological Vaught conjecture for abelian Polish groups by Sami. Assume G is an abelian Polish group and X a Polish G-space. Define an equivalence relation E on X by xEy ⇐⇒ Gx = Gy , where Gx = {g ∈ G : g · x = x} is the stabilizer group of x ∈ X. X ⊆ E. Exercise 9.5.3 Show that E is a Π11 equivalence relation and EG Deduce that if there are perfectly many E-equivalence classes then there are perfectly many G-orbits.

Exercise 9.5.4 Assume there are only countably many E-equivalence classes. Show that each E-equivalence class is Borel and that for each E-equivalence X  C is Borel. (Hint: Use Theorem 8.2.1.) class C the relation EG Exercise 9.5.5 Show that either there are only countably many G-orbits in X or there are perfectly many G-orbits.

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Chapter 10 Turbulent Actions of Polish Groups

In this chapter we turn back to Polish group actions and orbit equivalence relations. The Borel reducibility hierarchy of analytic equivalence relations has taken shape with the results we have established in the previous chapters. However, it is also noticeable that the results were obtained with a variety of machineries ranging from Baire category methods, measure theoretic methods, group theory, and descriptive set theory. There are still many pairs of equivalence relations we have introduced so far but have not mentioned their reducibility relation, either because it is still an open problem or because its known proof goes beyond the limitation of length or scope of this book. It is thus clear that a sweeping method to prove reducibility or, more challengingly, nonreducibility would be much desirable. The greatest success up to date is Hjorth’s theory of turbulence, which we present in this chapter. It will turn out that orbit equivalence relations from turbulent actions are not reducible to any S∞ -orbit equivalence relation. This powerful theorem has many applications in the classification problems of mathematics, and its potential is still being discovered in current research. The results in this entire chapter are due to Hjorth [70], unless otherwise indicated. Expositions of further results of the theory of turbulence can be found in References [70] and [101].

10.1

Homomorphisms and generic ergodicity

In this section we review the Baire category method for showing nonsmoothness of orbit equivalence relations and introduce some new concepts. Let G be a Polish group and X a Polish G-space. Recall from Section 6.1 X is generically ergodic iff every G-invariant Borel set is either meager that EG X or comeager. We showed in Proposition 6.1.9 that EG is generically ergodic X iff it has a dense orbit, and in Proposition 6.1.10 that if EG is generically X ergodic and has no comeager orbits, then EG is not smooth. The proof of Proposition 6.1.10 (similar to that of Proposition 6.1.6) is by contradiction. We now turn this proof into a positive statement by working a little harder.

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Proposition 10.1.1 Let G be a Polish group, X a Polish G-space, and f : X → 2ω a Baire X y. Suppose measurable G-invariant map, that is, f (x) = f (y) whenever xEG X EG is generically ergodic. Then there is a comeager set C ⊆ X on which f is constant. Proof. First by Proposition 6.1.9 (iii) there is an invariant dense Gδ set Y ⊆ X such that every orbit in Y is dense. Now Y is still a Polish G-space. Thus it suffices to find a comeager set C ⊆ Y on which f is constant. Let C ⊆ Y be comeager such that f  C is continuous. By the continuity of the action the map x → g · x is a homeomorphism of X onto X for all g ∈ G. Thus we have that ∀g ∈ G ∀∗ x ∈ Y g · x ∈ C. Then by the Kuratowski–Ulam theorem, ∀∗ x ∈ Y ∀∗ g ∈ G g · x ∈ C. Therefore we may find and fix an x ∈ Y such that ∀∗ g ∈ G g · x ∈ C. We claim that [x]G ∩ C is dense. For this let y ∈ Y be arbitrary and U ⊆ Y be open with y ∈ U . Since [x]G is dense the set {g ∈ G : g · x ∈ U } is nonempty. But it is also open, so there is g ∈ G such that g · x ∈ U ∩ C. This shows that [x]G ∩ C ∩ U = ∅. Now f  C takes the constant value f (x). To see this, let z ∈ C. By the density of [x]G ∩ C there is a sequence (xn )n∈ω in [x]G ∩ C such that xn → z as n → ∞. However, f is continuous on C, and therefore f (xn ) → f (z) as n → ∞. But f (xn ) = f (x) by the invariance assumption, so f (z) = f (x). In view of its proof the hypotheses of the proposition can be weakened to assume a Baire measurable G-invariant map f defined on a comeager set. Proposition 10.1.2 Let G be a Polish group, X a Polish G-space, C ⊆ X a comeager set, and X is generically f : C → 2ω a Baire measurable G-invariant map. Suppose EG  ergodic. Then there is a comeager set C ⊆ X on which f is constant. Proof. As in the above proof there is an invariant comeager set C0 such that every orbit in C0 is dense. Now C ∩ C0 is still comeager, we may let C  ⊆ C ∩ C0 be comeager such that f  C  is continuous. The same proof as above gives that f is constant on C  . A number of things can be extracted from these propositions and their proofs. We start with some definitions. Definition 10.1.3 Let X, Y be sets and E, F equivalence relations on X, Y , respectively. (1) A map f : X → Y is a homomorphism from E to F if for any x1 , x2 ∈ X, x1 Ex2 ⇒ f (x1 )F f (x2 ).

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(2) If X is a Polish space, then a generic homomorphism from E to F is a map f from a comeager subset C0 of X to Y so that for some comeager set C1 ⊆ C0 f is a homomorphism from E  C1 to F . Definition 10.1.4 Let X, Y be Polish spaces and E, F equivalence relations on X, Y , respectively. We say that E is generically F -ergodic if for every Baire measurable generic homomorphism f from E to F there is a comeager set C ⊆ X such that f (x1 )F f (x2 ) for any x1 , x2 ∈ C. With the new terminology Proposition 10.1.2 can be succinctly stated as Any generically ergodic orbit equivalence relation is generically id(2ω )-ergodic. It turns out that these notions are actually equivalent. Lemma 10.1.5 X is generically Let G be a Polish group and X a Polish G-space. Then EG ergodic iff it is generically id(2ω )-ergodic. X Proof. Suppose EG is generically id(2ω )-ergodic. Let {Un }n∈ω be a countable base for X consisting of nonempty open sets. Define f : X → 2ω by

f (x)(n) = 1 ⇐⇒ ∃g ∈ G g · x ∈ Un ⇐⇒ x ∈ [Un ]G . Since [Un ]G is open for each n ∈ ω, f is a Borel map. It is clear that f is X a homomorphism from EG to id(2ω ). It follows that there is a comeager set C ⊆ X on which f is constant. We claim that for any x ∈ C and n ∈ ω, f (x)(n) = 1. Assume not and let x ∈ C and n ∈ ω be such that f (x)(n) = 0. Then x ∈ [Un ]G . Since C is comeager and [Un ]G is open, C ∩ [Un ]G = ∅. Let y ∈ C ∩ [Un ]G . Then f (y)(n) = 1. Now x, y ∈ C and f (x) = f (y), contradiction. Now by the claim for any x ∈ C and n ∈ ω, [x]G ∩ Un = ∅ since x ∈ [Un ]G . This means that for any x ∈ C, [x]G is dense. Hence the notion of generic F -ergodicity is a generalization of generic ergodicity. This is why we kept using the terminology even if generic ergodicity means no more than the existence of a dense orbit. Of course generic ergodicity is ubiquitous, since every orbit is dense in its closure, which is an invariant subset and hence a Polish G-space. To infer nonsmoothness as we did in Proposition 6.1.10 we need nonexistence of comeager orbits. Under the assumption of generic ergodicity every invariant Borel set is either meager or comeager, and in particular so is every orbit. Thus the nonexistence of comeager orbits is equivalent to the assumption that every orbit is meager. To summarize, if a Polish G-space is generically X is nonsmooth. But we already ergodic and every orbit is meager, then EG

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proved more in Theorem 6.2.1. There it was shown that if a Polish G-space X . This has only meager orbits and is generically ergodic, then E0 c EG discussion motivates the following definition. Definition 10.1.6 Let X be a Polish space and E an equivalence relation on X. We say that E is properly generically ergodic if every E-equivalence class is meager and E is generically ergodic. If F is an equivalence relation on a Polish space Y , then E is properly generically F -ergodic if every E-equivalence class is meager and E is generically F -ergodic. With this terminology Theorem 6.2.1 can be stated as If an orbit equivalence relation E is properly generically ergodic then E0 c E; in particular, E is not smooth. In general proper generic F -ergodicity will be the condition we are after because it is enough to guarantee nonreducibility. Lemma 10.1.7 Let X, Y be Polish spaces and E, F be equivalence relations on X, Y , respectively. Suppose E is properly generically F -ergodic. Then there is no Baire measurable reduction from E to F . In particular E ≤B F . Moreover, for any comeager subset C ⊆ X, E  C ≤B F . Proof. If f is a Baire measurable reduction from E  C to F for some comeager C ⊆ X then in particular it is a generic homomorphism from E to F , and by generic F -ergodicity there is a comeager set C0 ⊆ C such that f  C0 ⊆ [y]F for some y ∈ Y . But since f is a reduction on C this implies that C0 ⊆ [x]E for some x ∈ X. This contradicts the assumption that every E-class is meager. In later sections of this chapter we will consider, as we did in Proposition 10.1.2, structural properties of the actions that can guarantee proper generic F -ergodicity. In doing this we will assume that the orbit equivalence relations we consider are all properly generically ergodic. Exercise 10.1.1 Let X, Y be Polish spaces, E, F be equivalence relations on X, Y , respectively, and f : C → Y for some comeager subset C ⊆ X. Show that f is a generic homomorphism iff the set {(x, y) ∈ E : (f (x), f (y)) ∈ F } is meager in C × C. Exercise 10.1.2 Let E be an equivalence relation on a Polish space. Show that the following are equivalent: (i) E is generically id(2)-ergodic;

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(ii) E is generically id(ω)-ergodic; (iii) Every E-invariant set with the Baire property is either meager or comeager. Moreover, if E = EG where G is a group of homeomorphisms on X, then E is generically id(ω)-ergodic iff E is generically ergodic. Exercise 10.1.3 Let G be a countably infinite group acting by shift on 2G . Show that the orbit equivalence relation EG is properly generically ergodic. Exercise 10.1.4 Show that if E is (properly) generically F1 -ergodic and F2 ≤B F1 , then E is (properly) generically F2 -ergodic. In particular, if F has perfectly many equivalence classes then any (properly) generically F -ergodic equivalence relation is also (properly) generically ergodic.

10.2

Local orbits of Polish group actions

The key idea of the Hjorth theory of turbulence is to consider local orbits of Polish group actions. The analysis of the local orbits reveals a crucial difference between the actions of S∞ and those inducing orbit equivalence relations generically ergodic with respect to certain S∞ -orbit equivalence relations. Throughout this section we let G be a Polish group, X a Polish G-space, {Un }n∈ω a countable base for X, and {Vn }n∈ω a countable open nbhd base for 1G . We also let G0 be a countable dense subgroup of G and enumerate it as {γn }n∈ω . Definition 10.2.1 For x ∈ X, U ⊆ X open with x ∈ U and V ⊆ G with 1G ∈ V , the local U -V orbit of x, denoted O(x, U, V ), is the set of y ∈ U for which there exist l ∈ ω, x = x0 , x1 , . . . , xl = y ∈ U , and g0 , . . . , gl−1 ∈ V , such that xi+1 = gi · xi for all i < l. Note that the local orbits are in general Σ11 . The following lemma collects some basic facts about local orbits. They are easy to check and we leave the proof to the reader. Lemma 10.2.2 Let U ⊆ X be open, x, y, z ∈ U , and V ⊆ G open with 1G ∈ V . Then the following hold: (i) x ∈ O(x, U, V ). (ii) If U  ⊆ U and 1G ∈ V  ⊆ V are open, then O(x, U  , V  ) ⊆ O(x, U, V ).

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(iii) If y ∈ O(x, U, V ) and z ∈ O(y, U, V ) then z ∈ O(x, U, V ). (iv) If y ∈ O(x, U, V ) then O(y, U, V ) ⊆ O(x, U, V ). (v) If V = V −1 then y ∈ O(x, U, V ) iff x ∈ O(y, U, V ). (vi) If y ∈ O(x, U, V ) then there is an open V  with 1G ∈ V  such that V  · y ∈ O(x, U, V ). Example 10.2.3 Let Y be a perfect Polish space. Consider the action of S∞ on Y ω by shift: (g · x)(n) = x(g −1 (n)). Now let U ⊆ Y ω be open and x ∈ U . Then there is a basic open set U0 ⊆ U such that x ∈ U0 . We may assume U0 = W0 × W1 × · · · × Wm × Y ω for open sets W0 , W1 , . . . , Wm ⊆ Y . Now let V ⊆ S∞ be the basic open set V = {g ∈ S∞ : g(i) = i ∀i ≤ m}. Then V · U0 = U0 and O(x, U, V ) ⊆ {x(0)} × · · · × {x(m)} × Y ω . Note that O(x, U, V ) is nowhere dense, and in fact so is O(x, U, V ). The equivalence relation induced by the above action was considered in Exercises 8.3.4 and 8.3.5, where it was denoted as id(Y )∗ and was shown to be Borel bireducible with id(Y )+ , or =+ . It will turn out that generically =+ -ergodic orbit equivalence relations behave differently: their local orbits will be somewhere dense. It is clear that a set is nowhere dense iff its closure is. So for our purposes we are going to investigate the closure of local orbits. Next we define codes for them and study their properties. Notation 10.2.4 For a local orbit O(x, U, V ), let ϕ0 (x, U, V ) = {k ∈ ω : Uk ∩ O(x, U, V ) = ∅}. Note that ϕ0 (x, U, V ) is an element of 2ω coding the closure of O(x, U, V ) in the sense that ϕ0 (x, U, V ) = ϕ0 (x , U  , V  ) iff O(x, U, V ) = O(x , U  , V  ). Lemma 10.2.5 For any open U ⊆ X and 1G ∈ V ⊆ G, the map ϕ0 (·, U, V ) : X → 2ω is continuous. In particular, ϕ0 : X × ω × ω → 2ω is continuous. Proof. Let {Wm }m∈ω be a countable base for G. Assume without loss of generality that Uk ⊆ U . We show that {x ∈ U : O(x, U, V ) ∩ Uk = ∅} is open. For this let Ol (x, U, V ) be the set of y ∈ U for which there are x =

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x0 , x1 , . . . , xl = y ∈ U and g0 , . . . , gl−1 ∈ V such that xi+1 = gi ·xi for all i < l. Then by continuity of the action we have that if y ∈ Ol (x, U, V )∩Uk then there are basic open Un with x ∈ Un ⊆ U and basic open Wm0 , Wm1 , . . . , Wml−1 with gi ∈ Wmi for all i < l, such that (Wml−1 . . . Wm0 ) · Un ⊆ Uk , and for all i < l, (Wmi . . . Wm0 ) · Un ⊆ U. This implies that O(x, U, V ) ∩ Uk = ∅ ⇐⇒ ∃l ∈ ω Ol (x, U, V ) ∩ Uk = ∅ ⇐⇒ ∃l ∈ ω ∃n ∈ ω ∃m0 , . . . , ml−1 ∈ ω [x ∈ Un ∧ (Wml−1 . . . Wm0 ) · Un ⊆ Uk ∧ ∀i < l (Wmi . . . Wm0 ) · Un ⊆ U ]. It is now clear that the set {x ∈ U : O(x, U, V ) ∩ Uk = ∅} is a countable union of basic open sets. Lemma 10.2.6 For any open U ⊆ X, x ∈ U , and open 1G ∈ V ⊆ G, {ϕ0 (x0 , U, V ) : x0 ∈ [x]G ∩ U } = {ϕ0 (γn · x, U, V ) : n ∈ ω, γn · x ∈ U }. In particular, the set {ϕ0 (x0 , U, V ) : x0 ∈ [x]G ∩ U } is countable. Proof. Let g0 ∈ G and x0 = g0 · x ∈ [x]G ∩ U . By continuity of the action there is an open set V  = V −1 ⊆ V such that V  ·x0 ⊆ U . Thus for all g ∈ V  , g · x0 ∈ O(x0 , U, V ) and x0 = g −1 · (g · x0 ) ∈ O(g · x0 , U, V ). By Lemma 10.2.2 (iv) O(x0 , U, V ) = O(g ·x0 , U, V ) for all g ∈ V  . Now let γn ∈ V  g0 ∩G0 . Then γn · x = gg0 · x = g · x0 for some g ∈ V  . Hence O(x, U, V ) = O(γn · x, U, V ). Theorem 10.2.7 X is generically Let G be a Polish group and X a Polish G-space. Suppose EG + = -ergodic. Then for all open U ⊆ X and 1G ∈ V ⊆ G open there is a comeager set C ⊆ X such that for all x ∈ C and x0 ∈ [x]G ∩ U , O(x0 , U, V ) is somewhere dense. Proof.

Fix U and V and consider the map ϕ : X → (2ω )ω defined by   1 ϕ0 (γn · x, U, V ), if gn · x ∈ U , ϕ(x)(n) = (0, 0, . . . ), otherwise.

ϕ is Borel. We claim that there is a comeager set C0 such that ϕ  C0 is a X  C0 to =+ . Let homomorphism from EG C0 = X − G0 · (U − U ).

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Note that U − U is nowhere dense, and by the continuity of the action, for each n ∈ ω, gn · (U − U ) is also nowhere dense. Thus G0 · (U − U ) is meager. This shows that C0 is comeager. To see that ϕ  C0 is a homomorphism let X  x . If [x]G ∩ U = ∅ then ϕ(x)(n) = (0, 0, . . . ) = ϕ(x )(n) for all n ∈ ω, so xEG ϕ(x) = ϕ(x ). Therefore we assume [x]G ∩ U = ∅. By Lemma 10.2.6 {ϕ0 (γn · x, U, V ) : n ∈ ω, γn · x ∈ U } = {ϕ0 (γn · x , U, V ) : n ∈ ω, γn · x ∈ U }. If both G0 · x ⊆ U and G0 · x ⊆ U then we already have that ϕ(x) =+ ϕ(x ). Otherwise if both G0 · x ⊆ U and G0 · x ⊆ U then we also have that ϕ(x) =+ ϕ(x ). Assume we are in the remaining cases, and by symmetry assume G0 · x ⊆ U and G0 · x ⊆ U . Then by the continuity of the action [x]G = G · x ⊆ G0 · x ⊆ U but for some γn ∈ G0 , γn · x ∈ U . This shows that γn · x ∈ U − U and hence x ∈ γn−1 · (U − U ) ⊆ G0 · (U − U ), contradicting our assumption that x ∈ C0 . X Applying generic =+ -ergodicity of EG we obtain a comeager set C ⊆ C0 such that for any x, y ∈ C, ϕ(x) =+ ϕ(y). Let x ∈ C and x0 ∈ [x]G ∩ U . We verify that O(x0 , U, V ) is somewhere dense. Assume not. Let A = O(x0 , U, V ). Then A is nowhere dense and C  = X − G0 · A is comeager. Let y ∈ C∩C  . Then by ϕ(x) =+ ϕ(y) there is γn ∈ G0 such that ϕ0 (γn ·y, U, V ) = ϕ0 (x0 , U, V ). Hence γn · y ∈ O(γn · y, U, V ) = O(x0 , U, V ) = A. This implies that y ∈ G0 · A, contradicting our assumption that y ∈ C  . Exercise 10.2.1 Prove Lemma 10.2.2. Exercise 10.2.2 Show that for any open U ⊆ G, x ∈ U , open 1G ∈ V ⊆ G, and g ∈ G, g · O(x, U, V ) = O(g · x, g · U, gV g −1 ). Exercise 10.2.3 Let U ⊆ X be open and V ⊆ G be open with 1G ∈ V = V −1 . Define ∼U,V on U by x ∼U,V y ⇐⇒ O(x, U, V ) = O(y, U, V ). Show that ∼U,V is an equivalence relation with countably many equivalence classes.

10.3

Turbulent and generically turbulent actions

In Theorem 10.2.7 a purely topological condition is identified from the consideration of generic =+ -ergodicity. This condition was the main part of Hjorth’s definition of turbulence, which we now give.

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Definition 10.3.1 Let G be a Polish group and X a Polish G-space. The action of G on X is turbulent if (T1) every orbit is meager, (T2) every orbit is dense, and (T3) every local orbit is somewhere dense, that is, for any open U ⊆ G, x ∈ U , and open 1G ∈ V ⊆ G, O(x, U, V ) is somewhere dense. To facilitate our discussions we introduce some more definitions. Definition 10.3.2 Let G be a Polish group and X a Polish G-space. The action of G on X is generically turbulent if it is turbulent on a comeager subset of X. Definition 10.3.3 Let G be a Polish group and X a Polish G-space. (1) The action of G on X is preturbulent if for all x, y ∈ X, U ⊆ X open with x ∈ U and open 1G ∈ V ⊆ G, O(x, U, V ) ∩ [y]G = ∅. (2) The action of G on X is generically preturbulent if it is preturbulent on a comeager subset of X. (3) The action of G on X is properly generically preturbulent if it is generically preturbulent and every orbit is meager. It is easy to see that preturbulence is weaker than the condition (T2)+(T3) but implies (T2). It follows that generic turbulence implies proper generic preturbulence. The main objective of this section is to show that the converse is true. Lemma 10.3.4 Let Y = {(xn ) ∈ (2ω )ω : ∀n = m ∈ ω xn = xm }. Then =+ Y ∼B =+ . Proof. For any z ∈ 2ω and k ∈ ω, let ζ(z, k) = 0k 1 z, that is, ζ(z, k) ∈ 2ω is such that ζ(z, k)(i) = 1 ⇐⇒ i = k ∨ ( i ≥ k + 1 ∧ z(i − k − 1) = 1 ). It is easy to check that for all z, z  ∈ 2ω , ζ(z, k) = ζ(z  , k  ) iff z = z  and k = k . Now for any x = (xn ) ∈ (2ω )ω let Sx = {ζ(xn , k) : k ∈ ω}. Then Sx is a countably infinite set, and for x, x ∈ (2ω )ω , Sx = Sx iff x =+ x . Let η(x) ∈ (2ω )ω be an element encoding Sx without repetitions. Then η(x) =+ η(x ) iff x =+ x . This η is a Borel reduction of =+ to =+ Y .

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Theorem 10.3.5 Let G be a Polish group and X a Polish G-space. If the action of G on X is X is generically =+ -ergodic. preturbulent, then EG Proof. Let {Vn }n∈ω a countable open nbhd base for 1G . X Let f be a Baire measurable generic homomorphism from EG to =+ Y , where Y is as in Lemma 10.3.4. Let C ⊆ X be a comeager Borel set so that X  C to =+ Y and that C ⊆ C ∗G , f  C is a continuous homomorphism from EG ∗ that is, for any x ∈ C, ∀ g ∈ G g · x ∈ C. To see that such a C exists, let C0 X be a dense Gδ set so that f  C0 is a continuous homomorphism from EG  C0 + ∗G ∗G to =  Y . Note that C0 is comeager and Borel. Let C = C0 ∩ C0 . Then C ⊆ C0 is comeager Borel, and C ⊆ C ∗G . X Since f  C is a homomorphism from EG  C to =+ Y , for any x ∈ C, k ∈ ω,  and x ∈ [x]G ∩ C, there is a unique l ∈ ω such that (f (x))k = (f (x ))l . Fix x ∈ C and k ∈ ω. We claim that for some l ∈ ω there is a nonempty open W ⊆ G so that for all g ∈ W with g · x ∈ C, (f (g · x))l = (f (x))k . To see this, let

Hl = {g ∈ G : g · x ∈ C ∧ (f (g · x))l = (f (x))k } for each l ∈ ω. Then l Hl = {g ∈ G : g · x ∈ C} is a comeager set in G. Hence for some l ∈ ω Hl is nonmeager and therefore comeager in some nonempty open set W ⊆ G. Fix such an l ∈ ω and W ⊆ G. Now suppose g ∈ W with g · x ∈ C. Then there is a sequence (gn ) of elements of W ∩ Hl such that gn → g as n → ∞. Then gn · x ∈ C and (f (gn · x))l = (f (x))k for all n ∈ ω. By continuity of f on C, and of the action, (f (g · x))l = (f (x))k . This proves the claim. It follows from the claim that for any x ∈ C there are l ∈ ω, g ∈ G and an open 1G ∈ V ⊆ G such that for any h, h ∈ V so that hg · x, h g · x ∈ C, we have (f (hg · x))l = (f (h g · x))l . In view of this fact, define, for each x ∈ C and l ∈ ω, N (x, l) = n + 1 where n is the least such that for any h ∈ Vn with h · x ∈ C, we have (f (h · x))l = (f (x))l ; if no such n exists then let N (x, l) = 0. Note that N (x, l) is a Borel function from C × ω to ω, since by continuity of f  C, ∀h ∈ Vn ( h · x ∈ C ⇒ (f (h · x))l = (f (x))l ) ⇐⇒ ∀∗ h ∈ Vn ( h · x ∈ C ⇒ (f (h · x))l = (f (x))l ) is Borel. Now let D ⊆ C be a comeager Borel set such that N  (D × ω) is continuous and that D ⊆ D∗G . Such a D exists by a similar argument as that for the existence of C. The above claim implies that for any x ∈ D and k ∈ ω there are x ∈ [x]G ∩ D and l ∈ ω such that N (x , l) > 0 and (f (x ))l = (f (x))k . Let x, y ∈ D. We show that f (x) =+ f (y). By symmetry it suffices to show that (f (x))k ∈ {(f (y))n : n ∈ ω} for any k ∈ ω. For this fix k ∈ ω. There are x ∈ [x]G ∩ D and l ∈ ω such that N (x , l) > 0 and ((f (x ))l = (f (x))k . Hence without loss of generality we may assume that

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x = x and k = l. Thus N (x, k) > 0. Suppose N (x, k) = m + 1. Since N  (D × ω) is continuous, there is an open set U ⊆ X with x ∈ U such that for all z ∈ U ∩ D, N (z, k) = m + 1. This means that for all z ∈ U ∩ D and g ∈ Vm with g · z ∈ D, (f (g · z))k = (f (z))k . Let W ⊆ G be open with 1G ∈ W 3 ⊆ Vm , and let V = W ∩ W −1 . Now by preturbulence O(x, U, V ) ∩ [y]G = ∅. This means that there is y0 ∈ [y]G ∩ U and sequence (xi )i∈ω of elements in U and (gi )i∈ω of elements of V such that x0 = x, xi+1 = gi · xi for all i ∈ ω, and y0 ∈ {xi : i ∈ ω}. Let d be a compatible complete metric on X. We define sequences (xi )i∈ω of elements of U and (i )i∈ω of elements of V such that for all i ∈ ω, i · xi = xi , d(xi , xi ) < 2−i , and xi ∈ D ∩ U . For this fix i ∈ ω. If i = 0 let 0 = 1G and x0 = x0 = x. Assume i > 0. Let hi−1 = gi−1 gi−2 · · · g0 . Then xi = hi−1 · x. Since x ∈ D, we have that for comeager many g ∈ G, g · x ∈ D. It follows that there are comeager many g ∈ V hi−1 such that g · x ∈ D. Let i ∈ V be such that hi−1 · x ∈ D ∩ U and that d(i hi−1 · x, hi−1 · x) < 2−i . This means that d(i · xi , xi ) < 2−i . So if we let xi = i · xi the required properties are fulfilled. Now note that y0 ∈ {xi : i ∈ ω}. Also since xi ∈ D ∩ U , N (xi , k) = m + 1. Thus for all g ∈ Vm with g · xi ∈ D, (f (g · xi ))k = (f (xi ))k . For each i ∈ ω, −1  ∈ V 3 ⊆ W 3 ⊆ Vm , we have that xi+1 = i+1 gi −1 i · xi . But since i+1 gi i (f (xi+1 ))k = (f (xi ))k . This means that for all i ∈ ω, (f (xi ))k = (f (x))k . In order to apply the continuity of f  D we need δ ∈ Vm with the following properties: (i) δ · y0 ∈ D, (ii) for all i ∈ ω, δ · xi ∈ D, and (iii) for all i ∈ ω, (f (δ · xi ))k = (f (xi ))k = (f (x))k . Granting such a δ we have that δ · y ∈ {δ · xi : i ∈ ω}, and then by the continuity of f  D, (f (δ · y0 ))k = (f (x))k , which finishes the proof. However, since y ∈ D ⊆ D∗G , the set of δ such that δ · y0 ∈ D is comeager. Similarly, since each xi ∈ D, the set of δ such that δ · xi ∈ D is comeager for each i ∈ ω. Finally, since for any z ∈ U ∩D, N (z, k) = m+1, we have (f (δ·xi ))k = (f (xi ))k whenever δ ∈ Vm and δ · xi ∈ D. The set of such δ is comeager in Vm . By taking δ to be in the intersection of all these sets, we obtain the required properties (i) through (iii). Corollary 10.3.6 Let G be a Polish group and X a Polish G-space. Then the action of G on X X is generically =+ -ergodic. is generically preturbulent iff EG Proof. Suppose the action is preturbulent on a dense Gδ set C ⊆ X. Then it is also preturbulent on the invariant dense Gδ set C ∗G . It follows from TheX X orem 10.3.5 that EG is generically =+ -ergodic. Conversely, if EG is gener-

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ically =+ -ergodic then by Theorem 10.2.7 there is a comeager set C ⊆ X such that for all x ∈ C, open U ⊆ X with x ∈ U , and open 1G ∈ V ⊆ G, X O(x, U, V ) contains some nonempty open set. Note that EG is generically ergodic (Exercise 10.1.4), thus there is a dense orbit. It follows that there is a comeager set D ⊆ C all of whose orbits are dense. Then for any y ∈ D, O(x, U, V ) ∩ [y]G = ∅. Corollary 10.3.7 Let G be a Polish group and X a Polish G-space. Then the following are equivalent: (a) The action of G on X is generically turbulent. (b) The action of G on X is properly generically preturbulent. X is properly generically =+ -ergodic. (c) EG

Proof. It is obvious that (a) implies (b). The equivalence of (b) and (c) is immediate from the preceding theorem. Then (c) implies (a) because of Theorem 10.2.7. Exercise 10.3.1 Show that if an action satisfies (T2) and (T3) of Definition 10.3.1 then it is preturbulent. Show that in a preturbulent action every orbit is dense. Let G be a Polish group and X a Polish G-space. Call a point x ∈ X turbulent if for all open U ⊆ X with x ∈ U and open 1G ∈ V ⊆ G, O(x, U, V ) is somewhere dense. Call an orbit [x]G turbulent if every y ∈ [x]G is turbulent. Exercise 10.3.2 Show that x is turbulent iff for all open U ⊆ X with x ∈ U and open 1G ∈ V ⊆ G, there is an open U  ⊆ U with x ∈ U  ⊆ O(x, U  , V ). Exercise 10.3.3 Let bases for X and G be given. Show that x is turbulent iff for all basic open U ⊆ X with x ∈ U and basic open 1G ∈ V ⊆ G, O(x, U, V ) is somewhere dense. Exercise 10.3.4 Show that the set of all turbulent points is Gδ . Exercise 10.3.5 Show that if x is turbulent and y ∈ [x]G then y is turbulent. Thus an orbit is turbulent iff it contains a turbulent point. Exercise 10.3.6 Show that if there is a dense turbulent orbit then there is a comeager set of points all of whose orbits are dense turbulent. Exercise 10.3.7 Let G be a Polish group and X a Polish G-space. Show that the following are equivalent:

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(i) The action of G on X is generically turbulent. (ii) There is a dense turbulent orbit in X and every orbit is meager. (iii) There is a comeager set of points all of whose orbits are dense, meager, and turbulent.

10.4

The Hjorth turbulence theorem

In this section we prove the Hjorth turbulence theorem stating that any orbit equivalence relation of a turbulent action is properly generically Eturbulent for any S∞ -orbit equivalence relation E. This is the main theorem of this chapter. We need the following lemma in the main proof. The lemma gives a generic uniform property for homomorphisms of orbit equivalence relations. Lemma 10.4.1 Let G, H be Polish groups, X a Polish G-space, Y a Polish H-space, and f a X Y to EH . Then for any open Baire measurable generic homomorphism from EG 1H ∈ W ⊆ H there is a comeager set C ⊆ X such that for any x ∈ C there is an open 1G ∈ V ⊆ G such that ∀∗ g ∈ V ∃h ∈ W ( f (g · x) = h · f (x) ). Proof.

Fix an open 1H ∈ W ⊆ H. Let A ⊆ X be defined as x ∈ A ⇐⇒ x ∈ ( f −1 (W · f (x)) ) G .

It is straightforward to check that the lemma states that A is comeager. Let X C0 ⊆ X be a dense Gδ such that f is a continuous homomorphism from EG  Y ∗G ∗G ∗G C0 to EH . Then C0 is comeager and Borel. We claim that C0 ∩ C0 ⊆ A . For this fix also x ∈ C0 ∩ C0∗G . Let W0 ⊆ H be open such that 1H ∈ W0 = W0−1 ⊆ W02 ⊆ W . Then there is a countable set R ⊆ H such that W0 R = H. X Y  C0 to EH , we have that Since f  C0 is a homomorphism from EG [x]G ∩ C0 ⊆ f −1 (H · f (x)) = f −1 (W0 R · f (x)) =



f −1 (W0 r · f (x)).

r∈R

For each r ∈ R let Gr = {g ∈ G : g · x ∈ C0 ∩ f −1 (W0 r · f (x))}. Note that W0 r · f (x) is Σ11 in Y , from which it follows that C0 ∩ f −1 (W0 r · f (x)) is Σ11 in C0 and Gr is Σ11 in G, and they all have the Baire property. Then

for each r ∈ R let Or ⊆ G be open such that Gr Or is meager. Then r∈R Or is

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dense open in G and each Gr is comeager in Or . Let D = r∈R (Gr ∩ Or ). Then D is comeager in G. We next verify that for all g ∈ D, g · x ∈ A. For this fix g ∈ D. Say g ∈ Gr ∩ Or . Then g · x ∈ C0 ∩ f −1 (W0 r · f (x)). Let V ⊆ G be open with 1G ∈ V and V g ⊆ Or . Then for comeager many g  ∈ V , g  g ∈ Gr ∩ Or , and so g  g · x ∈ C0 ∩ f −1 (W0 r · f (x)). This means that there are h0 , h0 ∈ W0 such that f (g · x) = h0 r · f (x) and f (g  g · x) = h0 r · f (x). Thus f (g  g · x) = h0 h−1 · f (g · x). Since h0 h0 ∈ W02 ⊆ W , we have that 0  f (g g · x) ∈ W · f (g · x), or g  · (g · x) ∈ f −1 (W · f (g · x)). This gives that g · x ∈ A, hence the claim is proved. By the claim A∗G is comeager, and hence ∀∗ x ∈ X ∀∗ g ∈ G g · x ∈ A. By Kuratowski–Ulam, ∀∗ g ∈ G ∀∗ x ∈ X g · A. Let g be any such element. Then ∀∗ x g · x ∈ A. This states that g −1 · A is comeager. And by continuity of the action, A is comeager. Theorem 10.4.2 Let G be a Polish group, X a Polish G-space, and Y a Polish S∞ -space. If X is generically ESY∞ -ergodic. the action of G on X is preturbulent, then EG Proof. Fix a countable base {Un }n∈ω for X with U0 = X and a countable nbhd base {Vn }n∈ω for 1G in G with V0 = G. Let dY < 1 be a compatible complete metric on Y . Let D be the complete metric on S∞ given by −1 D(h1 , h2 ) = d(h1 , h2 ) + d(h−1 1 , h2 ), where d is the usual metric on S∞ inherω ited from the Baire space ω , that is,  −n 2 , if n ∈ ω is the least such that h1 (n) = h2 (n), d(h1 , h2 ) = 0, otherwise. Let e = 1S∞ . For k ∈ ω let Nk = {h ∈ S∞ : d(h, e) < 2−k } = {h ∈ S∞ : h(i) = i ∀i ≤ k}. Then {Nk }k∈ω is a nbhd base for e in S∞ . X Let f be a Baire measurable generic homomorphism from EG to ESY∞ . By Lemma 10.4.1 for any Nk ⊆ S∞ there are comeager many x ∈ X such that there is some basic open Vm ⊆ G with ∀∗ g ∈ Vm ∃h ∈ Nk f (g · x) = h · f (x). Then there is a comeager set of x ∈ X such that for any Nk there is Vm so that the above displayed property holds. Let C0 ⊆ X be such a comeager set that is Borel and such that f  C0 is a continuous homomorphism from X Y EG  C0 to EH . In view of this define, for x ∈ X and k ∈ ω, N (x, k) = m+ 1 if m is the least so that the above property holds; and N (x, k) = 0 if there is no such m. Then for any x ∈ C0 and k ∈ ω, N (x, k) > 0. Note that N  (C0 × ω) is Baire measurable since the defining condition is Σ11 . Since the action of G on X is preturbulent, by Theorems 10.3.5 and 10.2.7, for all Un ⊆ X and Vm ⊆ G there is a comeager set of x ∈ X such that for all x0 ∈ [x]G ∩ Un , O(x, Un , Vm ) is somewhere dense. It follows that there is

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a comeager set of x ∈ X such that for all Un and Vm , for all x0 ∈ [x]G ∩ Un , O(x, Un , Vm ) is somewhere dense. X Also by Theorem 10.3.5 EG is generically =+ -ergodic, and therefore it is generically ergodic. It follows that there is a comeager set of x ∈ X in which every orbit is dense. Taking the intersections of these comeager sets, we obtain a comeager C1 ⊆ C0 such that (i) for any x ∈ C1 , [x]G ∩ C1 is dense; X Y  C1 to EH ; (ii) f  C1 is a continuous homomorphism from EG

(iii) N  (C1 × ω) → (ω − {0}) is continuous; and (iv) for all Un ⊆ X, Vm ⊆ G, and x ∈ C1 ∩ Un , O(x, Un , Vm ) is somewhere dense. Finally let C = C1 ∩ C1∗G . Then C is comeager, and we claim that for all x, y ∈ C, f (x)ESY∞ f (y). To prove this, fix x, y ∈ C. We construct g, h ∈ S∞ such that g · f (x) = h · f (y). The elements g, h will be approximated by two D-Cauchy sequences (gi ) and (hi ) in S∞ . Each gi or hi arises from finding xi or yi respectively so that gi · f (x) = f (xi ) or hi · f (y) = f (yi ). To guarantee that (f (xi ))i∈ω and (f (yi ))i∈ω converge to the same point in Y we will construct sequences of basic open sets (Unx (i) )i∈ω and (Uny (i) )i∈ω so that Unx (i+1) ⊆ Uny (i) ⊆ Unx (i) for all i ∈ ω and so that diamY f (Unx (i) ) → 0 as i → ∞. Finally these basic open sets Unx (i) and Uny (i) will be found using the turbulence condition (iv); in doing this we also construct sequences of basic open nbhds Vmx (i) and Vmy (i) . In summary we construct sequences of elements indexed by i ∈ ω, nx (i), ny (i), mx (i), my (i) ∈ ω, xi , yi ∈ C, gi , hi ∈ S∞ , so that the following are satisfied for all i ∈ ω: (1) x0 = x, y0 = y; g0 = h0 = e; (2) Unx (i+1) ⊆ Uny (i) ⊆ Unx (i) ; (3x ) O(xi , Unx (i) , Vmx (i) ) is dense in Uny (i) ; (3y ) O(yi , Uny (i) , Vmy (i) ) is dense in Unx (i+1) ; (4x ) xi+1 ∈ Unx (i+1) ∩ C ∩ O(xi , Unx (i) , Vmx (i) ); (4y ) yi+1 ∈ Uny (i+1) ∩ C ∩ O(yi , Uny (i) , Vmy (i) ); (5) diamY f (Unx (i) ∩ C) ≤ 2−i ; (6) gi · f (x) = f (xi ); hi · f (y) = f (yi ); (7) D(gi , gi+1 ) ≤ 2−i ; D(hi , hi+1 ) ≤ 2−i ; (8x ) for i > 0, kx (i) = max{gi (l), gi−1 (l) : l ≤ i}, for all z ∈ Unx (i) ∩ C, N (z, kx (i)) = mx (i) + 1;

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Invariant Descriptive Set Theory (8y ) for ky (i) = max{hi (l), h−1 i (l) : l ≤ i}, for all z ∈ Uny (i) ∩ C, N (z, ky (i)) = my (i) + 1.

Granting the construction, (gi ) and (hi ) are D-Cauchy by (7); and dY (gi · f (x), hi · f (y)) < 2−i by (2), (4), (5), and (6). Let gi → g and hi → h as i → ∞. Then g · f (x) = h · f (y), as required. To begin the construction, we let nx (0) = mx (0) = 0, g0 = e, x0 = x. Then Unx (0) = X and Vmx (0) = G. Note that O(x, X, G) = [x]G is dense in X. Also let y0 = y and h0 = e. Then ky (0) = 0. By the continuity of N on C × ω there is some Un such that y ∈ Un and for all z ∈ Un ∩C, N (z, 0) = N (y, 0) = m+ 1 for some m ∈ ω. Let ny (0), my (0) be such m, n, respectively. Then (3x ) and (8y ) are fulfilled. This finishes the definition for i = 0. By induction assume we have completed the definition for the index i. We indicate how to define nx (i + 1), mx (i + 1), xi+1 , gi+1 . The definition for ny (i + 1), my (i + 1), yi+1 , hi+1 is similar. Since yi ∈ C we have by (iv) that O(yi , Uny (i) , Vmy (i) ) is somewhere dense. Let W ⊆ Uny (i) be nonempty open such that O(yi , Uny (i) , Vmy (i) ) is dense in W and such that diamY f (W ∩ C) ≤ 2−(i+1) . This guarantees requirements (2), (3y ), and (5). We now choose xi+1 to satisfy (4x ). For notational simplicity let U = Unx (i) , V = Vmx (i) , and z0 = xi . Note first that O(z0 , U, V ) is dense in W since it is dense in Uny (i) and W ⊆ Uny (i) . Let z1 , . . . , zl ∈ U , γ0 , . . . , γl−1 ∈ V be such that zj+1 = γj · zj for all j < l and zl ∈ W . In particular the sequence z0 , z1 , . . . , zl witnesses that zl ∈ O(x, U, V ) ∩ W . We will find 0 = 1G , . . . , l ∈ G such that, letting zj = j · zj for all j ≤ l, we have that z0 , z1 , . . . , zl ∈ C ∩ U , zl ∈ W , and i+1 γi −1 ∈V. i This guarantees that the sequence z0 , z1 , . . . , zl of elements of C witnesses −1 that zl ∈ O(x, U, V ) ∩ W ∩ C. For this let 1G ∈ V  = V  be open such   that for all j ≤ l, V γi V ⊆ V . It suffices to choose 1 , . . . , l ∈ V  so that j · zj ∈ C ∩ U and zl ∈ W . But for each j ≤ l the set of such j is comeager in a nonempty open subset of V  , hence the required j can be chosen as required. This construction will eventually guarantee (4x ). Again for notational simplicity we suppress the  and assume that z0 = xi , z1 , . . . , zl ∈ U ∩ C, γ0 , . . . , γl−1 ∈ V are defined such that zj+1 = γj · zj for all j < l and that zl ∈ W . We indicate how to fulfill (7). Since zj ∈ U ∩ C for all j ≤ l, by (8x ) of index i, for k = kx (i) = max{gi (λ), gi−1 (λ) : λ ≤ i}, N (zj , k) = mx (i) + 1. This means that ∀∗ γ ∈ V = Vmx (i) ∃p ∈ Nk f (γ · zj ) = p · f (zj ). By repeating the construction in the preceding paragraph we may assume that γ0 , . . . , γl−1 are chosen from such comeager sets that there are f (γj · zj ) = pj · f (zj ) for some pj ∈ Nk for all j < l. This gives that f (zj+1 ) = pj · zj and therefore f (zl ) = pl−1 . . . p0 · f (z0 ) = (pl−1 . . . p0 ) · f (xi ). Note that pl−1 . . . p0 ∈ Nk .

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In view of the above argument, we may let xi+1 = zl as above. Then gi+1 = (pl−1 . . . p0 )gi , which implies that d(gi+1 , gi ), d(gi+1 , gi ) ≤ 2−(i+1)) , and so D(gi+1 , gi ) ≤ 2−i . This fulfills (6) and (7). −1 (λ) : λ ≤ i + 1} and mx (i + 1) = Finally, let kx (i + 1) = max{gi+1 (λ), gi+1 N (xi+1 , kx (i+1))−1. By the continuity of N  (C×ω), let nx (i+1) be such that xi+1 ⊆ Vnx (i+1) ⊆ W and for all z ∈ Vnx (i+1) ∩ C, N (z, kx (i + 1)) = mx (i + 1). Then (4x ) and (8x ) are fulfilled. We refer to the following corollary as the Hjorth turbulence theorem. Corollary 10.4.3 Let G be a Polish group and X a Polish G-space. If the action of G on X is properly generically ESY∞ -ergodic for any Polish X is turbulent, then EG X ≤B ESY∞ . S∞ -space Y ; in particular, EG The following corollary is a continuation of Corollary 10.3.7. Corollary 10.4.4 Let G be a Polish group and X a Polish G-space. Then the following are equivalent: (a) The action of G on X is generically turbulent. X is properly generically ESY∞ -ergodic for all Polish S∞ -space Y . (d) EG

Exercise 10.4.1 Let G be a Polish group, X a Polish G-space, H a closed subgroup of S∞ , and Y a Polish H-space. Suppose =+ ≤B F . Show that X the action of G on X is generically turbulent iff EG is properly generically F -ergodic.

10.5

Examples of turbulence

In this section we will show that the equivalence relations c0 , p (1 ≤ p < +∞) we introduced in Sections 8.4 and 8.5 are not Borel reducible to any S∞ orbit equivalence relation. This is by showing that the actions are actually turbulent. Definition 10.5.1 A subset S of Rω is strongly dense if S = Rω but for every n ∈ ω and (x0 , . . . , xn−1 ) ∈ Rn , there is y ∈ S such that yi = xi for all i < n. Theorem 10.5.2 Let G ⊆ Rω be a strongly dense Polishable subgroup of Rω . Then the translation action of G on Rω is turbulent.

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Proof. First it is easy to check that every orbit is dense. Thus the action is generically ergodic, and every invariant Borel set, in particular, every orbit, is either meager or comeager. However, every orbit is a homeomorphic copy of G; thus if any orbit is comeager then every orbit is comeager, contradicting the assumption that G is a proper subset of Rω (and therefore there is more than one orbit). This shows that every orbit is meager. It remains to check the condition (T3) from the definition of turbulence. It suffices to show that for all x ∈ Rω , basic open U ⊆ Rω with x ∈ U and open 1G ∈ V ⊆ G, O(x, U, V ) is somewhere dense. For this let x ∈ U , where U = {y ∈ Rω : |yi − xi | <  ∀i < n} for some n ∈ ω and  > 0. Let τ be the Polish group topology on G. Let 1G ∈ V ⊆ G be τ -open. We claim that O(x, U, V ) is indeed dense in the entire U . For this let y ∈ U be arbitrary, and let U0 be a basic open set containing y. Without loss of generality we may assume that for some δ <  and m ≥ n, U0 = {z ∈ Rω : |zi − yi | < δ ∀i < m}. Consider the projection πm : G → Rm where for all g ∈ G, πn (g) = (g0 , . . . , gm−1 ). By Polishability of G, πm is a Borel homomorphism from the Polish group (G, τ ) into Rm . Since G is strongly dense, πm is onto. Thus by Theorem 2.3.3 πm is both continuous and open. Let W = πm (V ). Then (0, . . . , 0) ∈ W and W ⊆ Rm is open. Thus for some η > 0, we have W0 = {w ∈ Rm : |wi | < η ∀i < m} ⊆ W. Now let N be large enough such that |xi −yi | < N η for all i < m. Let w ∈ Rm be such that wi = (yi − xi )/N for all i < m. Then w ∈ W0 . Let g ∈ V be such that πm (g) = w. Then the sequence x, g + x, 2g + x, . . . , N g + x witnesses that N g+x ∈ O(x, U, V ). Finally N g+x ∈ U0 , and thus O(x, U, V )∩ U0 = ∅. Corollary 10.5.3 For any S∞ -orbit equivalence relation E and any 1 ≤ p < +∞, c0 , p ≤B E. Proof. Rω .

The Polishable groups c0 , p (1 ≤ p < +∞) are strongly dense in

In fact a large collection of Polish group actions have been examined and proved to be turbulent. The details of these proofs very often depend on

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the nature of the group and the action involved, and offer little insight into further development of the theory of turbulence. The theory of turbulence establishes an important criterion of classifiability for classification problems in mathematics.

10.6

Orbit equivalence relations and E1

In this section we apply the techniques developed in this chapter and prove a theorem of Kechris and Louveau [102] about the hypersmooth equivalence relation E1 . The proof we present is due to Hjorth [70]. Recall from Section 8.1 that the equivalence relation E1 is defined on 2ω×ω . By the Borel isomorphism between 2ω and R, E1 is obviously Borel isomorphic to the following equivalence relation on Rω , which we still denote by E1 : xE1 y ⇐⇒ ∃n ∈ ω ∀m ≥ n xm = ym . Note that E1 can be written as an increasing sequence of smooth equivalence relations Fn where xFn y ⇐⇒ ∀m ≥ n xm = ym . Each Fn can be viewed as an orbit equivalence relation of an action of Rn on Rω given by  gm + xm , if m < n, (g · x)m = xm , otherwise. The action is apparently continuous. The increasing union of Rn for all n ∈ ω, which we would denote by R<ω , is customarily denoted as c00 in Banach space theory. c00 is an Fσ subgroup of Rω , and therefore is a standard Borel group. Similar to Lemma 9.3.3 one can show that c00 is not Polishable (see Exercise 10.6.1). Theorem 10.6.1 (Kechris–Louveau) X Let G be a Polish group and X a Borel G-space. Then E1 ≤B EG . Proof. As usual we assume without loss of generality that X is a Polish X . Then G-space. Assume f : Rω → X is a Borel reduction from E1 to EG X for each n ∈ ω, f is a Borel homomorphism from Fn to EG . By applying Lemma 10.4.1 to f for all n ∈ ω, we can obtain a dense Gδ subset C of Rω such that (i) f  C : C → X is continuous,

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(ii) for all n ∈ ω, all open nbhd W of 1G in G, and x ∈ C, there is an open nbhd V of the identity of Rn such that ∀∗ h ∈ V ∃g ∈ W ( f (h · x) = g · f (x) ). Let {Oi }i∈ω be a sequence of dense open subsets of Rω such that C = i∈ω Oi . n ω Rn For each n ∈ ω let Cn = C ∗R . Then each Cn is comeager in R and ω invariant. It follows that for any n ∈ ω, m≥n Cm is comeager in R and Rn -invariant. Let dG be a compatible complete metric on G. We are ready to construct sequences of elements indexed by i ∈ ω, xi ∈ Rω , hi ∈ Hi , Ui open in Rω , gi ∈ G, such that the following conditions hold: (1) xi+1 = hi · xi ; (2) xi ∈ Ui ∩ C ∩

m≥i

Cm ;

(3) hij > 0 for all j < i; (4) hij < 2−i for all j < i; (5) Ui+1 ⊆ Ui ⊆ Oi ; (6) gi · f (x0 ) = f (xi ); (7) dG (gi , gi+1 ) < 2−i . Granting the construction, we will arrive at a contradiction as follows. By (1) and (4) limi xi exists in Rω , which we denote by x. By (3), x0 and x are E1 -inequivalent. By (5) x ∈ C. However, by (7) there is g∞ ∈ G with gi → g∞ as i → ∞. Thus by (2), (6), and the continuity of f  C, g∞ · f (x0 ) = lim gi · f (x0 ) = lim f (xi ) = f (x). i

i

X f (x), and by our assumption that f is a reduction, This shows that f (x0 )EG 0 x E1 x, a contradiction. It remains to construct the sequences by induction on i. To begin with, let x0 ∈ C ∩ m≥0 Cm be arbitrary. Since x0 ∈ O0 , let U0 be any open set in Rω with x0 ∈ U0 ⊆ O0 . Then let g0 = 1G . In the inductive step we assume that xi , Ui , and gi have been defined. We define hi , Ui+1 and gi+1 to satisfy the conditions (1) through (7) for appropriate indices. Let W be an open nbhd of 1G such that for all g ∈ W , dG (ggi , gi ) < 2−i . By (ii) let V0 be an open nbhd of the identity in Ri such that

∀∗ h ∈ V0 ∃g ∈ W ( f (h · xi ) = g · f (xi ) ).

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Let D ⊆ V0 be a comeager set in V0 witnessing the displayed property. Let V ⊆ V0 be also an open nbhd of the identity of Rn such that V · xi ⊆ Ui and −i i for all h ∈ V and j < i, hj < 2 . Now note that the set D ∩ {h ∈ V : h · x ∈ C ∩ m≥i Cm } is comeager in V . Since {h ∈ V : ∀j < i hj > 0 } is an open subset of V , we have that  Cm ∧ ∀j < i hj > 0} = ∅. D ∩ {h ∈ V : h · xi ∈ C ∩ m≥i

Let hi be an arbitrary element of this nonempty set. Then we have that hi · xi ∈ Ui ∩ C ∩ m≥i Cm . Let xi+1 = hi · xi . Conditions (1), (3), and (4) are satisfied. Since hi ∈ D, there is g ∈ W such that f (hi · xi ) = g · f (xi ). Fix such a g, and let gi+1 = ggi . Then g · f (xi ) = ggi · f (x0 ) = gi+1 · f (x0 ), and −i since g ∈ W , dG (gi+1 , gi ) = dG (gg i , gi ) < 2 . Thus (6) and (7) are fulfilled. Finally since xi+1 ∈ Ui ∩ C = Ui ∩ i Oi , we may find open set Ui+1 such that xi+1 ∈ Ui+1 ⊆ Ui+1 ⊆ Ui ∩ Oi . This gives (2) and (5). We have thus finished the construction as required. Of course it follows from the theorem that c00 is not Polishable, since otherwise E1 would be itself an orbit equivalence relation of a Polish group action. Exercise 10.6.1 Give a direct proof that c00 is not Polishable.

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Part III

Countable Model Theory

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Chapter 11 Polish Topologies of Infinitary Logic

In this part of the book we study S∞ -orbit equivalence relations. In Sections 2.4 and 3.6 we have already seen connections between S∞ -orbit equivalence relations and isomorphism relations for countable models. In this part we will establish a full duality between the two subjects. Historically many of the model theoretic results predated and in fact motivated their group action counterpart. On the other hand, the study of isomorphism relation of countable structures also benefited from the perspective of Polish group actions. Since the realization that they are the same subject our understanding of countable model theory has advanced significantly.

11.1

A review of first-order logic

Recall that some notion of logic has been reviewed in Sections 2.4 and 3.6, where we gave a characterization of closed subgroups of S∞ by automorphism groups of countable structures and discussed the logic actions of S∞ on Mod(L) for countable relational languages L. In this section we review more concepts of logic. Let L = {Ri }i∈I be a countable relational language, where Ri is an ni -ary relation symbol. Fix once and for all variable symbols v0 , v1 , . . . , vn , . . . and let V = {vi : i ∈ ω}. An atomic L-formula is an expression of the form Ri (x1 , . . . , xni ), where i ∈ I and x1 , . . . , xni ∈ V. We will use the logical connectives ¬, ∧, ∨, ∃, ∀ and necessary parentheses to obtain more complicated formal expressions. A negated atomic L-formula is an expression of the form ¬ϕ, where ϕ is an atomic L-formula. The set of L-formulas is the smallest set F of expressions satisfying the following closure properties: (i) if ϕ is an atomic L-formula then ϕ ∈ F;

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(ii) if ϕ, ψ ∈ F then ¬ϕ, ϕ ∧ ψ, ϕ ∨ ψ ∈ F; (iii) if ϕ ∈ F and x ∈ V, then ∃xϕ, ∀xϕ ∈ F. It is clear that any formula is of finite length and the set F is countable. As in the usual practice of formal mathematics, when forming more complicated L-formulas we use parentheses to eliminate any possible ambiguity in parsing the formula, and omit them when there is no danger of confusion. Thus any L-formula can be uniquely classified as one of the six forms mentioned in (i) through (iii). This allows us to prove statements about L-formulas by induction on their forms as well as make inductive definitions. For example, the concept of a subformula of an L-formula ϕ is inductively defined on the form of ϕ as follows: (i) ϕ is a subformula of ϕ; (ii) if ϕ = ¬ψ then ψ is a subformula of ϕ; if ϕ = ψ ∧ ψ  or ϕ = ψ ∨ ψ  then both ψ and ψ  are subformulas of ϕ; (iii) if ϕ = ∃xψ or ϕ = ∀xψ then ψ is a subformula of ϕ; (iv) if φ is a subformula of ϕ and ψ is a subformula of φ, then ψ is a subformula of ϕ. Example 11.1.1 Let R be a binary relation symbol. The following formula ϕ attempts to describe R as an equivalence relation: ∀v0 R(v0 , v0 ) ∧ ∀v0 ∀v1 (¬R(v0 , v1 ) ∨ R(v1 , v0 )) ∧∀v0 ∀v1 ∀v2 (¬R(v0 , v1 ) ∨ ¬R(v1 , v2 ) ∨ R(v0 , v2 )). The clause on symmetry ¬R(v0 , v1 ) ∨ R(v1 , v0 ) is a subformula of ϕ. The example shows that it is desirable to introduce logical connectives → and ↔ as informal substitutes of their formal counterparts. With these the symmetry clause can be replaced by a more readable expression R(v0 , v1 ) → R(v1 , v0 ). A variable can occur in different subformulas of an L-formula. The occurrences sometimes have different properties. An occurrence in ϕ of a variable x is bound if it happens in a subformula of the form ∃xψ or ∀xψ; otherwise the occurrence is free. A variable x is a free variable of ϕ if there is a free occurrence of x in ϕ. An L-formula is an L-sentence if it does not have any free variables. Example 11.1.2 The formula ϕ defined in Example 11.1.1 is in fact a sentence. Consider the following formula ψ: ϕ ∧ ∃v1 ¬R(v0 , v1 ).

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The explicit occurrence of v0 in ψ is free, making v0 a free variable of ψ, despite the fact that v0 also occurs bound in the subformula ϕ. The formula ψ attempts to address a hypothetical element whose equivalence class is not the whole domain. In the above example the variable v0 occurs both free and bound in the formula ψ, which is legitimate but confusing. It is clear that we may avoid this by replacing all bound occurrences of v0 by a new variable symbol, say v3 , without changing the intended meaning of the formula. In general, it is not hard to show that any formula is logically equivalent to one with none of the variables occurring both free and bound. A formula is called quantifier-free if it does not contain any quantifiers ∃ or ∀. A formula is in prenex normal form if it is of the form Q1 x1 . . . Qk xk ϕ where Q1 , . . . , Qk ∈ {∃, ∀} are quantifiers and ϕ is quantifier-free. It is well known that any formula is logically equivalent to one in prenex normal form. Before continuing we need to deal with a nuisance. Suppose in the above examples we need to describe a property of the equivalence relation that every equivalence class contains exactly one element. Then the natural formula to employ is ∀v0 ∀v1 (R(v0 , v1 ) → v0 = v1 ). However, under our definitions this is not a legitimate L-formula, since the equality symbol is not specified to be in L. Even if the symbol = is added to the language, it is impossible to describe the desired properties of the equality relation with a formula. Therefore, our approach is to add = as a default symbol beyond those in L and make sure that it is always interpreted as the true equality relation. To emphasize this distinction, we call an (L ∪ {=})formula an Lωω -formula and similarly handle other concepts defined above. Note that this is not the standard practice in the literature. Some authors choose to specify that L is a language with equality and continue to call our Lωω -formulas L-formulas. What we have just defined is the syntax of first-order logic Lωω . In this notation the first subscript ω stands for the fact that only finitary (formally binary) conjunctions ∧ and disjunctions ∨ are allowed in any formula, and the second subscript ω stands for the fact that only finitely many quantifiers ∃ or ∀ are allowed in any formula. We now turn to the semantics of first-order logic. Consider an L-structure M = (|M |, {RiM }i∈I ) as defined in Section 2.4. An assignment is a function  : V → |M |. For a variable x ∈ V and an element a ∈ |M |, we define a new assignment [a/x] by letting  a, if y = x, [a/x](y) = (y), otherwise.

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Given an Lωω -formula ϕ, we define a satisfaction relation (M, ) |= ϕ by induction on the form of ϕ as follows: (i) If ϕ is x = y for x, y ∈ V, then (M, ) |= ϕ iff (x) = (y). (ii) If ϕ is Ri (x1 , . . . , xni ) for i ∈ I and x1 , . . . , xni ∈ V, then (M, ) |= ϕ iff RiM ((x1 ), . . . , (xni )) holds. (iii) If ϕ is ¬ψ, then (M, ) |= ϕ iff (M, ) |= ψ. (iv) If ϕ is φ ∧ ψ, then (M, ) |= ϕ iff both (M, ) |= φ and (M, ) |= ψ. (v) If ϕ is φ ∨ ψ, then (M, ) |= ϕ iff either (M, ) |= φ or (M, ) |= ψ. (vi) If ϕ is ∃xψ where x ∈ V, then (M, ) |= ϕ iff there is a ∈ |M | such that (M, [a/x]) |= ψ. (vii) If ϕ is ∀xψ where x ∈ V, then (M, ) |= ϕ iff for all a ∈ |M |, (M, [a/x]) |= ψ. We say that M satisfies ϕ, or M models ϕ, and denote it by M |= ϕ, if (M, ) |= ϕ for all assignments . These definitions make precise our usual interpretation of formulas by statements about the model. It is easy to check that if ϕ is a sentence, that is, every variable in ϕ is bound, then for any assignment , (M, ) |= ϕ iff M |= ϕ. The following notation will be handy. If  is an assignment, x1 , . . . , xk ∈ V are distinct, and a1 , . . . , ak ∈ |M | arbitrary, we define [a1 /x1 , . . . , ak /xk ] = [a1 /x1 ] . . . [ak /xk ]. If ϕ is a formula with its free variables among x1 , . . . , xk , we will write ϕ as ϕ(x1 , . . . , xk ) to emphasize it. For a1 , . . . , ak ∈ |M |, we write M |= ϕ[a1 /x1 , . . . , ak /xk ], or simply M |= ϕ[a1 , . . . , ak ], if for any assignment , (M, [a1 /x1 , . . . , ak /xk ]) |= ϕ. It is not hard to check that the definition makes sense since the satisfaction relation (M, ) |= ϕ(x1 , . . . , xk ) does not depend on the values of  outside {x1 , . . . , xk }. An Lωω -theory, or simply a theory, T is a set of Lωω -sentences. A nonempty L-structure M is a model of T , and we denote M |= T , if M |= ϕ for all ϕ ∈ T . The theory T is consistent if there exists a model of T , and inconsistent otherwise. The famous compactness theorem of G¨odel states that a theory T is consistent iff every finite subset of T is consistent. On the other hand, the theory of an L-structure M , denoted Th(M ), is defined by Th(M ) = {ϕ : ϕ is an Lωω -sentence and M |= ϕ}. A theory T is complete if T = Th(M ) for some L-structure M . Note that by our definition for any Lωω -sentence ϕ and complete theory T , either ϕ ∈ T or ¬ϕ ∈ T .

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In a refined analysis we define Lωω -types. For n ∈ ω, an Lωω -n-type, or simply an n-type, S is a set of Lωω -formulas such that for some distinct variables x1 , . . . , xn ∈ V all formulas in S have free variables among x1 , . . . , xn . In this situation we write S(x1 , . . . , xn ) to emphasize the hypothesis that all formulas in S have free variables among x1 , . . . , xn . A type is an n-type for some n ∈ ω. If S(x1 , . . . , xn ) is an n-type, M an L-structure, and a1 , . . . , an ∈ M , we say that (a1 , . . . , an ) realizes S, and write M |= S[a1 , . . . , an ], if for all ϕ(x1 , . . . , xn ) ∈ S, M |= ϕ[a1 , . . . , an ]. An n-type S(x1 , . . . , xn ) is consistent with a theory T if there is a model M of T and a1 , . . . , an ∈ |M | such that M |= S[a1 , . . . , an ], and inconsistent with T otherwise. In this case we say that S is a type of T . A type is consistent (inconsistent) if it is consistent (inconsistent, respectively) with the empty theory. A type S(x1 , . . . , xn ) is a complete type of a theory T if for all formulas ϕ(x1 , . . . , xn ), either S ∪ {ϕ} or S ∪ {¬ϕ} is inconsistent with T . A type is complete if it is a complete type of the empty theory. For an L-structure and a1 , . . . , an ∈ |M |, the type of the tuple (a1 , . . . , an ) is defined as tp(a1 , . . . , an ) = {ϕ(v0 , . . . , vn−1 ) : M |= ϕ[a1 , . . . , an ]}. If M is a model of a theory T and a1 , . . . , an ∈ |M |, then the type of the tuple (a1 , . . . , an ) is obviously a complete type consistent with T . Exercise 11.1.1 Let L = {<}. Find Lωω -formulas whose intended interpretations are the following: (1) < is a linear order. (2) < is dense linear order without endpoints. (3) < is a linear order of cardinality n for some fixed n ∈ ω. Exercise 11.1.2 Find a relational language L and an Lωω -sentence ϕ such that the models of ϕ are exactly Boolean algebras. Exercise 11.1.3 Let R be a 3-ary relation symbol and L = {R}. Find an Lωω -sentence ϕ such that all models M of ϕ are groups with the multiplication defined by a · b = c iff M |= R[a, b, c]. Exercise 11.1.4 Show that if ϕ(x1 , . . . , xk ) is a formula whose free variables are among x1 , . . . , xk and 1 , 2 are assignments such that 1 (xj ) = 2 (xj ) for j = 1, . . . , k, then (M, 1 ) |= ϕ iff (M, 2 ) |= ϕ. Exercise 11.1.5 Let L be the empty language. (a) Find an Lωω -theory T such that the models of T are exactly infinite sets. (b) Use the compactness theorem to show that there is no Lωω -sentence ϕ such that the models of ϕ are exactly infinite sets. (Hint: Assume such ϕ exists. Let T be given by (a) and consider T ∪ {¬ϕ}.)

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Invariant Descriptive Set Theory

Model theory of infinitary logic

In this section we define the infinitary logic Lω1 ω and review more concepts that eventually will lead to connections with invariant descriptive set theory. For a more comprehensive treatment of infinitary logic the reader should consult Reference [3] or [105]. We continue to consider a countable relational language % & L. To define the infinitary logic we use the new logical connectives and . Definition 11.2.1 Let L be a countable relational language. The set Fω1 of Lω1 ω -formulas is the smallest set of expressions satisfying the following closure properties: (i) if ϕ is an atomic Lωω -formula then ϕ ∈ Fω1 ; (ii) if ϕ ∈ Fω1 then ¬ϕ ∈ Fω1 ; (iii) if ϕ ∈ Fω1 and x ∈ V then ∃xϕ, ∀xϕ ∈ Fω1 ; % & (iv) if Φ is a countable subset of Fω1 then Φ, Φ ∈ Fω1 . Similar to the finitary case we can define the notion of a subformula by induction on the form of a formula ϕ as follows: (i) ϕ is a subformula of ϕ; (ii) if ϕ = ¬ψ, or ϕ = ∃xψ, or ϕ = ∃xψ for some x ∈ V, then ψ is a subformula of ϕ; % % (iii) if ϕ = Φ or ϕ = Φ for some countable Φ ⊆ Fω1 , and ψ ∈ Φ, then ψ is a subformula of ϕ; (iv) if φ is a subformula of ϕ and ψ is a subformula of φ then ψ is a subformula of ϕ. Also in a similar fashion one can define the notion of free and bound variables. Then an Lω1 ω -formula is an Lω1 ω -sentence if it does not have any free variables. The following simple lemma can be proved using induction. Lemma 11.2.2 If ϕ is an Lω1 ω -formula with only finitely many free variables, then so does any subformula of ϕ. In particular, if ψ is a subformula of an Lω1 ω -sentence then ψ has only finitely many free variables. The semantics of Lω1 ω is also similar to the finitary case, with the only obvious modification being the following:

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If M is an L-structure,  an assignment, and Φ a countable subset of Lω1 ω -formulas, then ' (M, ) |= Φ iff for all ϕ ∈ Φ, (M, ) |= ϕ, and (M, ) |=

(

Φ iff for some ϕ ∈ Φ, (M, ) |= ϕ.

We also use other concepts, notation, and conventions of first-order logic whenever they are applicable to Lω1 ω . Obviously the infinitary logic has more T is countable, and therefore is expressive power. Note that any Lωω -theory % logically equivalent to the infinitary sentence T . The set Fω1 has the size of the continuum, but every Lω1 ω -formula has only countably many subformulas. Definition 11.2.3 A set F of Lω1 ω -formulas is a fragment if it satisfies the following closure properties: (i) all Lωω -formulas are in F ; (ii) if ϕ ∈ F and ψ is a subformula of ϕ, then ψ ∈ F ; (iii) if ϕ, ψ ∈ F and x ∈ V, then ¬ϕ, ϕ ∧ ψ, ϕ ∨ ψ, ∃xϕ, ∀xϕ ∈ F . (iv) if ϕ ∈ F and ψ is obtained from ϕ by a change of variables, then ψ ∈ F . If A ⊆ Fω1 then the smallest fragment containing A is called the fragment generated by A and is denoted F (A). If A ⊆ Fω1 is countable, then so is F (A). In particular, if A contains a single formula ϕ, then we write F (ϕ) for F ({ϕ}). Apparently the set of all Lωω -formulas is the smallest fragment. We now turn to some concepts in the model theory of Lω1 ω . Definition 11.2.4 Let M, N be L-structures and F a fragment. (1) An Lω1 ω -theory is a set of Lω1 ω -sentences. An F -theory is a set of sentences in F . (2) The F -theory of M , denoted ThF (M ), is the set of all sentences ϕ ∈ F such that M |= ϕ. (3) M and N are F -elementarily equivalent, denoted M ≡F N , if ThF (M ) = ThF (N ), that is, for any sentence ϕ ∈ F , M |= ϕ iff N |= ϕ. (4) An injection f : |M | → |N | is an F -elementary embedding if for any formula ϕ(x1 , . . . , xn ) ∈ F all of whose free variables are among x1 , . . . , xn ∈ V, and for any a1 , . . . , an ∈ M , M |= ϕ[a1 , . . . , an ] ⇐⇒ N |= ϕ[f (a1 ), . . . , f (an )].

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(5) M is a substructure, or a submodel, of N if |M | ⊆ |N | and for any Ri ∈ L and a1 , . . . , ani ∈ M , M |= Ri [a1 , . . . , ani ] ⇐⇒ N |= Ri [a1 , . . . , ani ]. (6) M is an F -elementary substructure of N , denoted M ≺F N , if M is a substructure of N and the identity embedding is an F -elementary embedding. All concepts defined here generalize some similar ones in the first-order logic. We denote these first-order concepts by dropping the subscript F in the corresponding notation. Since L is a relational language, any subset in an L-structure gives rise to a substructure. Thus in our context, substructures correspond uniquely to subsets of structures. We therefore adopt the following notation. If M is a substructure of N , we write M ⊆ N . Conversely, if N is an L-structure and S is a subset of |N | then we also use S to denote the unique substructure of N determined by S. The following lemma collects some easy facts about the concepts defined above. Lemma 11.2.5 Let L be a countable relational language, F a fragment, and M, N, P Lstructures. Then the following hold: (a) If f is an F -elementary embedding from M into N , then f : M ∼ = f (M ) and f (M ) ≺F N . (b) If M ≺F N , then M ≡F N . (c) If M ≺F P and P ≺F N , then M ≺F N . Proof. (a) It follows from the F -elementarity for atomic formulas that f : M∼ = f (M ). Now suppose ϕ(x1 , . . . , xn ) ∈ F and b1 , . . . , bn ∈ f (M ). Let a1 = f −1 (b1 ), . . . , an = f −1 (bn ). Then f (M ) |= ϕ[b1 , . . . , bn ] iff M |= ϕ[a1 , . . . , an ] since f is an isomorphism. But M |= ϕ[a1 , . . . , an ] iff N |= ϕ[b1 , . . . , bn ] by the assumption that f is an F -elementary embedding. Thus we have f (M ) |= ϕ[b1 , . . . , bn ] iff N |= ϕ[b1 , . . . , bn ]. This shows that f (M ) ≺F N . (b) and (c) are obvious. The following lemma is usually referred to as the Tarski–Vaught criterion. Lemma 11.2.6 (Tarski–Vaught) Let L be a countable relational language, F a fragment, N an L-structure, and M ⊆ N . Then M ≺F N iff for all formulas ϕ(x1 , . . . , xn , y) ∈ F and a1 , . . . , an ∈ M , if there is b ∈ N such that N |= ϕ[a1 , . . . , an , b] then there is b0 ∈ M such that N |= ϕ[a1 , . . . , an , b0 ].

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Proof. For (⇒) suppose there is b ∈ N such that N |= ϕ[a1 , . . . , an , b]. Then ∃yϕ ∈ F and N |= ∃yϕ[a1 , . . . , an ], and therefore M |= ∃yϕ[a1 , . . . , an ] by elementarity. This gives some b0 ∈ M such that M |= ϕ[a1 , . . . , an , b0 ]. But by elementarity again we have N |= ϕ[a1 , . . . , an , b0 ] as desired. For (⇐), we show by induction on the form of a formula ψ(x1 , . . . , xn ) ∈ F that for any a1 , . . . , an ∈ M , M |= ψ[a1 , . . . , an ] iff N |= ψ[a1 , . . . , an ]. The case when ψ is atomic is immediate, since M ⊆ N . The case when ϕ is ¬ψ is trivial. If ψ % is Φ for a countable set Φ ⊆ F , then every formula in Φ has free variables among x1 , . . . , xn , and by the inductive hypotheses M |= ψ[a1 , . . . , an ] ⇐⇒ for all φ ∈ Φ, M |= φ[a1 , . . . , an ] ⇐⇒ for all φ ∈ Φ, N |= φ[a1 , . . . , an ] ⇐⇒ N |= ψ[a1 , . . . , an ]. & The case when ψ is Φ is similar. Now if ψ is ∃yϕ, then the free variables of ϕ are among x1 , . . . , xn , y. Suppose M |= ψ[a1 , . . . , an ]. Then there is b0 ∈ M with M |= ϕ[a1 , . . . , an , b0 ], and by the inductive hypothesis N |= ϕ[a1 , . . . , an b0 ], which gives that N |= ψ[a1 , . . . , an ]. Conversely, suppose N |= ψ[a1 , . . . , an ]. Then there is b ∈ N such that N |= ϕ[a1 , . . . , an , b]. By our assumption there is b0 ∈ M such that N |= ϕ[a1 , . . . , an , b] and by the inductive hypothesis M |= ϕ[a1 , . . . , an , b0 ], which gives that M |= ψ[a1 , . . . , an ]. This shows that M |= ψ[a1 , . . . , an ] iff N |= ψ[a1 , . . . , an ]. Finally if ψ is ∀yϕ, then we apply the above argument to ¬ϕ(x1 , . . . , xn , y) and get M |= ∃y¬ϕ[a1 , . . . , an ] iff N |= ∃y¬ϕ[a1 , . . . , an ]. Thus M |= ψ[a1 , . . . , an ] ⇐⇒ M |= ∃y¬ϕ[a1 , . . . , an ] ⇐⇒ N | = ∃y¬ϕ[a1 , . . . , an ] ⇐⇒ N |= ψ[a1 , . . . , an ].

Thus the elementarity of a submodel lies on the existence of witnesses for existential formulas. Exercise 11.2.1 Show that every Lω1 ω -formula has only countably many subformulas. Exercise 11.2.2 Prove Lemma 11.2.2. Exercise 11.2.3 Recall that for a prime number p a p-group is a group in which every element has order pn for some n ∈ ω. Find a countable relational language L and an Lω1 ω -sentence ϕ such that the models of ϕ are exactly the abelian p-groups. (Compare Exercise 11.1.3.) Exercise 11.2.4 Let L = {<} and α < ω1 . Find an Lω1 ω -sentence ϕα such that for any L-structure M , M |= ϕα iff ot(M ) = α.

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Exercise 11.2.5 Show that if M is a substructure of N then for any quantifierfree formula ϕ(x1 , . . . , xn ) and a1 , . . . , an ∈ M , M |= ϕ[a1 , . . . , an ] iff N |= ϕ[a1 , . . . , an ]. Exercise 11.2.6 Let M be an L-structure and (Mn )n∈ω a sequence of substructures of M . Suppose M = n Mn and Mn ≺F Mn+1 for all n ∈ ω. Show that Mn ≺F M for all n ∈ ω. Exercise 11.2.7 Let L be a countable relational language, F a fragment, N an L-structure, and M ⊆ N . Show that M ≺F N iff for all formulas ϕ(x1 , . . . , xn , y) ∈ F and a1 , . . . , an ∈ M , M |= ∃yϕ[a1 , . . . , an ] iff N |= ∃yϕ[a1 , . . . , an ].

11.3

Invariant Borel classes of countable models

In this section we consider the canonical topology and invariant Borel subsets of Mod(L) following the approach of Vaught [161]. Let L = {Ri }i∈I be a countable relational language, with each Ri an ni -ary relation symbol. Recall from Section 3.5 that the bijection  ni 2ω = XL −→ Mod(L) i∈I

x −→ Mx

associates with each element of the product space XL a countable L-structure. With this natural correspondence Mod(L) becomes a compact Polish space, and we call this topology on Mod(L) the canonical topology. For a refined analysis we use the following notation. Definition 11.3.1 For any Lω1 ω -formula ϕ(x1 , . . . , xn ) and tuple a = (a1 , . . . , an ) ∈ ω n , let Mod(ϕ, a) = {M ∈ Mod(L) : M |= ϕ[a1 , . . . , an ]}. In particular, if ϕ is a sentence, we let Mod(ϕ) = {M ∈ Mod(L) : M |= ϕ}. For a set A of Lω1 ω -formulas, the topology generated by A, denoted tA , is the topology on Mod(L) generated by the subbase BA = {Mod(ϕ, a) : ϕ ∈ A, a ∈ ω <ω }.

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We denote by Batom the subbase given by all atomic and negated atomic Lωω -formulas, and tatom the topology generated by Batom . With this notation the canonical topology on Mod(L) given by XL is just tatom . Similarly, denote by Bqf the subbase given by all quantifier-free Lωω -formulas, and tqf the topology generated by Bqf . Since quantifier-free formulas are exactly the Boolean combinations of atomic formulas, and in particular atomic and negated atomic formulas are quantifier-free, the topologies tatom and tqf are the same. In the following lemma we give another alternative subbase. For n ∈ ω we let n denote the tuple (0, 1, . . . , n − 1) ∈ ω n . Note that 0 is the empty tuple ∅. Lemma 11.3.2 Let B0 = { Mod(ϕ, n) : n ∈ ω, ϕ is a quantifier-free Lωω -formula }. Then B0 is a subbase for the canonical topology on Mod(L). Proof. We show that for any quantifier-free Lωω -formula ψ(x1 , . . . , xk ) and any tuple a = (a1 , . . . , ak ) ∈ ω k , there is a quantifier-free formula ϕ and n ∈ ω such that Mod(ϕ, n) = Mod(ψ, a). First note that we may assume without loss of generality that a1 , . . . , ak are distinct. In fact, suppose ai = aj for i < j. Then we let a be obtained from the tuple a by removing aj , that is, a = (a1 , . . . , aj−1 , aj+1 , . . . , ak ). Also let ψ  be obtained from ψ by replacing all occurrences of xj by xi . Then for any N ∈ Mod(L), N |= ψ[a] iff N |= ψ  [a ]. Therefore repetitions in the tuple a can be eliminated by iteratively applying this procedure. Next we may assume that a1 < · · · < ak . In fact, if aj1 < · · · < ajk is an enumeration of the tuple a in the increasing order, then we may let a = (aj1 , . . . , ajk ) and ψ  be obtained from ψ by replacing all occurrences of xi by xji for all i = 1, . . . , k. For any N ∈ Mod(L), we still have that N |= ψ[a] iff N |= ψ  [a ]. Finally let n = ak +1 and ϕ be obtained from ψ by replacing all occurrences of xi by vai for all i = 1, . . . , k. Note that all (free) variables of ψ are among x1 , . . . , xk , and it follows that all variables of ϕ are among va1 , . . . , vak . In particular, the free variables of ϕ are among v0 , . . . , vn−1 . It is again obvious that Mod(ψ, a) = Mod(ϕ, n). Note that we actually showed that B0 = Bqf . Lemma 11.3.3 For any Lω1 ω -formula ϕ(x1 , . . . , xn ) and a = (a1 , . . . , an ) ∈ ω n , Mod(ϕ, a) is a Borel subset of Mod(L).

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Proof. This is proved by induction on all Lω1 ω -formulas with finitely many free variables. If ϕ is atomic then Mod(ϕ, a) is open for any a. The inductive cases follow from the following computations:   & % Mod(ψ, a), Mod( Φ, a) = Mod(ψ, a), Mod( Φ, a) = ψ∈Φ

Mod(∃xψ, a) =



Mod(ψ, a b),

b∈ω

ψ∈Φ

Mod(∀xψ, a) =



Mod(ψ, a b).

b∈ω

Also recall from Section 3.5 that the isomorphism relation among L-structures is an orbit equivalence relation induced by the logic action of S∞ . Specifically, for g ∈ S∞ , M ∈ Mod(L), and any n-ary R ∈ L, Rg·M (a1 , . . . , an ) ⇐⇒ RM (g −1 (a1 ), . . . , g −1 (an )) for any a = (a1 , . . . , an ) ∈ ω n . In the notation of logic, the action is determined by specifying that g · M |= R[a] ⇐⇒ M |= R[g −1 (a)]. The logic action is clearly continuous. Definition 11.3.4 Let L be a countable relational language. An invariant Borel class of countable L-structures is an S∞ -invariant Borel subset of Mod(L). If ϕ is an Lω1 ω -sentence then Mod(ϕ) is Borel by Lemma 11.3.3, and isomorphism invariant since if M ∼ = N then M |= ϕ iff N |= ϕ. Thus for any sentence ϕ, Mod(ϕ) is an invariant Borel class. Next we show that the converse is true. For this we use the Vaught transforms and the following notation. Let [ω]n denote the set of tuples (a0 , . . . , an−1 ) ∈ ω n where a0 , . . . , an−1 are distinct. Note that [ω]0 = {∅}. For each a = (a0 , . . . , an−1 ) ∈ [ω]n , let Na ⊆ S∞ be the set of all elements f ∈ S∞ such that f (ai ) = i for i < n. Note that N∅ = S∞ . For any set B ⊆ Mod(L) and n ∈ ω, define B [∗n] , B [ n] ⊆ Mod(L) × [ω]n by     B [ n] = (M, a) : M ∈ B Na , B [∗n] = (M, a) : M ∈ B ∗Na . Lemma 11.3.5 If B ⊆ Mod(L) is Borel, then for any n ∈ ω, there are Lω1 ω -formulas ϕn (x1 , . . . , xn ) and ψn (x1 , . . . , xn ) such that (M, a) ∈ B [ n] ⇐⇒ M ∈ Mod(ϕn , a), (M, a) ∈ B [∗n] ⇐⇒ M ∈ Mod(ψn , a).

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Proof. Let B0 be the subbase defined in Lemma 11.3.2. Let B be the collection of all sets B ⊆ Mod(L) such that the conclusion of the lemma holds. We show that B0 ⊆ B and that B is a σ-algebra. In the definition of the formulas we use the abbreviation (∀y1 . . . yk )= ψ to stand for the formula ∀y1 . . . ∀yk (

'

yi = yj → ψ ).

1≤i<j≤k

For B0 ⊆ B we let ϕ be quantifier-free and n ∈ ω, and check that Mod(ϕ, n) ∈ B. Note that for any M ∈ Mod(L), a ∈ [ω]n and g ∈ Na , g · M |= ϕ[n] ⇐⇒ M |= ϕ[g −1 (n)] ⇐⇒ M |= ϕ[a], where the right-hand side is independent of g. Thus (M, a) ∈ Mod(ϕ, n)[ n] ⇐⇒ ∃∗ g ∈ Na g · M ∈ Mod(ϕ, n) ⇐⇒ M |= ϕ[a]. Therefore we can take ϕn = ϕ. Similarly we can also take ψn = ϕ. This shows that Mod(ϕ, a) ∈ B. m Next suppose Bm ∈ B for m ∈ ω and let ϕm n and ψn be witnesses. Supm m pose the free variable of ϕ and ψ are among x , . . . , xn . We show that 1 n &n

m B ∈ B. Let ϕ = ϕ . Then n m∈ω m m∈ω n

 (M, a) ∈

[ n]



Bm

⇐⇒ ∃∗ g ∈ Na g · M ∈

m∈ω



Bm

m∈ω

⇐⇒ ∃m ∈ ω ∃∗ g ∈ Na g · M ∈ Bm ⇐⇒ ∃m ∈ ω M |= ϕm a] n [ ⇐⇒ M |= ϕn [a]. The ∗-transform is a bit more complicated. Fix n ∈ ω. For each k ∈ ω we let ψn,k = (∀y1 . . . yk )= (

'

1≤i≤n 1≤j≤k

and put ψn =

% k∈ω

xi = yk →

(

ϕm n+k (x1 , . . . , xn , y1 , . . . , yk ) ),

m∈ω

ψn,k . Note that the free variables of ψn,k and ψn are still

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among x1 , . . . , xn . Now we have 

[∗n]



(M, a) ∈

Bm

m∈ω

⇐⇒ ∀∗ g ∈ Na ∃m ∈ ω g · M ∈ Bm ⇐⇒ ∀b



ab ∈ [ω]n+k → ∃m ∈ ω ∃∗ g ∈ Nab g · M ∈ Bm

 ⇐⇒ ∀b



a b ∈ [ω]

n+k

→ M |=

(



 ϕm ab] n+k [

m∈ω

⇐⇒ M |= ψn [a]. It remains to show that if B ∈ B then Mod(L) − B ∈ B. For this let ϕn and ψn be the witnesses of B ∈ B. Then it is easy to see that (M, a) ∈ (Mod(L) − B)[ n] ⇐⇒ M |= ¬ψn [a] and (M, a) ∈ (Mod(L) − B)[∗n] ⇐⇒ M |= ¬ϕn [a].

Note that the lemma also applies in the case n = 0. Theorem 11.3.6 Let L be a countable relational language and C ⊆ Mod(L). Then C is an invariant Borel class iff there is an Lω1 ω -sentence ϕ such that C = Mod(ϕ). Proof. The implication (⇐) is clear. For (⇒) let C be an invariant Borel class. Then C = C S∞ and C [ 0] = C ×{∅}. By Lemma 11.3.5 for n = 0, there is a sentence ϕ such that (M, ∅) ∈ C [ 0] iff M ∈ Mod(ϕ). Thus C = Mod(ϕ).

Theorem 11.3.6 has far-reaching consequences on the isomorphism relations of invariant Borel classes. Notation 11.3.7 Let L be a countable relational language. For any Lω1 ω -sentence ϕ we let ∼ =ϕ denote the isomorphism relation ∼ = Mod(ϕ). Since Mod(ϕ) is an invariant Borel subset of Mod(L), it is a Borel S∞ space with the action inherited from the logic action. Thus ∼ =ϕ is an S∞ -orbit equivalence relation. The following theorem establishes the converse.

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Theorem 11.3.8 For any Borel S∞ -space X there is a countable relational language L and an Lω1 ω -sentence ϕ such that ESX∞ is Borel isomorphic to ∼ =ϕ . In particular, ESX∞ ∼B ∼ =ϕ . Proof. By Theorem 3.6.1 there is a countable relational language L and a Borel S∞ -embedding j : X → Mod(L). It follows that j(X) is an invariant Borel class and that j is a Borel S∞ -isomorphism between X and j(X). Now by Theorem 11.3.6 there is an Lω1 ω -sentence ϕ such that j(X) = Mod(ϕ). Then j is a Borel isomorphism between ESX∞ and ∼ = j(X), or ∼ =ϕ . The following theorem also shows that being an isomorphism relation is closed under Borel reductions for Borel orbit equivalence relations. Theorem 11.3.9 Let G be a Polish group, X a Borel G-space, L a countable relational language, X ≤B ∼ and ϕ an Lω1 ω -sentence. Suppose that ∼ =ϕ is Borel and EG =ϕ . Then X there is an Lω1 ω -sentence σ such that Mod(σ) ⊆ Mod(ϕ) and EG ∼B ∼ =σ . X Proof. Let f : X → Mod(ϕ) be a Borel reduction from EG to ∼ =ϕ . By Theorem 5.2.3 f is a faithful Borel reduction. And thus Y = [f (X)]∼ = is a Borel subset of Mod(ϕ). By Theorem 11.3.6 there is an Lω1 ω -sentence σ such X that Mod(σ) = Y . Then by Corollary 5.2.4 EG ∼B ∼ =σ .

The most important open problems on isomorphism relations are about the number of nonisomorphic countable models in invariant Borel classes. The Vaught conjecture is the statement Any complete first-order theory either has only countably many nonisomorphic countable models or has perfectly many nonisomorphic countable models. And the Lω1 ω -Vaught conjecture states that For any countable relational language L and Lω1 ω -sentence ϕ, ∼ϕ either has only countably many classes or has perfectly many = classes. The Vaught conjecture is a special case of the Lω1 ω -Vaught conjecture since any complete first-order theory (that is, Lωω -theory) T is logically equivalent % to ϕ = T . In both statements “perfectly many” refers to a perfect subset of Mod(ϕ). Recall that TVC(G), the topological Vaught conjecture for G, is the statement (among several equivalent formulations) X has either only countably many orbits For any Borel G-space EG or perfectly many orbits.

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It is thus immediate from Theorem 11.3.8 that the Lω1 ω -Vaught conjecture is equivalent to TVC(S∞ ). Exercise 11.3.1 Let B00 = { Mod(ϕ, n) : n ∈ ω, ϕ is atomic or negated atomic }. Show that B00 = Batom . Exercise 11.3.2 In the proof of Lemma 11.3.5 show directly that B is closed under countable intersection. Exercise 11.3.3 For each of the following equivalence relations E find a countable relational language L and an Lω1 ω -sentence ϕ such that ∼ =ϕ ∼B E: (a) id(ω), (b) id(2ω ), (c) E0 .

11.4

Polish topologies generated by countable fragments

The technique of change of topology in the context of Polish group actions, as we presented in Chapter 4 following the approaches of References [8] and [5], has been a powerful tool in invariant descriptive set theory. This technique has its natural motivation in the model theoretic context (for example, see References [152], [9], and [124]), and was used to obtain stronger results for countable models than for general orbit equivalence relations (for example, see References [77] and [82]). In the preceding section we have mostly considered the canonical topology on Mod(L) and its various (sub)bases. From Chapter 4 we know abstractly that there are many other Polish topologies we can put on Mod(L). In particular, for any invariant Borel class C ⊆ Mod(L) there exists a Polish topology finer than the canonical topology such that C becomes clopen but the logic action is still continuous. In this section we consider some natural bases for such topologies. First note that by Definition 11.3.1 for any set of Lω1 ω -formulas A we can alway associate a subbase BA and therefore a topology tA generated by BA . However, the definition only uses formulas in A with only finitely many free variables. That is to say, if we let A be the set of formulas in A with only finitely many free variables, then BA = BA and tA = tA . Thus in our future discussions we may always assume that the set A consists of only formulas with only finitely many free variables.

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Theorem 11.4.1 Let L be a countable relational language and F a countable fragment of Lω1 ω . Then BF is a clopen base for the topology tF and (Mod(L), tF ) is a Polish S∞ -space. Proof. For any formula ϕ(x1 , . . . , xn ), let S(ϕ) be the set of all quantifierfree Lωω -formulas, all subformulas of ϕ, and their negations. We first prove by induction on the form of ϕ that BS(ϕ) is a clopen base for a Polish topology. If ϕ is quantifier-free then B% S(ϕ) = Bqf is a clopen base for tqf . Also S(¬ϕ) = S(ϕ). We next consider ϕ = Φ for a countable set Φ, and note that S(ϕ) =

{ϕ, ¬ϕ} ∪ ψ∈Φ S(ψ). By the inductive hypotheses each BS(ψ) is a clopen base for a Polish topology finer than the canonical topology. It follows from

Lemma 4.2.2 that ψ∈Φ S(ψ) is also a clopen base for a Polish topology τ . Now fix a = (a1 , . . . , an ) ∈ ω n . Note that Mod(ϕ, a) = ψ∈Φ Mod(ψ, a) and is therefore τ -closed. By Lemma 4.2.1 the topology τ ∪ {Mod(ϕ, a)} is Polish. It follows from Lemma 4.2.2 again that τ ∪ {Mod(ϕ, a) : a ∈ ω n } generates a Polish topology, % for which BS(ϕ) is a clopen base. & This proves the inductive case for ϕ = Φ. The argument for ϕ = Φ is similar. n , . . . , x , y). For each  a ∈ ω , Mod(ϕ, a) = Next we consider ϕ = ∃yψ(x 1 n

 Mod(ψ,  a b), and is therefore t -open. However, S(ψ) b∈ω  Mod(¬ψ, a b) Mod(¬ϕ, a) = b∈ω

is tS(ψ) -closed. By a similar argument as above using Lemmas 4.2.1 and 4.2.2 we get that BS(ϕ) is a clopen base for a Polish topology. The case ϕ = ∀yψ(x1 , . . . , xn , y) is similar. Now that F is a countable fragment, we have that  BF = S(ϕ) ϕ∈F

is a countable union. Thus by Lemma 4.2.2 again BF generates a Polish topology. It is a clopen base since F is closed under negation and finitary logical connectives ∧ and ∨. To show the continuity of the logic action with respect to tF , let g ∈ S∞ , M ∈ Mod(L), Mod(ϕ, a) ∈ BF , and suppose g · M ∈ Mod(ϕ, a). Note that g −1 · Mod(ϕ, a) = Mod(ϕ, g −1 (a)). Thus if we let b = g −1 (a) and Na,b = {f ∈ S∞ : f (b) = a}, then g ∈ N  is a basic open set in S∞ , M ∈ Mod(ϕ, b), and  a,b

for any f ∈ Na,b and N ∈ Mod(ϕ, b), f · N ∈ Mod(ϕ, f (b)) = Mod(ϕ, a). Thus topologies generated by countable fragments are concrete examples of finer Polish topologies on Mod(L). If C is an invariant Borel class and ϕ is an

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Lω1 ω -sentence such that C = Mod(ϕ), then the fragment F (ϕ) generated by ϕ is countable and generates a Polish topology tF (ϕ) in which C is clopen and such that (Mod(L), tF (ϕ) ) is a Polish S∞ -space. In particular, (Mod(ϕ), tF (ϕ) ) is a Polish S∞ -space whose orbit equivalence relation is ∼ =ϕ . As we will see, model theoretic properties of countable structures in a countable fragment F correspond nicely to topological properties in the topology tF . For simplicity we use [M ], rather than [M ]∼ = , to denote the isomorphism class of M . Theorem 11.4.2 Let L be a countable relational language and F a countable fragment of Lω1 ω . tF

Then for any M, N ∈ Mod(L), M ≡F N iff [M ]

tF

= [N ] .

Proof. For each sentence ϕ ∈ F , Mod(ϕ) is a basic open set in tF . TheretF tF fore, if [M ] = [N ] , then for any sentence ϕ ∈ F , [M ] ∩ Mod(ϕ) = ∅ iff [N ]∩Mod(ϕ) = ∅. Since Mod(ϕ) is invariant, this implies that M ∈ Mod(ϕ) iff N ∈ Mod(ϕ). Hence M ≡F N . Conversely, suppose M ≡F N . We show that for any Mod(ϕ, a) ∈ BF , [M ] ∩ Mod(ϕ, a) = ∅ iff [N ] ∩ Mod(ϕ, a) = ∅. This tF tF of course implies that [M ] = [N ] . For this suppose ϕ(x1 , . . . , xn ) ∈ F , a ∈ ω n , and assume [M ]∩Mod(ϕ, a) = ∅. By symmetry it suffices to show that [N ]∩Mod(ϕ, a) = ∅. First note that we may assume without loss of generality that a ∈ [ω]n by a standard argument of change of variables as in the proof of Lemma 11.3.2. Without loss of generality we may also assume M ∈ Mod(ϕ, a). Thus M |= ϕ[a], and in particular M |= (∃x1 . . . xn )= ϕ(x1 , . . . , xn ), where (∃x1 . . . xn )= ϕ is an abbreviation of ∃x1 . . . ∃xn (

'

xi = xj ∧ ϕ ).

1≤i<j≤n

Now the sentence (∃x1 . . . xn )= ϕ(x1 , . . . , xn ) ∈ F , and by M ≡F N we get that N |= (∃x1 . . . xn )= ϕ(x1 , . . . , xn ). Hence there is b ∈ [ω]n such that N |= ϕ[b]. Let g ∈ S∞ be such that g(b) = a. Then g · N |= ϕ[a], or g · N ∈ Mod(ϕ, a). This shows that [N ] ∩ Mod(ϕ, a) = ∅ as desired. By this theorem F -theories of countable models correspond to the tF closure of their isomorphism classes. This has an immediate corollary on ω-categoricity, which is defined below. Definition 11.4.3 Let L be a countable relational language. An Lω1 ω -theory T is ω-categorical if M ∼ = N whenever M, N ∈ ϕ∈T Mod(ϕ). Thus an ω-categorical theory has a unique countable model up to isomorphism.

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Corollary 11.4.4 Let L be a countable relational language and F a countable fragment of Lω1 ω . Then [M ] is closed in tF iff ThF (M ) is ω-categorical. Proof. First suppose [M ] is closed in tF and let N ≡F M . Then by tF tF Theorem 11.4.2, [N ] ⊆ [N ] = [M ] = [M ]. This shows that N ∈ [M ] ∼ and hence N = M . Conversely, if ThF (M ) is ω-categorical then [M ] = ϕ∈ThF (M) Mod(ϕ), which is tF -closed. Next we consider F -elementary embeddability. For this we recall that the canonical metric on S∞ (the one inherited from the Baire space ω ω ) is  −n 2 , if n ∈ ω is the least such that f (n) = g(n), d(f, g) = 0, if f = g. This metric is left-invariant but not complete. Theorem 11.4.5 (Becker) Let L be a countable relational language and F a countable fragment of Lω1 ω . Let d be the canonical metric on S∞ . Then for any M, N ∈ Mod(L), the following are equivalent: (a) There is an F -elementary embedding from M into N . (b) There is a d-Cauchy sequence (gn )n∈ω in S∞ such that gn · M → N in tF . Proof. (a)⇒(b): Let j : M → N be an F -elementary embedding. We show that for any ϕ(x1 , . . . , xn ) ∈ F , a ∈ ω n , and m ∈ ω, if N ∈ Mod(ϕ, a) then there is fm ∈ S∞ such that fm (i) = j(i) for all i < m and fm ·M ∈ Mod(ϕ, a). As before we may assume that a ∈ [ω]n . We may also assume that k ≤ n and a1 , . . . , ak ∈ {j(0), . . . , j(m − 1)} and ak+1 , . . . , an ∈ {j(0), . . . , j(m − 1)}. Then by our assumption N |= ψ[a1 , . . . , ak ], where ψ(x1 , . . . , xk ) is the formula ' (∃xk+1 . . . xn )= ( xi = xi ∧ ϕ ). 1≤i≤k k+1≤i ≤n

Since ψ ∈ F it follows from F -elementarity that M |= ψ[j −1 (a1 ), . . . , j −1 (ak )]. Let bi = j −1 (ai ) for 1 ≤ i ≤ k. Then there are bk+1 , . . . , bn ∈ ω such that b ∈ [ω]n and M |= ϕ[b]. Now let fm ∈ S∞ by such that fm (i) = j(i) for all i < m, and fm (bk+1 ) = ak+1 , . . . , fm (bn ) = an . Then fm (bi ) = ai for all 1 ≤ i ≤ n. Therefore fm · M |= ϕ[a], or fm · M ∈ Mod(ϕ, a).

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sequence of Now since tF is first countable we may let Un be a decreasing basic open sets of the form Mod(ϕ, a) ∈ BF such that n Un = {N }. By the above claim there is gn ∈ S∞ such that gn (i) = j(i) for i < n and gn ·M ∈ Un . Clearly gn · M → N in tF . However, for any n < n , d(gn , gn ) ≤ 2−(n+1) , hence (gn )n∈ω is d-Cauchy. (b)⇒(a): Let (gn )n∈ω be a d-Cauchy sequence in S∞ such that gn · M → N in tF . Then for any k ∈ ω there is nk such that for all n, m ≥ nk , d(gn , gm ) < 2−k . This implies in particular that for all n, m ≥ nk , gn (k) = gm (k). Thus we may let j(k) = limn gn (k), and j is well defined. It is easy to see that j is an injection. We verify that j is an F -elementary embedding from M into N . Let ϕ(x1 , . . . , xn ) ∈ F and a ∈ ω n . Assume M |= ϕ[a] but N |= ϕ[j(a)], that is, N ∈ Mod(ϕ, j(a)). Since gn ·M → N in tF and Mod(ϕ, j(a)) is clopen, there is N ∈ ω such that for all m ≥ N , gm ·M ∈ Mod(ϕ, j(a)). But let m > N be large enough such that j(ai ) = gm (ai ) for all 1 ≤ i ≤ n. Then we have gm ·M |= ϕ[gm (a)], and since j(a) = gm (a), gm ·M |= ϕ[j(a)], a contradiction. This shows that M |= ϕ[a] implies N |= ϕ[j(a)]. For the converse note that the same argument gives that M |= ¬ϕ[a] implies N |= ¬ϕ[j(a)], and therefore N |= ϕ[j(a)] implies M |= ϕ[a]. Thus j is an F -elementary embedding. tF

Clause (b) of this theorem obviously implies that N ∈ [M ] , and thus tF tF [N ] ⊆ [M ] . It is curious to observe that clause (b) does not trivially tF tF imply that [M ] = [N ] , since given d-Cauchy sequence (gn ) such that gn · M → N it is not clear how to construct another d-Cauchy sequence (hn ) so that hn · N → M (note that (gn−1 ) is not necessarily d-Cauchy). However, by combining the above two theorems this implication is true, since it follows from the existence of an F -elementary embedding from M into N that M ≡F N . In the following exercises let L be a countable relational language and F a countable fragment of Lω1 ω . Exercise 11.4.1 Let B0F = {Mod(ϕ, n) : ϕ ∈ F, n ∈ ω}. Show that B0F = BF . Exercise 11.4.2 Show that for any U ∈ BF , [U ]∼ = = S∞ · U is clopen in tF . Exercise 11.4.3 Show that ≡F is a closed equivalence relation on (Mod(L), tF ).

11.5

Atomic models and Gδ orbits

By now it should have been clear that for the study of countable models the connection between methods of logic and topology is natural and intrinsic. In

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this section we continue to explore this connection and establish some deeper results. To do this we will have to rely on some model theoretic results we state without proof. We start with some more definitions. Definition 11.5.1 Let L be a countable relational language and F a countable fragment of Lω1 ω . (1) For any M ∈ Mod(L) and a ∈ ω n , the F -type of a over M is tpF (a/M ) = {ϕ(x1 , . . . , xn ) ∈ F : M |= ϕ[a]}. An F -type is a set of formulas Φ such that for some M ∈ Mod(L) and a ∈ ω n , Φ ⊆ tpF (a/M ). An F -type Φ is complete if Φ = tpF (a/M ) for some M and a ∈ ω n . (2) An F -type Φ is realized in M if Φ ⊆ tpF (a/M ) for some a ∈ ω n ; otherwise it is omitted in M . The following definitions are specific about a complete F -theory T . Definition 11.5.2 Let L be a countable relational language and F a countable fragment of Lω1 ω . (1) An F -theory T is complete if T = ThF (M ) for some M ∈ Mod(L). (2) If T is a complete theory and M ∈ Mod(L), then M is a model of T , denoted M |= T , if M |= ϕ for all ϕ ∈ T . The set of all models of T in Mod(L) is denoted by Mod(T ). (3) Let T be a complete F -theory. An F -type is an F -type of T if it is realized in some countable model of T . It is a complete F -type of T if it is a complete type realized in a countable model of T . (4) Let T be a complete F -theory and Φ an F -type of T . Φ is principal if there is a complete F -type Ψ of T such that Φ ⊆ Ψ and there is a formula ϕ ∈ Ψ such that for any M ∈ Mod(T ) and ψ ∈ Φ, M |= ∀x1 . . . ∀xn (ϕ → ψ); otherwise Φ is nonprincipal. Note that if F is a countable fragment % so is any complete F -theory or F type. For any F -type Φ if we let ϕ = Φ, then M |= ∀x1 . . . ∀xn (ϕ → ψ) for all M ∈ Mod(T ); however ϕ ∈ Φ and in general ϕ ∈ F . Thus in the definition of principal types it is essential to refer to the theory T and the fragment F . The following lemma is a standard fact in model theory and is the main reason to consider principal types. Lemma 11.5.3 Let L be a countable relational language, F a countable fragment of Lω1 ω , and T a complete F -theory. Then every principal F -type of T is realized in any M ∈ Mod(T ).

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Proof. Let Φ be a principal F -type of T , witnessed by complete F -type Ψ of T and formula ϕ ∈ Ψ. Let M ∈ Mod(T ) and a ∈ ω n be such that Φ ⊆ Ψ = tpF (a/M ). Then M |= ∃x1 . . . ∃xn ϕ. Now ∃x1 . . . ∃xn ϕ is a sentence in F , and thus in ThF (M ) = T since T is a complete F -theory. It follows that for any N ∈ Mod(T ), N |= ∃x1 . . . ∃xn ϕ. Let b ∈ ω n be such that N |= ϕ[b]. But then by principality N |= ∀x1 . . . ∀xn (ϕ → ψ) for all ψ ∈ Φ. Hence N |= ψ[b] for all ψ ∈ Φ. This means that Φ is realized in N as witnessed by b. It turns out that the converse of the lemma is also true. This is implied by the following theorem known as the omitting types theorem, which we state without proof. A proof for the first-order case of the omitting types theorem can be found in most textbooks in model theory, for example, Reference [84]; for the case of infinitary logic the proof is similar (see References [3] and [105]). Theorem 11.5.4 Let L be a countable relational language, F a countable fragment of Lω1 ω , and T a complete F -theory. For any countable set {Φn }n∈ω of nonprincipal F -types of T there exists M ∈ Mod(T ) such that M omits all types Φn for n ∈ ω. We now turn to atomic models. Definition 11.5.5 Let L be a countable relational language, F a countable fragment of Lω1 ω , and T a complete F -theory. A model M ∈ Mod(T ) is F -atomic if all F -types of T realized in M are principal. Note that when referring to F -atomic models it is unnecessary, although sometimes only convenient, to specify the complete F -theory since a model M can only be F -atomic with respect to ThF (M ). Theorem 11.5.6 Let L be a countable relational language, F a countable fragment of Lω1 ω , and T a complete F -theory. (i) If M, N ∈ Mod(T ) and M is F -atomic, then there is an F -elementary embedding from M into N . (ii) If M, N ∈ Mod(T ) are both F -atomic, then M ∼ = N. Proof. We only prove (i). The proof of (ii) is by a back-and-forth argument similar to (i) and is left as an exercise. Suppose M is F -atomic. For each n ∈ ω we let Φn (v0 , . . . , vn−1 ) = tpF (n/M ) and ϕn (v0 , . . . , vn−1 ) ∈ F be the witness for the principality of Φn . Let

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N ∈ Mod(T ). We define a distinct sequence (bn )n∈ω in |N | such that for all n ∈ ω, N |= ϕn [b0 , . . . , bn−1 ]. To begin with we note that N |= Φ0 and that by Lemma 11.5.3 Φ1 is realized in N , hence there is b0 ∈ ω such that N |= ϕ1 [b0 ]. In general suppose b0 , . . . , bn−1 have been defined so that N |= ϕn [b0 , . . . , bn−1 ]. Since M |= ϕn [0, . . . , n − 1] and M |= ϕn+1 [0, . . . , n − 1, n], we have that M |= ψ[0, . . . , n − 1], where ψ is the formula ' vn = vi ∧ ϕn ∧ ϕn+1 ). ∃vn ( i
Since ψ ∈ F , we have ψ ∈ Φn , and by principality of Φn , N |= ∀v0 . . . ∀vn−1 (ϕ → ψ). This implies that ' N |= ∃vn ( vn = vi ∧ ϕn ∧ ϕn+1 )[b0 , . . . , bn−1 ] i
since N |= ϕ[b0 , . . . , bn−1 ]. Therefore there is bn ∈ ω distinct from b0 , . . . , bn−1 such that N |= ϕn+1 [b0 , . . . , bn−1 , bn ]. This finishes the definition of the sequence (bn ). We claim that the map j(n) = bn , n ∈ ω, is an F -elementary embedding from M into N . To show this it suffices to show that for any n ∈ ω and formula φ(v0 , . . . , vn−1 ) ∈ F , M |= φ[n] iff N |= φ[j(n)]. For this fix n ∈ ω and φ ∈ F . It suffices to show that M |= φ[n] implies N |= φ[j(n)]. So we suppose M |= φ[n]. This means that φ ∈ Φn and therefore by principality of Φn witnessed by ϕn , N |= ∀v0 . . . ∀vn−1 (ϕn → φ). Now N |= ϕn [j(n)], and it follows that N |= φ[j(n)], as required. Theorem 11.5.7 (Miller–Suzuki) Let L be a countable relational language and F a countable fragment of Lω1 ω . Then M is F -atomic iff [M ] is Gδ in tF . Proof. Suppose M is F -atomic, and for each n ∈ ω, let {Φnm }m∈ω enumerate the set {tpF (a/M ) : a ∈ ω n }. Note that {Φnm : n, m ∈ ω} in fact enumerate all complete principal F -types of T = ThF (M ). For each n, m ∈ ω, let ϕnm ∈ F be a formula witnessing that Φnm is principal. Let ϕ be the Lω1 ω -sentence ' ( ∀x1 . . . ∀xn ϕnm . n∈ω

m∈ω

Then any model N ∈ Mod(T ) of ϕ is F -atomic. To see this let N |= ϕ. Then for any a ∈ ω n there is m ∈ ω such that N |= ϕnm [a]. It follows that for any ψ ∈ Φnm , N |= ψ[a], and therefore tpF (a/N ) = Φnm is principal. Since N realizes only principal types, it is F -atomic. By Theorem 11.5.6 if N ∈ Mod(T ) and N |= ϕ then N ∼ = M . Thus for N ∈ Mod(T ), N |= ϕ iff N ∼ = M . This implies that    N ∈ [M ] ⇐⇒ N |= ϕ ⇐⇒ N ∈ Mod(ϕnm , a), n∈ω  a∈ω n m∈ω

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and thus [M ] is Gδ in tF .

Conversely, assume that [M ] is Gδ in tF . Let [M ] = n∈ω m∈ω Un,m ,  for ϕn,m ∈ F (see Exercise 11.4.1), and where each Un,m = Mod(ϕn,m , m) such that Un,m ⊆ Un,m whenever m < m , and Un,m ⊇ Un ,m whenever n < n . This can be achieved by a proof similar to that of Lemma 11.3.2 and by using finitary conjunctions (as n increases) and disjunctions (as m increases). Then [M ] = [M ]∗S∞ , and by a computation similar to that in the proof of Lemma 11.3.5 we have that N ∈ [M ]∗S∞ ⇐⇒ ∀∗ g ∈ S∞ ∀n ∈ ω ∃m ∈ ω g · N ∈ Un,m ⇐⇒ ∀n ∈ ω ∀∗ g ∈ S∞ ∃m ∈ ω g · N ∈ Un,m ⇐⇒ ∀n ∈ ω ∀k ∈ ω ∀∗ g ∈ S∞ ∃m ∈ ω g · N ∈ Umax{n,k},m ⇐⇒ ∀n ∈ ω ∀b ∈ [ω]k ∃m ≥ max{n, k} ∃∗ g ∈ Nb g · N ∈ Umax{n,k},m ⇐⇒ ∀n ∈ ω ∀b ∈ [ω]n ∃m ≥ n ∃∗ g ∈ Nb g · N ∈ Un,m ⇐⇒ ∀n ∈ ω ∀b ∈ [ω]n ∃m ≥ n ∃c (bc ∈ [ω]m ∧ ∃∗ g ∈ Nbc g · N ∈ Mod(ϕn,m , m)  ) ⇐⇒ ∀n ∈ ω ∀b ∈ [ω]n ∃m ≥ n ∃c (bc ∈ [ω]m ∧ M ∈ Mod(ϕn,m , bc) ). Thus for m ≥ n if we let ψn,m (x1 , . . . , xn ) ∈ F be the formula ∃xn+1 . . . ∃xm (

'

xi = xj ∧ ϕn,m ),

1≤i<j≤m

& % then N ∈ [M ] iff N |= n∈ω ∀x1 . . . ∀xn m≥n ψn,m . Now let Φn = {¬ψn,m : m ∈ ω} for n ∈ ω. Then Φn ⊆ F . If Φn is an F -type of T then by Lemma 11.5.3 it is nonprincipal since it is not realized in M . Let S ⊆ ω be such that Φn is an F -type iff n ∈ S. We are ready to verify that M is F -atomic. Assume not, and let Ψ be a nonprincipal F -type realized in M . Then the collection {Φn : n ∈ S} ∪ {Ψ} is a countable set of nonprincipal types of T = ThF (M ). By the Omitting Types Theorem 11.5.4 there is an N ∈ Mod(T ) omitting all types in the collection. For n ∈ S there are also no a ∈ % ω n such that Φn ⊆ & tpF (a/N ) since Φn is not an F -type of T . Therefore N |= n∈ω ∀x1 . . . ∀xn m≥n ψn,m . This

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shows that N ∈ [M ], or N ∼ = M . However, N omits the F -type Ψ, whereas M realizes it, hence N ∼  M , a contradiction. = As a corollary we give a model theoretic characterizations for smoothness of isomorphism relations. Theorem 11.5.8 Let L be a countable relational language and ϕ an Lω1 ω -sentence. Then the following are equivalent: (i) ∼ =ϕ is smooth. (ii) There is a countable fragment F (containing ϕ) such that for any M ∈ Mod(ϕ), ThF (M ) is ω-categorical. (iii) There is a countable fragment F (containing ϕ) such that every M ∈ Mod(L) is F-atomic. Proof. (i)⇒(ii): Let X be a Polish space and f : Mod(ϕ) → X be Borel such that M ∼ = N iff f (M ) = f (N ). Let {Un }n∈ω be a countable base for X. Then M∼ = N ⇐⇒ ∀n ∈ ω M ∈ f −1 (Un ) ↔ N ∈ f −1 (Un ). Now each f −1 (Un ) is an invariant Borel class, so there is an Lω1 ω -sentence ψn such that f −1 (Un ) = Mod(ψn ). Let F be a countable fragment generated by {ϕ, ψ0 , ψ1 , . . . }. Then in the topology τF on Mod(ϕ), each isomorphism class [M ] is closed. Then by Corollary 11.4.4, ThF (M ) is ω-categorical. (ii)⇒(iii) follows immediately from Corollary 11.4.4 and Theorem 11.5.7. (iii)⇒(i): By Theorem 11.5.7 for any M ∈ Mod(L), [M ] is Gδ in τF on Mod(ϕ). By Theorem 6.4.4 ∼ =ϕ is smooth. Exercise 11.5.1 Prove Theorem 11.5.6 (ii). Exercise 11.5.2 Assuming the statement of Theorem 11.5.7, deduce that if M and N are both F -atomic models of the same complete F -theory, then M∼ = N.

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Chapter 12 The Scott Analysis

The Scott analysis is a framework to determine the isomorphism type of any countable structure through descriptions of the structure in infinitary logic. As an abstract solution to the isomorphism problem, it provides useful information about invariant Borel classes of countable models. A generalized Scott analysis on arbitrary Polish G-spaces provides a scenario of how the orbit equivalence relations might be reduced to isomorphism relations.

12.1

Elements of the Scott analysis

In this section we present the original Scott analysis for countable models. In the setup we have a countable relational language L and a countable Lstructure M . The objective is to obtain an Lω1 ω -sentence ϕM , known as the canonical Scott sentence for M , with the property that for any countable L-structure N , if N |= ϕM , then N ∼ = M . This sentence is built up by a transfinite process. At each stage of the process the formulas obtained are considered approximations of the final product. By the Scott analysis we refer to this whole process and the partial information these approximations provide. Definition 12.1.1 Let L be a countable relational language. The quantifier rank of an Lω1 ω formula is inductively defined as follows: qr(ϕ) = 0 if ϕ is atomic, qr(¬ϕ) % = qr(ϕ), & qr( Φ) = qr( Φ) = sup{qr(ϕ) : ϕ ∈ Φ}, qr(∃xϕ) = qr(∀xϕ) = qr(ϕ) + 1. Definition 12.1.2 Let L be a countable relational language, M, N ∈ Mod(L), a, b ∈ ω n for some n ∈ ω, and α < ω1 . We say that (M, a) and (N, b) are α-equivalent, denoted (M, a) ≡α (N, b), if for all Lω1 ω -formula ϕ(x1 , . . . , xn ) with qr(ϕ) ≤ α, M |= 273 © 2009 by Taylor & Francis Group, LLC

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ϕ[a] iff N |= ϕ[b]. We also say that M and N are α-equivalent, denoted M ≡α N , if for all Lω1 ω -sentence ϕ with qr(ϕ) ≤ α, M |= ϕ iff N |= ϕ. Thus M ≡α N can be interpreted as the special case (M, ∅) ≡α (N, ∅). Definition 12.1.3 Let L be a countable relational language and M ∈ Mod(L). For any tuple a ∈ ω n and α < ω1 , the canonical Scott formula of rank α for a in M , a v ) where v = (v0 , . . . , vn−1 ) ∈ Vn , is inductively defined as denoted ϕM, α ( follows: % a (v ) = { θ(v ) : θ is atomic or negated atomic and M |= θ[a] }, ϕM, 0 a a v ) = ϕM, v) ∧ ϕM, α ( α+1 (

'



a b  (∃vn )ϕM, (v vn ) ∧ (∀vn ) α

b∈ω a v) = ϕM, λ (

'

(



a b  ϕM, (v vn ), α

b∈ω

a ϕM, v ), if λ is a limit. α (

α<λ a M, a It is easy to check that qr(ϕM, a]. α ) = α and M |= ϕα [

Lemma 12.1.4 Let L be a countable relational language, M, N ∈ Mod(L), a, b ∈ ω n for some n ∈ ω, and α < ω1 . Then the following are equivalent: (i) (M, a) ≡α (N, b). a (ii) N |= ϕM, α [b]. 

a b (iii) ϕM, = ϕN, α α .

Proof. The implication (i)⇒(ii) easily follows from the definitions. We show (ii)⇒(iii) and (iii)⇒(i). First we show (ii)⇒(iii) by induction on α. The cases of α = 0 and α being a limit are both straightforward from the definitions. a  For the successor case, assume N |= ϕM, α+1 [b]. Thus  N |=

a ϕM, α [b]

and N |=



'

a c (∃vn )ϕM, α

[b] and N |= (∀vn )

c∈ω

(

a c  ϕM, [b]. α

c∈ω 

M, a a M, a b By N |= ϕM, = ϕN, α [b] and the inductive hypothesis ϕα α . To show ϕα+1 = 

b ϕN, α+1 it suffices to prove that 

a {ϕM, α

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c

 d

b : c ∈ ω} = {ϕN, α

: d ∈ ω}.

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a c  [b], and therefore there is To see this let first c ∈ ω. Then N |= (∃vn )ϕM, α  a c  a c [ b d]. By the inductive hypothesis, ϕM, = d ∈ ω such that N |= ϕM, α α N,b d ϕα . Conversely, let d ∈ ω be arbitrary. Then ( a c   ϕM, [b d]. N |= α c∈ω 

a c  Thus there is c ∈ ω such that N |= ϕM, [b d], and by the inductive hyα M, a c N,b d pothesis ϕα = ϕα . This finishes the proof of (ii)⇒(iii). 

a b = ϕN, Next we show (iii)⇒(i) by induction on α. For α = 0, assume ϕM, 0 0 . Then for any atomic or negated atomic formula θ, M |= θ[a] iff N |= θ[b]. Let ψ(x1 , . . . , xn ) be a quantifier-free formula. We need to verify that M |= ψ[a] iff N |= ψ[b] by induction on the form of ψ. If ψ is atomic%then there is nothing to prove. The case ψ = ¬ψ  is easy. Suppose ψ = Φ. Then for   any ψ  ∈ Φ, M |= hypothesis, and it % ψ [a] iff N%|= ψ [b] by the inductive & follows that M |= Φ iff N |= Φ. The case ψ = Φ follows similarly. This completes the case α = 0. % b a a M, a = ϕN, = β<α ϕM, Suppose next α is a limit, and ϕM, α α . Since ϕα β a and for each β < α, qr(ϕM, ) = β, we must have that for each β < α, β M, a N,b ϕ = ϕ . By the inductive hypotheses, it follows that (M, a) ≡β (N, b) β

β

for all β < α. We check that (M, a) ≡α (N, b). For this let ϕ(x1 , . . . , xn ) be a formula with qr(ϕ) ≤ α. We again use induction on the form % of α. The cases that ϕ is atomic and that ϕ = ¬ϕ are easy. Suppose ϕ = Φ. Then for every ϕ ∈ Φ, qr(ϕ ) ≤ α. By the inductive hypotheses, for any φ ∈ Φ, & M |= φ [a] iff N |= φ [b]. Hence M |= ϕ[a] iff N |= ϕ[b]. The case ϕ = Φ is similar. This completes the case α is a limit. a N,b M, a Finally suppose ϕM, α+1 = ϕα+1 . It follows in particular that N |= ϕα [b],  a b = ϕN, and since (ii)⇒(iii), we have that ϕM, α α . By the inductive hypothesis,  (M, a) ≡α (N, b). Also by the proof of (ii)⇒(iii), we have that 

a {ϕM, α

c

 d

b : c ∈ ω} = {ϕN, α

: d ∈ ω}.

Let ϕ be a formula with qr(ϕ) = α + 1. Suppose ϕ = ∃xψ. If M |= ϕ[a], then for some c ∈ ω we have M |= ψ[a c]. By the above equality there is d ∈ ω b d a c = ϕN, , and by the inductive hypothesis, (M, a c) ≡α such that ϕM, α α   (N, b d). Since qr(ψ) = α, it follows that N |= ψ[b d] and N |= ϕ[b]. By symmetry we have shown that M |= ϕ[a] iff N |= ϕ[b]. The case α = ∀xψ is similar. We have thus completed the successor case and the proof of (iii)⇒(i).

The proof of Lemma 12.1.4 is routine and uneventful, but the content of the lemma is rather surprising. Note that there are uncountably many Lω1 ω formulas of any rank; however the lemma says that for every rank it takes just

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one formula to summarize all the truth of a tuple on that rank. It is obvious from the definition of α-equivalence that ≡β ⊆ ≡α if α ≤ β. By the lemma it 



a b a b follows that if ϕM, = ϕN, then ϕM, = ϕN, for all β ≥ α. α α β β

Definition 12.1.5 Let L be a countable relational language and M, N ∈ Mod(L). For any n ∈ ω and a, b ∈ ω n , we say that (M, a) and (N, b) are ∞-equivalent or Lω1 ω equivalent, denoted by (M, a) ≡∞ (M, b), if (M, a) ≡α (N, b) for all α < ω1 . We write (M, a) ∼ = (N, b) if there is an isomorphism π from M onto N such that π(a) = b. We write M ≡∞ N for (M, ∅) ≡∞ (N, ∅). Lemma 12.1.6 (Karp) Let L be a countable relational language and M, N ∈ Mod(L). For any n ∈ ω and a, b ∈ ω n , then (M, a) ≡∞ (N, b) iff (M, a) ∼ = (N, b). In particular, ∼ M ≡∞ N iff M = N . ∼ (N, b) Proof. By an easy induction on α one can show that if (M, a) =  then (M, a) ≡α (N, b). Then (⇐) follows. To show (⇒), assume (M, a) ≡∞ (N, b). Without loss of generality we may assume a = (a0 , . . . , an−1 ), b = (b0 , . . . , bn−1 ) ∈ [ω]n , since for any i = j < n, ai = aj iff bi = bj by (M, a) ≡0 (N, b). By induction on m ∈ ω we define an+m , bn+m ∈ ω such that for all m ∈ ω, (a) if m is even, an+m is the least element of ω − {a0 , . . . , an+m−1 }; if m is odd, bn+m is the least element of ω − {b0 , . . . , bn+m−1 }; (b) letting am = (a0 , . . . , an+m−1 ) and bm = (b0 , . . . , bn+m−1 ), we have (M, am ) ≡∞ (N, bm ). Before carrying out the construction we note the following facts. For any  be the least ordinal α < ω1 such that k ∈ ω and tuples c, d ∈ ω k , we let α(c, d)   = 0 if (M, c) ≡∞ (N, d).  (M, c) ≡α (M, d), if such an α exists; and let α(c, d) Let  : k ∈ ω, c, d ∈ ω k }. α(M, N ) = sup{α(c, d) Then α(M, N ) < ω1 . It is straightforward from the definition that if (M, c) ≡β  for some β ≥ α(M, N ), then (M, c) ≡∞ (N, d).  Thus to satisfy (b) we (N, d) m  only need to make sure that (M, a ) ≡α(M,N ) (N, bm ). Now for m = 0 let an be the least element of ω − {a0 , . . . , an−1 }. Since (M, a) ≡∞ (N, b), and in particular, (M, a) ≡α(M,N )+1 (N, b), we have that a  N |= ϕM, α(M,N )+1 [b]. It follows that ' a c  N |= (∃vn )ϕM, α(M,N ) [b], c∈ω

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and in particular 

a an  N |= (∃vn )ϕM, α(M,N ) [b].

Let bn ∈ ω be such that 1

a  N |= ϕM, α(M,N ) [b bn ].

Then by Lemma 12.1.4 (M, a1 ) ≡α(M,N ) (N, b1 ), and therefore (M, a1 ) ≡∞ (N, b1 ) as required by (c). The case m = 1 is similar with the roles of a’s and b’s reversed. The general even case is similar to the case m = 0 and the odd case is similar to that of m = 1. This completes the construction of the sequences (an+m )m∈ω and (bn+m )m∈ω . Define π(ai ) = bi for all i ∈ ω. We check that π is an isomorphism from M onto N . It follows from (a) that π is a bijection from ω onto ω. Suppose R ∈ L is an l-ary relation symbol and M |= R[an1 , . . . , anl ]. Let m ∈ ω be large enough such that {n1 , . . . , nl } ⊆ {0, . . . , n + m}. Then N |= R[bn1 , . . . , bnl ] since (M, am ) ≡0 (N, bm ). By symmetry we have that M |= R[an1 , . . . , anl ] iff N |= R[bn1 , . . . , bnl ]. Definition 12.1.7 Let L be a countable relational language and M ∈ Mod(L). The Scott rank of M , denoted by sr(M ), is the least ordinal α such that for any n ∈ ω and b a M,b a = ϕM, then ϕM, tuples a, b ∈ ω n , if ϕM, α α α+1 = ϕα+1 . The canonical Scott sentence of M , denoted by ϕM , is ' a M, a ∀v0 . . . ∀vn−1 ( ϕM, ϕM,∅ sr(M) ∧ sr(M) (v0 , . . . , vn−1 ) → ϕsr(M)+1 (v0 , . . . , vn−1 ) ). n∈ω  a∈ω n

Note that qr(ϕM ) = sr(M ) + ω. Theorem 12.1.8 (Scott) Let L be a countable relational language and M, N ∈ Mod(L). Then the following are equivalent: (i) M ∼ = N. (ii) N |= ϕM . (iii) ϕM = ϕN . Proof.

a ∈ ωn. We first verify that M |= ϕM . Of course M |= ϕM,∅ sr(M) . Let  

a M,b By the definition of the Scott rank, for any b ∈ ω n , if ϕM, sr(M) = ϕsr(M) then

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a M,b ϕM, sr(M)+1 = ϕsr(M)+1 . By Lemma 12.1.4 it is equivalent to stating that if a  M, a  M |= ϕM, sr(M) [b] then M |= ϕsr(M)+1 [b]. Thus we have that a M, a M |= ∀v ( ϕM, sr(M) → ϕsr(M)+1 ).

Hence M |= ϕM . Now the implications (i)⇒(iii)⇒(ii) are obvious. It remains to show (ii)⇒(i). For this assume N |= ϕM . We claim that for any α ≥ sr(M ) + 1, ' a M, a N |= ∀v0 . . . ∀vn−1 ( ϕM, sr(M) (v0 , . . . , vn−1 ) → ϕα (v0 , . . . , vn−1 ) ). n∈ω  a∈ω n

This is proved by induction on α ≥ sr(M ) + 1. The case α = sr(M ) + 1 is immediate from the assumption N |= ϕM . The case α is a limit follows easily from the inductive hypotheses. We only consider the case α = β + 1 where a  β ≥ sr(M ) + 1. For this let n ∈ ω and a, b ∈ ω n such that N |= ϕM, sr(M) [b]. We need to show that N |= ϕM,a [b]. By the inductive hypothesis N |= ϕM,a [b]. β+1

β

a  Let c ∈ ω. Since N |= ϕM, sr(M)+1 [b] by N |= ϕM , we can find d ∈ ω such  a c  that N |= ϕM, sr(M) [b d]. By the inductive hypothesis it follows that N |=  ϕM,a c [b d]. A similar, symmetric argument gives that for any d ∈ ω there β



a c  is c ∈ ω such that N |= ϕM, [b d]. This finishes the proof for the successor β case. By the n = 0 case of the claim we obtain that N |= ϕM,∅ for all α ≥ α sr(M ) + 1. By Lemma 12.1.4 N ≡∞ M , and hence by Lemma 12.1.6 N ∼ = M.

Exercise 12.1.1 Show that if (M, a) ≡sr(M) (M, b) then (M, a) ≡∞ (M, b). A tuple a is Lω1 ω -definable in M if there is an Lω1 ω -formula θ(x) such that M |= ϕ[a] and M |= ¬ϕ[b] for all b = a. We say that (M, a) is rigid if there is no nontrivial automorphism π of M with π(a) = a. Similarly, M is rigid if it has no nontrivial automorphism. Exercise 12.1.2 Show that a is Lω1 ω -definable in M iff for all π ∈ Aut(M ), π(a) = a. Exercise 12.1.3 Show that M is rigid, that is, has no nontrivial automorphism, iff every element of M is Lω1 ω -definable in M . Exercise 12.1.4 Show that if (M, a) is rigid for some a then M has only countably many automorphisms.

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The Scott Analysis

12.2

279

Borel approximations of isomorphism relations

Throughout this section we fix a countable relational language L. We consider Mod(L) with the canonical topology and investigate the descriptive complexity of the sets and relations arising in the Scott analysis. Lemma 12.2.1 For any α < ω1 the relation ≡α is a Borel equivalence relation with every equivalence class Π01+2·α . Proof. By Lemma 12.1.4 and the definition of canonical Scott formulas we can inductively characterize ≡α as follows: (M, a) ≡0 (N, b) ⇐⇒ for all atomic formula θ, M |= θ[a] iff N |= θ[b]. (M, a) ≡α+1 (N, b) ⇐⇒ ∀c ∈ ω ∃d ∈ ω (M, a c) ≡α (N, b d) ∧∀d ∈ ω ∃c ∈ ω (M, a c) ≡α (N, b d). (M, a) ≡λ (N, b) ⇐⇒ ∀α < λ (M, a) ≡α (N, b), if λ is a limit. The lemma follows from this characterization by an easy induction. Again since there are uncountably many formulas of any given quantifier rank, the lemma is not clear from the definition of ≡α . By Lemma 12.1.6 the isomorphism relation on Mod(L) is α<ω ≡α . Thus ≡α can be viewed as Borel approximations of the isomorphism relation. We will show that these approximations are canonical in the sense that for any invariant Borel class Mod(ϕ) if ∼ =ϕ is Borel then it coincides with ≡α  Mod(ϕ) for some α < ω1 . Before proving this we need the following observation. Lemma 12.2.2 If an invariant Borel class C is a Π0α subset of Mod(L), then there is a formula ϕ with qr(ϕ) ≤ ω · α such that C = Mod(ϕ). Proof. Examine the proof of Lemma 11.3.5 and count the quantifier rank of the formulas involved. We get that if B ⊆ Mod(L) is Π0α then for each n ∈ ω there are formulas ϕn and ψn with qr(ϕn ), qr(ψn ) ≤ ω · α such that (M, a) ∈ B [ n] ⇐⇒ M ∈ Mod(ϕn , a), (M, a) ∈ B [∗n] ⇐⇒ M ∈ Mod(ψn , a). Now if C is an invariant Borel class then C [∗0] = C × {∅} and C = Mod(ϕ0 ), where qr(ϕ0 ) ≤ ω · α.

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0 In particular if a single isomorphism class [M ]∼ = is Πα then there is ϕ with qr(ϕ) ≤ ω · α so that for any N ∈ Mod(L), if N |= ϕ then N ∼ = M . Thus the sentence has the similar property as the canonical Scott sentence. We next prove that the quantifier rank of such a sentence is a good approximation of the Scott rank.

Lemma 12.2.3 Let M ∈ Mod(L) and α < ω1 . Suppose for all N ∈ Mod(L), if M ≡α N then M∼ = N . Then sr(M ) ≤ α + ω. Proof.

Let F be the countable fragment of Lω1 ω generated by {ϕM,∅ : β< β

a α + ω}. Then for every n ∈ ω, a ∈ ω n and k ∈ ω, ϕM, α+k ∈ F since it is a M,∅ subformula of ϕα+k+n . Let T = ThF (M ). We note that for any N ∈ Mod(T ), N ∼ since ϕM,∅ ∈ F, = M . This is because, if N ∈ Mod(T ), then N |= ϕM,∅ α α ∼ and by Lemma 12.1.4 N ≡α M ; therefore N = M by our assumption. Thus M is the unique model of T up to isomorphism. We claim that M is an F atomic model of T . To see this, assume otherwise, that there is a nonprincipal F -type Φ of T realized in M . Then by the omitting types theorem there is a model N ∈ Mod(T ) omitting the type Φ, and thus N ∼  M , contradiction. = Let n ∈ ω and a ∈ ω n be arbitrary. Then tpF (a/M ) is principal. Hence there is an F -formula ψa ∈ tpF (a/M ) such that for all φ ∈ tpF (a/M ), N ∈ Mod(T ) and b ∈ ω n , if N |= ψa [b] then N |= φ[b]. By our construction qr(ψa ) < α + ω. Let βa = qr(ψa ). We now have that for all b ∈ ω n , if (M, a) ≡βa (M, b), then (M, a) ≡F (M, b), and therefore (M, a) ≡α+ω (M, b). We now claim that for all n ∈ ω and a, b ∈ ω n , if (M, a) ≡α+ω (M, b) then (M, a) ∼ = (M, b). This implies that sr(M ) ≤ α + ω. To prove the claim we assume (M, a) ≡α+ω (M, b) and use a back-and-forth construction. We indicate one step of the construction. For any c ∈ ω, we have that (M, a) ≡βa c +1 (M, b), and hence there is d ∈ ω such that (M, a c) ≡βa c (M, b d). This implies that (M, a c) ≡α+ω (M, b d). Now it is clear that the back-and-forth construction can sustain to give an automorphism π of M so that π(a) = b.

Theorem 12.2.4 Let L be a countable relational language and ϕ an Lω1 ω -sentence. Then the following are equivalent: ∼ϕ is Borel. (i) = (ii) There is α < ω1 such that ∼ =ϕ coincides with ≡α  Mod(ϕ). (iii) There is α < ω1 such that sr(M ) < α for all M ∈ Mod(ϕ). Proof. The implication (ii)⇒(i) is obvious by Lemma 12.2.1. Next we show (iii)⇒(ii). Note that (iii) implies that there is α < ω1 such that sr(M )+ω < α

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for all M ∈ Mod(ϕ). Now for M, N ∈ Mod(ϕ), if M ≡α N , then N |= ϕM and therefore N ∼ = M by Scott’s Theorem 12.1.8. This shows that ≡α  Mod(ϕ) coincides with ∼ =ϕ . Finally we show (i)⇒(iii). Assume ∼ =ϕ is Π0β for some β < ω1 . Then in 0 particular every orbit is Πβ . By Lemma 12.2.2 for every M ∈ Mod(ϕ) there is a sentence ψM with qr(ψM ) ≤ ω · β such that [M ]∼ = = Mod(ψM ). Let α = ω · β + ω + 1 = ω · (β + 1) + 1. By Lemma 12.2.3 sr(M ) < α. We next define a cofinal family of Borel isomorphism relations known as the Friedman–Stanley tower. The basic idea is to start with the simplest equivalence relation, the identity relation on a standard Borel space, and iterate the Friedman–Stanley jump operation transfinitely. Recall that for equivalence relation E on a standard Borel space X, the Friedman–Stanley jump E + is an equivalence relation on X ω defined as xE + y ⇐⇒ {[xn ]E : n ∈ ω} = {[yn ]E : n ∈ ω} . To define the Friedman–Stanley tower we need first to fix some notation for exponentiation of countable ordinals. We note that the notation we introduce is not standard, and the purpose to introduce it is to avoid confusion with product spaces. Notation 12.2.5 For any α < ω1 we define the ordinal (α) inductively as follows: (0) = ω, (α + 1) = (α) · ω, (λ) = supα<λ (α), if λ is a limit. The usual notation for (α) is ω 1+α ; there  is obviously a danger of confusion with the product space such as ω α = i<α ω. The equivalence relations we (α) are considering are on product spaces $ω . We fix a natural homeomorphism # (α)ω (α+1) between the space ω and ω . Each of the ordinals (α) is a limit itself, and if α ≤ β then (α) + (β) = (β). If λ is a limit, then we also have (λ) = α<λ (α). This induces a  homeomorphism between the space ω (λ) and the product space α<λ ω (α) . Notation 12.2.6 For any α < ω1 we define the as follows: =0+ = (α+1)+ = = =λ+ =

equivalence relation =α+ on ω (α) inductively id(ω ω ), + (=α+ ) ,  α+ , if λ is a limit. α<λ =

By induction on α < ω1 it is easy to verify that =α+ is Borel bireducible to some S∞ -orbit equivalence relation and that each =α+ is a Π01+2·α equivalence relation on ω (α) .

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Lemma 12.2.7 Let L be a countable relational language. For any α < ω1 , ≡α is Borel reducible to =α+ . Proof. By induction on α < ω1 we show that the equivalence relation ≡α on Mod(L) × ω <ω is Borel reducible to =α+ . For α = 0, ≡0 is closed and hence smooth; therefore ≡0 is Borel reducible to id(ω ω ). Assume fα is a Borel reduction of ≡α on Mod(L) × ω <ω to =α+ on ω (α) . Then note that (M, a) ≡α+1 (N, b) ⇐⇒ ∀c ∈ ω ∃d ∈ ω (M, a c) ≡α (N, b d) ∀d ∈ ω ∃c ∈ ω (M, a c) ≡α (N, b d). <ω for we let fα+1 (M, a) be an element of $ any (M, a) ∈ Mod(L) × ω #Thus (α) ω ω such that fα+1 (M, a)(c) = fα (M, a c).

Then (M, a) ≡α+1 (N, b) iff fα+1 (M, a) =(α+1)+ fα+1 (N, b). This finishes the proof of the successor case. Assume λ is a limit and for all α < λ let fα be a Borel reduction from ≡α to =α+ . Then define fλ (M, a) = (fα (M, a))α<λ , and we have

(M, a) ≡λ (N, b) ⇐⇒ ∀α < λ (M, a) ≡α (N, b) ⇐⇒ ∀α < λ fα (M, a) =α+ fα (N, b) ⇐⇒ fλ (M, a) =λ+ fλ (N, b).

It follows immediately from this lemma and Theorem 12.2.4 that in the Borel reducibility hierarchy the Friedman–Stanley tower is cofinal for all Borel S∞ -orbit equivalence relations or Borel isomorphism relations. Corollary 12.2.8 Let L be a countable relational language and ϕ an Lω1 ω -sentence. Then ∼ =ϕ is Borel iff there is α < ω1 such that ∼ =ϕ is Borel reducible to =α+ . Corollary 12.2.9 Let G be a closed subgroup of S∞ and X a standard Borel G-space. Then X X is Borel iff there is α < ω1 such that EG ≤B =α+ . EG Exercise 12.2.1 Let L = {Ri }i∈ω , where each Ri is a unary relation symbol. Show that ∼ =L is Borel bireducible with =+ . Exercise 12.2.2 Show that for any α < ω1 =α+ is Borel bireducible with an S∞ -orbit equivalence relation on ω (α) .

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Exercise 12.2.3 Show that for any α < ω1 ≡α is continuously reducible to =α+ .

12.3

The Scott rank and computable ordinals

In the preceding section we already saw that the canonicity of the objects in the Scott analysis gives far-reaching results on the isomorphism relation of invariant Borel classes. The complexity of the isomorphism relation is in some precise sense equivalent to the uniformity of the complexity of the models involved. In this section we focus on sharper characterizations of the complexity of countable models. This was first done by Nadel [127], whose idea was to analyze the lightface content of the objects of the Scott analysis. It will turn out that these characterizations do give us further results about Borelness of invariant Borel classes. We now turn to Nadel’s study of the Scott analysis. We will use the notation and results reviewed in Section 1.6. In particular, the boundedness principles for Σ11 sets of computable well-orders and its relativizations will be used below. Fix a countable relational language L = {Ri }i∈I , where either I ⊆ ω is finite or I = ω, and the arity function i → ni is computable. We say that such an L is computably presented. Then there is a computable bijection between the Baire space ω ω and the space  ni 2ω . XL = i∈I CK(M)

If M ∈ Mod(L) we let xM ∈ XL be correspondent with M , and write ω1 CK(xM ) for ω1 . Now the canonicity of the Scott formulas of M implies that the CK(M) . α-equivalence within M is hyperarithmetic in M for α < ω1 Lemma 12.3.1 Let L be a countable relational language that is computably presented and CK(M)

M ∈ Mod(L). Then for any α < ω1

and n ∈ ω, the set

{(a, b) ∈ ω n × ω n : (M, a) ≡α (M, b)} ⊆ (ω n )2 is Δ11 (M ). Proof. This is an effective version of Lemma 12.2.1. For any i ∈ I and n ≥ ni , the set {a ∈ ω n : M |= Ri [a]} is clearly computable in M . Since L is computably presented, it follows that for all n ∈ ω and a, b ∈ ω n , (M, a) ≡0 (M, b) ⇐⇒ ∀i ∈ I (ni ≤ n → (M |= Ri [a] ↔ M |= Ri [b])).

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CK(M) Thus the set {(a, b) : (M, a) ≡0 (M, b)} is Π01 (M ). Now for any α < ω1 we may find x ∈ WOM such that ot(<x ) = α. Now the inductive proof of Lemma 12.2.1 can be repeated up to α, giving that {(a, b) : (M, a) ≡α (M, b)} is Π01+2·α (M ). This shows that the sets are all Δ11 (M ).

We now indicate how to eliminate the hypothesis that L is computably presented in the above lemma. Suppose L is an arbitrary countable relational language. Let x ∈ ω ω code its arity function. Then L is computable in x. There is a fixed bijection between the Baire space ω ω and XL that is also computable in x. Note that for every M ∈ Mod(L), x is computable in M . Thus by relativizing the above lemma, the assumption that L is presented computably can be removed. We also need the following uniform version of Lemma 12.3.1. Lemma 12.3.2 Let L be a countable relational language and M ∈ Mod(L). Then for any n ∈ ω both sets {(x, a, b) ∈ ω ω × (ω n )2 : x ∈ WOM → (M, a) ≡ot(<x ) (M, b)} and {(x, a, b) ∈ ω ω × (ω n )2 : x ∈ WOM → (M, a)≡ot(<x ) (M, b)} are Σ11 (M ). Proof. This clearly follows from the proof of Lemma 12.3.1 and the fact that WOM is Π11 (M ). Theorem 12.3.3 (Nadel) Let L be a countable relational language and M ∈ Mod(L). Then sr(M ) ≤ CK(M)

ω1

. CK(M)

. By the definition of the Scott rank, and by symProof. Let λ = ω1 metry, it suffices to show that if n ∈ ω, a, b ∈ ω n and (M, a) ≡λ (M, b), then for any c ∈ ω there is d ∈ ω such that (M, a c) ≡λ (M, b d). Assume not, and let n ∈ ω, a, b ∈ ω n , c ∈ ω such that (M, a) ≡λ (M, b) but ∀d ∈ ω ∃α < λ (M, a c) ≡α (M, b d). For each d ∈ ω we let αd < λ be the least such ordinal. Then note that for any α ≥ αd , (M, a c) ≡α (M, b d). Define S = { (d, x) ∈ ω × ω ω : x ∈ WOM ∧ ot(<x ) = αd }. Then we have that (d, x) ∈ S iff the following conditions hold:

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(i) x ∈ LO and x is computable in M , (ii) for all y ∈ Δ11 (M ), if y ∈ WOM then either (iia) (M, a c) ≡ot( α, then CK(M) sr(M ) < ω1 . Proof. The direction (⇒) follows from Theorem 12.2.4. We use the boundedness principle to show (⇐). Fix α0 < ω1 such that for all M ∈ Mod(ϕ), CK(M) either sr(M ) ≤ α0 or sr(M ) < ω1 . Fix x0 ∈ WO such that ot(<x0 ) = α0 . For all x ∈ WO let Px be the set of all M ∈ Mod(ϕ) such that for all n ∈ ω and a, b ∈ ω n , (M, a) ≡ot(<x ) (M, b) ⇒ (M, a) ≡ot(<x )+1 (M, b). Then each Px is Borel. Let P = {(M, x) ∈ Mod(ϕ) × ω ω : x ∈ WO ⇒ M ∈ Px }

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and Q = {(M, x) ∈ Mod(ϕ) × ω ω : x ∈ WO ⇒ M ∈ Px }. Then both P and Q are Σ11 . Let C = {(M, x) ∈ Mod(ϕ) × ω ω : ot(<x ) = sr(M )}. Then by our assumption (M, x) ∈ C iff either of the following conditions (i) and (ii) holds: (i) there is j : ω → ω an order-preserving injection of <x into <x0 , (M, x) ∈ P , and for all n ∈ ω, there is x ∈ LO and j  : ω → ω an order-preserving bijection from <x onto <x  n such that (M, x ) ∈ Q; (ii) for all y ∈ Δ11 (M ), if y ∈ WO then either (iia) (M, y) ∈ Q and there is j : ω → ω an order-preserving injection from α0 , from CK(M) which it follows that sr(M ) < ω1 . Hence if we take y ∈ WOM so that ot(
12.4

A topological variation of the Scott analysis

In this section we follow an approach of Hjorth [70] to investigate the Scott analysis. This leads to general results about S∞ actions, and much of the

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ideas presented in this section can be generalized to arbitrary Polish group actions. Throughout this section, and for the most part of the next one, we fix a Polish S∞ -space X and a countable base {Un }n∈ω for X. For each n ∈ ω let Vn = {g ∈ S∞ : ∀i < n g(i) = i}. Each Vn is a clopen subgroup, and {Vn }n∈ω forms a nbhd base for the identity of S∞ . Note that the collection {Vn g : n ∈ ω, g ∈ S∞ } is actually countable and is a base for S∞ . Similarly the collection {gVn : g ∈ S∞ , n ∈ ω} is also a countable base for S∞ . Definition 12.4.1 For x ∈ X and α < ω1 , the canonical Hjorth–Scott type of rank α for x with respect to Vn is inductively defined as follows: ϕ0 (x, Vn ) = {l ∈ ω : Vn · x ∩ Ul = ∅}, ϕα+1 (x, Vn ) = {(ϕα (ˆ x, Vm ))m≥n : x ˆ ∈ Vn · x}, ϕλ (x, Vn ) = (ϕα (x, Vn ))α<λ , if λ is a limit. The definition is obviously motivated by that of the canonical Scott formulas; however, there are important differences in details. First, apparently ϕ0 (x, Vn ) is an element of 2ω , but in general ϕα+1 (x, Vn ) is a set of countable sequences of types of rank α. We show below that ϕα+1 is a countable set, and with this observation we may code ϕα (x, Vn ) by elements of 2(α) . Lemma 12.4.2 For all α < ω1 , x, x1 , x2 ∈ X and n ∈ ω, the following hold: (a) if x1 ∈ Vn · x2 then ϕα (x1 , Vn ) = ϕα (x2 , Vn ); (b) ϕα+1 (x, Vn ) is countable. Proof. We first show (a) by induction on α < ω1 . Assume x1 ∈ Vn · x2 . Note that Vn is a subgroup of S∞ , and therefore Vn · x1 = Vn · x2 . This implies immediately that ϕ0 (x1 , Vn ) = ϕ0 (x2 , Vn ). For the successor case x, Vm ))m≥n : x ˆ ∈ Vn · x1 }. ϕα+1 (x1 , Vn ) = {(ϕα (ˆ Since Vn · x1 = Vn · x2 , this gives that ϕα+1 (x1 , Vn ) = {(ϕα (ˆ x, Vm ))m≥n : x ˆ ∈ Vn · x2 } = ϕα+1 (x2 , Vn ). The limit case follows easily from the inductive hypothesis. This finishes the proof of (a). Now for any x1 , x2 ∈ Vn ·x and m ≥ n, it follows from (a) that if x1 ∈ Vm ·x2 then ϕα (x1 , Vm ) = ϕα (x2 , Vm ). Since Vm is an open subgroup of Vn , there is a countable set g0 , g1 , · · · ∈ Vn such that Vn = k Vm gk . It follows that ϕα+1 (x, Vn ) = {(ϕα (gk · x, Vm ))m≥n : k ∈ ω},

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hence is a countable set. In view of the above lemma we may define ψα (x, Vn ) ∈ 2(α) to code ϕα (x, Vn ) so that ψ0 (x, Vn ) = ϕ0 (x, Vn ), ψα+1 (x, Vn ) = (ψα (gk · x, Vm ))m≥n,

k∈ω ,

ψλ (x, Vn ) = (ψα (x, Vn ))α<λ , if λ is a limit.

where (gk )k∈ω is a canonical countable sequence such that Vn = k Vm gk . Then by an easy induction we can see that ψα (x, Vn ) is an element of 2(α) . Moreover, ϕα (x1 , Vn ) = ϕα (x2 , Vn ) iff ψα (x1 , Vn ) =α+ ψα (x2 , Vn ). To summarize, the canonical Hjorth–Scott types are hereditarily countable sets describing the action of Vn on X. Its definition resembles, but is not literally equivalent to, the original Scott analysis. The main difference is in the definition of the types of rank 0. Despite the difference, the canonical Hjorth–Scott types give rise to equivalence relations approximating the orbit equivalence. These equivalence relations are Borel reducible and correspondent to the ones in the Friedman–Stanley tower. In the following we will recover all the crucial properties of elements of the original Scott analysis for the Hjorth–Scott types. We remark that there are logical methods which can make the two approaches literally equivalent. However, we prefer to give the full details of the topological approach, first because it can be independently presented, and secondly because we are going to generalize it in the next section. Lemma 12.4.3 For all α ≤ β < ω1 , x1 , x2 ∈ X and n ∈ ω, if ϕβ (x1 , Vn ) = ϕβ (x2 , Vn ) then ϕα (x1 , Vn ) = ϕα (x2 , Vn ). Proof. We fix α and prove the lemma by induction on β ≥ α. The statement is a tautology when β = α. For the successor case assume ϕβ+1 (x1 , Vn ) = ϕβ+1 (x2 , Vn ). In particular, there is x ˆ1 ∈ Vn · x1 such that ϕβ (ˆ x1 , Vm ) = x1 , Vn ) = ϕα (x2 , Vn ). ϕβ (x2 , Vm ) for all m ≥ n. By the inductive hypothesis ϕα (ˆ By Lemma 12.4.2 (a) ϕα (x1 , Vn ) = ϕα (ˆ x1 , Vn ). Hence ϕα (x1 , Vn ) = ϕα (x2 , Vn ). Finally for the limit case assume β is a limit, β > α, and ϕβ (x1 , Vn ) = ϕβ (x2 , Vn ). Then by definition ϕα (x1 , Vn ) = ϕα (x2 , Vn ). Thus as in the original Scott analysis we may define the equivalence relation ≡α on X × ω by (x, n) ≡α (y, m) ⇐⇒ n = m ∧ ϕα (x, Vn ) = ϕ(y, Vn ),

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and let ≡α on X be defined as x ≡α y ⇐⇒ (x, 0) ≡α (y, 0). Lemma 12.4.4 ≡α is a Π02+2·α equivalence relation that is Borel reducible to =α+ . Proof. that

≡0 is obviously smooth and apparently Gδ . For general α we have (x, m) ≡α (y, m) ⇐⇒ n = m ∧ ψ(x, Vn ) =α+ ψ(y, Vn ).

It is easy to verify that the mapping ψ  (x, n) = ψ(x, Vn ) is a Borel function from X × ω to 2(α) . We need to show that ESX∞ = α<ω1 ≡α . First we show that a rank similar to the Scott rank exists for every x ∈ X. Lemma 12.4.5 For all x ∈ X there is some γ(x) < ω1 such that for all α < ω1 , n ∈ ω, and x1 , x2 ∈ [x]S∞ , if ϕγ(x) (x1 , Vn ) = ϕγ(x) (x2 , Vn ) then ϕα (x1 , Vn ) = ϕα (x2 , Vn ). Proof. By Lemma 12.4.2 (a) for any x ˆ1 ∈ Vn · x1 , ϕα (ˆ x1 , Vn ) = ϕα (x1 , Vn ) for all α < ω . Let h , h , . . . enumerate a countable set such that G = V0 = 1 0 1

V h . Then for any g ∈ G there is k ∈ ω such that g · x ∈ Vn · (hk · x), k n k and therefore ϕα (g · x, Vn ) = ϕα (hk · x, Vn ) for all α < ω1 . On the other hand, by Lemma 12.4.3, if ϕα (hk · x, Vn ) = ϕα (hk · x, Vn ) then for all β ≥ α, ϕβ (hk · x, Vn ) = ϕβ (hk · x, Vn ). Thus for each pair k, k  ∈ ω we may let n γk,k  (x) be the least α such that ϕα (gk · x, Vn ) = ϕα (gk · x, Vn ) if it exists, n  and 0 otherwise. Let γ(x) = sup{γk,k  (x) : k, k , n ∈ ω}. Then γ(x) < ω1 is as required. Definition 12.4.6 The Hjorth–Scott rank of x ∈ X, denoted γ(x), is the least ordinal γ such that for all n ∈ ω, x1 , x2 ∈ [x]S∞ and α < ω1 , if ϕγ(x) (x1 , Vn ) = ϕγ(x) (x2 , Vn ) then ϕα (x1 , Vn ) = ϕα (x2 , Vn ). The Hjorth–Scott invariant of x ∈ X, denoted ϕ(x), is ϕγ(x)+2 (x, V0 ). By definition and Lemma 12.4.2(a) the Hjorth–Scott rank and invariant are both invariants of the orbit of x. The information for the Hjorth–Scott rank is contained in the Hjorth–Scott invariant, as the following lemma shows. Lemma 12.4.7 If ϕγ(x)+2 (y, V0 ) = ϕ(x) then γ(y) = γ(x).

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Proof. Let γ = γ(x). We first show γ(y) ≤ γ(x). For this let y1 , y2 ∈ [y]S∞ , n ∈ ω, and assume ϕγ (y1 , Vn ) = ϕγ (y2 , Vn ). We need to verify that ϕα (y1 , Vn ) = ϕα (y2 , Vn ) for all α < ω1 . The case α ≤ γ follows from Lemma 12.4.3. We prove the case α ≥ γ by induction on α. For the successor case we need to show that ϕα+1 (y1 , Vn ) = ϕα+1 (y2 , Vn ), or by definition {(ϕα (ˆ y1 , Vm ))m≥n : yˆ1 ∈ Vn · y1 } = {(ϕα (ˆ y2 , Vm ))m≥n : yˆ2 ∈ Vn · y2 }. By our assumption that ϕγ+2 (y, V0 ) = ϕ(x), we get that ϕγ+2 (y1 , V0 ) = ϕγ+2 (y2 , V0 ) = ϕγ+2 (y, V0 ) = ϕγ+2 (x, V0 ) by Lemma 12.4.2(a). Hence there are x1 , x2 ∈ V0 · x such that ϕγ+1 (y1 , Vn ) = ϕγ+1 (x1 , Vn ), ϕγ+1 (y2 , Vn ) = ϕγ+1 (x2 , Vn ). By Lemma 12.4.5 we have ϕγ (y1 , Vn ) = ϕγ (x1 , Vn ), ϕγ (y2 , Vn ) = ϕγ (x2 , Vn ). It follows that ϕγ (x1 , Vn ) = ϕγ (x2 , Vn ) by our assumption that ϕγ (y1 , Vn ) = ϕγ (y2 , Vn ). Now the definition of γ = γ(x) implies that ϕγ+1 (x1 , Vn ) = ϕγ+1 (x2 , Vn ). Thus it follows that ϕγ+1 (y1 , Vn ) = ϕγ+1 (y2 , Vn ), or {(ϕγ (ˆ y1 , Vm ))m≥n : yˆ1 ∈ Vn · y1 } = {(ϕγ (ˆ y2 , Vm ))m≥n : yˆ2 ∈ Vn · y2 }. By the inductive hypotheses {(ϕα (ˆ y1 , Vm ))m≥n : yˆ1 ∈ Vn · y1 } = {(ϕα (ˆ y2 , Vm ))m≥n : yˆ2 ∈ Vn · y2 }, and we are done with the successor case. The limit case is straightforward. This finishes the induction and the proof that γ(y) ≤ γ(x). Now since γ(y) + 2 ≤ γ(x) + 2 it follows that ϕ(y) = ϕγ(y)+2 (y, V0 ) = ϕγ(y)+2 (x, V0 ) by Lemma 12.4.5. Thus by symmetry we have that γ(x) ≤ γ(y). This shows that γ(y) = γ(x).

Theorem 12.4.8 (Hjorth) Let X be a Polish S∞ -space. Then xESX∞ y iff ϕ(x) = ϕ(y). Proof. It suffices to prove (⇐). So suppose ϕ(x) = ϕ(y). Then by Lemma 12.4.7 γ(x) = γ(y). Let γ be this common value. Let d be a compatible complete metric on X with d < 1. We construct sequences of elements gi , hi ∈ S∞ , xi , yi ∈ X, mi , ni ∈ ω for i ∈ ω so that the following hold:

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(i) g0 = h0 = 1S∞ ; (ii) xi = gi · x, yi = hi · y; (iii) xi , yi ∈ Umi , diam(Umi ) ≤ 2−i , Umi+1 ⊆ Umi ; ∈ Vni , ni < ni+1 ; (iv) gi+1 gi−1 ∈ Vni , hi+1 h−1 i (v) ϕγ (xi , Vni ) = ϕγ (yi , Vni ); (vi) if i is even then ki = sup{gi (j), gi−1 (j) : j < i} < ni ; (vii) if i is odd then li = sup{hi (j), h−1 i (j) : j < i} < ni . Granting the construction, (gi ) and (hi ) both have limits in S∞ . Let gi → g∞ and hi → h∞ as i → ∞. Then by the continuity of the S∞ -action, (ii) and (iii) we have g∞ · x = h∞ · y. Thus xESX∞ y. The construction is by induction on i. As (vi) and (vii) suggest the even and odd inductive steps form a back-and-forth argument. By symmetry it suffices to assume that i is even and that gi , hi , xi , yi , mi , ni have been defined to satisfy (i) through (vi). We define gi+1 , hi+1 , xi+1 , yi+1 , mi+1 , ni+1 to satisfy (ii) through (v) and (vii). First we note that ϕγ+1 (xi , Vni ) = ϕγ+1 (yi , Vni ). This is because, by ϕ(x) = ϕ(y), we have ϕγ+2 (xi , V0 ) = ϕγ+2 (yi , V0 ), and therefore there is yˆ ∈ V0 · yi = V0 · y such that ϕγ+1 (xi , Vni ) = ϕγ+1 (ˆ y , Vni ). By Lemma 12.4.5 and (v), y , Vni ). Now by the definition of γ = γ(y), ϕγ (yi , Vni ) = ϕγ (xi , Vni ) = ϕγ (ˆ ϕγ+1 (yi , Vni ) = ϕγ+1 (ˆ y , Vni ). By transitivity, ϕγ+1 (xi , Vni ) = ϕγ+1 (yi , Vni ) as claimed. We define hi+1 = hi . Then yi+1 = yi and li+1 = sup{hi (j), h−1 i (j) : j ≤ i}. Let ni+1 > max{ni , li+1 }. Then (vii) is satisfied. By the above claim there x, Vni+1 ) = ϕγ (yi , Vni+1 ). By Lemma 12.4.5 in is x ˆ ∈ Vni · xi such that ϕγ (ˆ particular ϕ0 (ˆ x, Vni+1 ) = ϕ0 (yi , Vni+1 ). Let mi+1 be such that yi ∈ Umi+1 and diam(Umi+1 ) ≤ 2−i−1 . Then Vni+1 · x ˆ ∩ Umi+1 = ∅. Let xi+1 ∈ Vni+1 · xˆ ⊆ Vni+1 Vni · xi = Vni · xi . Let gˆ ∈ Vni be such that gˆ · xi = xi+1 . Let gi+1 = gˆgi . Then (ii) through (iv) are satisfied. It remains to check that ϕγ (xi+1 , Vni+1 ) = ϕγ (yi+1 , Vni+1 ). Since xi+1 ∈ ˆ, by Lemma 12.4.2(a) we have ϕγ (xi+1 , Vni+1 ) = ϕγ (ˆ x, Vni+1 ). By our Vni+1 · x construction ϕγ (ˆ x, Vni+1 ) = ϕγ+1 (yi , Vni+1 ). Thus, since yi+1 = yi , we have ϕγ (xi+1 , Vni+1 ) = ϕγ (yi+1 , Vni+1 ), as required. The topological Scott analysis discussed in this section is the beginning of a train of powerful ideas that lead to remarkable results. In the next section we will give an example of how it can be used to obtain sharp results about S∞ actions and orbit equivalence relations. Much of this can be generalized to arbitrary Polish group actions. Exercise 12.4.1 Show that the function ψ  (x, n) = ψ(x, Vn ) from X × ω to 2(α) is Borel.

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Exercise 12.4.2 Replace S∞ by a closed subgroup G of S∞ and give a topological Scott analysis for Polish G-spaces.

12.5

Sharp analysis of S∞ -orbits

In this section we give an application of the topological Scott analysis in obtaining sharp results about S∞ -orbit equivalence relations. The theorems presented here are special cases of more general results obtained by the same methods (see References [70], [101], [77], and [82]). We first define a concept and prove some basic properties. Definition 12.5.1 Let X be a Polish space, E an equivalence relation on X, and α < ω1 . We say that E is potentially Π0α (or potentially Σ0α ) if there is a Polish topology τ on X finer than the original topology such that E is a Π0α (or Σ0α , respectively) subset of (X 2 , τ 2 ). Note that if E is potentially Π0α or potentially Σ0α for α < ω1 then E is Borel. Thus the concept makes sense only for Borel equivalence relations. The following lemmas are easy consequences of earlier theorems. Lemma 12.5.2 Let X be a Polish space and E an equivalence relation on X. Then the following are equivalent: (i) E is potentially Σ01 ; (ii) E ≤B id(ω); (iii) E is a Borel equivalence relation with only countably many equivalence classes. Lemma 12.5.3 Let X be a Polish space and E an equivalence relation on X. Then the following are equivalent: (i) E is potentially Π01 ; (ii) E is potentially Π02 ; (iii) E is smooth. Proof.

This is Theorem 6.4.4 (i) through (iii).

For orbit equivalence relations the potential classes are closely connected with the Borel reducibility.

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Lemma 12.5.4 X is Let G be a Polish group, X a Polish G-space, and α < ω1 . Then EG 0 0 0 potentially Πα (or potentially Σα ) iff there is a Polish space Y and a Πα (or X Σ0α , respectively) equivalence relation F on Y such that EG ≤B F . X Proof. If EG is potentially Π0α , as witnessed by the topology τ on X, then X let Y = (X, τ ) and F = E, and certainly EG ≤B F . Conversely, suppose X f is a reduction from X to Y witnessing that EG ≤G F . Let {Un } be a countable base for Y . Let τ be a finer topology on X such that every set in the countable collection {f (Un )} is clopen, and such that the action of G on (X, τ ) is still continuous. Then f is now continuous from (X, τ ) to Y , and X X is Π0α as a subset of (X 2 , τ 2 ). This shows that EG is potentially hence EG 0 0 Πα . The proof for potentially Σα equivalence relations is similar.

The following theorem is an application of the topological Scott analysis. Theorem 12.5.5 (Hjorth–Kechris–Louveau) X X Let X be a Polish S∞ -space. Then EG is potentially Π03 iff EG ≤B =+ . Proof. Since =+ is Π03 the implication (⇐) follows from Lemma 12.5.4. X Now suppose EG is potentially Π03 . By the technique of change of topology X is Π03 . we may assume without loss of generality that EG Fix a countable base {Ul } for X and clopen subgroups (Vn ) of S∞ as in the preceding section. Let τ be the original Polish topology on X. Let B be a countable base for S∞ . For n, l ∈ ω let Wn,l = {z ∈ X : Ul ∩ Vn · z = ∅}. Then each Wn,l is τ -closed. It follows that {Ul , Wn,l : n, l ∈ ω} generates a Polish topology τ  on X in which Wn,l is τ  -clopen. Let (gk ) enumerate a countable dense subgroup of S∞ . By Lemma 12.4.2 for any x ∈ X and n ∈ ω, ϕ1 (x, Vn ) = {(ϕ0 (gk · x, Vm ))m≥n : k ∈ ω}. Fix x ∈ X. For all n, k ∈ ω let Qn,k = {z ∈ X : ϕ0 (z, Vn ) = ϕ0 (gk · x, Vn )}. Then z ∈ Qn,k ⇐⇒ ∀l ∈ ω ( gk · x ∈ Wn,l ↔ z ∈ Wn,l ). Since Wn,l is τ  -clopen, Qn,k is τ  -closed. It follows that τ  ∪{Qn,k : n, k ∈ ω} generates again a Polish topology finer than τ  , which we denote by τ (x). It is easy to see that τ (x) is invariant, that is, if x ∈ [x]S∞ then τ (x ) = τ (x). Let C(x) be the closure of [x]S∞ in (X, τ (x)). Then (C(x), τ (x)) is again invariant. Since [x]S∞ is Π03 , there are closed sets Fi,j ⊆ (X, τ ) for i, j ∈ ω such that z ∈ [x]S∞ ⇐⇒ ∀i ∈ ω ∃j ∈ ω z ∈ Fi,j .

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This implies the following description of the orbit of x: z ∈ [x]S∞ ⇐⇒ ∀∗ g ∈ S∞ g · z ∈ [x]S∞ ⇐⇒ ∀∗ g ∈ S∞ ∀i ∈ ω ∃j ∈ ω g · z ∈ Fi,j ⇐⇒ ∀i ∈ ω ∀∗ g ∈ S∞ ∃j ∈ ω g · z ∈ Fi,j ⇐⇒ ∀i ∈ ω ∀V ∈ B ∃∗ g ∈ V ∃j ∈ ωg · z ∈ Fi,j ⇐⇒ ∀i ∈ ω ∀V ∈ B ∃j ∈ ω ∃V  ∈ B (V  ⊆ V ∧ ∀∗ g ∈ V  g · z ∈ Fi,j ) 

∗V ) ⇐⇒ ∀i ∈ ω ∀V ∈ B ∃j ∈ ω ∃V  ∈ B (V  ⊆ V ∧ z ∈ Fi,j ∗gVn ⇐⇒ ∀i ∈ ω ∀V ∈ B ∃j ∈ ω ∃n ∈ ω ∃g ∈ S∞ (gVn ⊆ V ∧ z ∈ Fi,j ).

In view of this, consider for each i ∈ ω and V ∈ B the set O(i, V ) = { y ∈ C(x) : ∃j ∈ ω ∃n ∈ ω ∃ˆ x ∈ [x]S∞ ∃g ∈ S∞ ∗gVn gVn ⊆ V ∧ xˆ ∈ Fi,j ∧ ϕ0 (y, Vn ) = ϕ0 (ˆ x, Vn ) }.

Then it follows from the above description that [x]S∞ ⊆ O(i, V ) for all i ∈ ω and V ∈ B. Hence O(i, , V ) ⊆ C(x) is dense in τ (x). Also from the definition of O(i, V ) it is clear that it is τ (x)-open. Thus O(i, V ) is comeager in (C(x), τ (x)). Let C0 = {O(i, V ) : i ∈ ω, V ∈ B}. Then C0 ⊆ (C(x), τ (x)) is comeager. ∗gVn ∗gVn and ϕ0 (y, Vn ) = ϕ0 (ˆ x, Vn ), then y ∈ Fi,j . This We note that if x ˆ ∈ Fi,j ∗gVn = (g −1 ·Fi,j )∗Vn , where g −1 ·Fi,j is τ -closed by the continuity is because Fi,j ∗gVn of the action. If y ∈ Fi,j , then there is l ∈ ω such that Ul ∩ g −1 · Fi,j = ∅ x, Vn ), it follows that for this l, and Vn · y ∩ Ul = ∅. Since ϕ0 (y, Vn ) = ϕ0 (ˆ Vn · xˆ ∩ Ul = ∅, hence x ˆ ∈ (g −1 · Fi,j )∗Vn , a contradiction. It follows immediate from the description of [x]S∞ that C0 ⊆ [x]S∞ . We have shown that (C(x), τ (x)) is a complete invariant of [x]S∞ . If (C(x), τ (x)) = (C(y), τ (y)), then from the above argument [x]S∞ and [y]S∞ are both comeager in this space, and hence have a nonempty intersection. This implies that [x]S∞ = [y]S∞ . It is clear from the construction that (C(x), τ (x)) depends only on ϕ1 (x, V0 ). In other words, if ϕ1 (x, V0 ) = ϕ1 (y, V0 ) then (C(x), τ (x)) = (C(y), τ (y)). We X have thus obtained that xEG y iff ϕ1 (x, V0 ) = ϕ1 (y, V0 ). Since ≡1 ≤B =+ , we X + have that EG ≤B = as required. The method illustrated in the proof of the theorem can be generalized. In general one can obtain a Polish topology τα (x) for α < ω1 to encode the

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canonical Hjorth–Scott type of rank α. This method can even be generalized to topological Scott analysis for arbitrary Polish group actions. In the rest of this section, we investigate potentially Σ03 S∞ -orbit equivalence relations and show that they are essentially countable. We need the following lemma of Kechris for essential countability. Lemma 12.5.6 (Kechris) Let X, Y be standard Borel spaces, E a Borel equivalence relation on X, and f : X → Y a Borel function. Suppose (i) for any x ∈ X, {f (y) : yEx} is countable, and (ii) for any x, y ∈ X, if f (x) = f (y) then xEy. Then E is essentially countable. Proof.

We may assume without loss of generality that X = Y = ω ω . Define

A = {y ∈ Y : ∃x ∈ X f (x) = y)} = f (X), B = {(y1 , y2 ) ∈ Y 2 : ∃x1 , x2 ∈ X (f (x1 ) = y1 ∧ f (x2 ) = y2 ∧ x1 Ex2 ) }, C = {(y1 , y2 ) ∈ Y 2 : ∀x1 , x2 ∈ X [(f (x1 ) = y1 ∧ f (x2 ) = y2 ) → x1 Ex2 ] }. Then B ∈ Σ11 , C ∈ Π11 , and by assumption (ii) B ∩ A2 = C ∩ A2 . By the Luzin separation theorem (Theorem 1.6.1) there is a Borel set D such that B ⊆ D ⊆ C, and hence D ∩ A2 = B ∩ A2 = C ∩ A2 . Note that the Σ11 set A has the following properties: (1) D is an equivalence relation on A, and (2) for all y1 ∈ A, the set {y2 ∈ A : (y1 , y2 ) ∈ D} is countable. We claim that there is a Borel set A0 ⊇ A such that (1) and (2) hold for A0 instead of A. To prove this claim let g : Y → LO be a Borel function with f −1 (WO) = Y − A. For any z ∈ A let αz = ot(
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and (2) hold for Y − Rz instead of A. First let z ∈ A. Then Rz = Y − A and so (1) and (2) hold for Y − Rz = A. On the other hand, let z ∈  A, then y ∈ Rz ⇐⇒ y ∈ A ∧ αy < αz ⇐⇒ there is n ∈ ω and j : ω → ω order-preserving from
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X is essentially countable. (iii) EG

Proof. (iii)⇒(i) is immediate from Lemma 12.5.4. (i)⇒(ii) is obvious. It remains to show (ii)⇒(iii). X Assume EG is potentially Σ03 . By the technique of change of topology we X is Σ03 . Let {Ul } be a countable may assume without loss of generality that EG base for X. Let {Wm } be a countable base for S∞ such that for all g ∈ S∞ and m ∈ ω, there is m ∈ ω such that Wm g = Wm . Note that the collection {Vn g : n ∈ ω, g ∈ S∞ } is such a countable base for S∞ . Fix x ∈ X. Since [x]S∞ is Σ03 there are Gδ sets Fn such that [x]S∞ = n Fn . We have that ∀∗ g ∈ S∞ ∃n ∈ ω g · x ∈ Fn . Thus in particular there is n ∈ ω and m ∈ ω such that ∀∗ g ∈ Wm g · x ∈ Fn . It follows that Fn ∩ Wm · x is a dense subset of Wm · x. Since it is also a Gδ subset of Wm · x it must be comeager. We therefore conclude that [x]S∞ is comeager in Wm · x since Fn ⊆ [x]S∞ . We have shown that for any x ∈ X there is m ∈ ω such that [x]S∞ is comeager in Wm · x. Let m(x) be the least such m ∈ ω. Now define a function θ from X into the Effros Borel space F (X) by f (x) = Wm(x) · x. We verify that θ is Borel. For this let a basic open set Ul be given. Then f (x) ∩ Ul = ∅ ⇐⇒ Wm(x) · x ∩ Ul = ∅ ⇐⇒ ∃m ∈ ω ( Wm · x ∩ Ul = ∅ ∧ ∀∗ y ∈ Wm · x yESX∞ x ∧ ∀m < m ∃∗ y ∈ Wm · x (x, y) ∈ ESX∞ ). This is Borel by Kuratowski–Ulam (see Exercise 3.2.6). We check that the function f satisfies the hypotheses of Kechris’ lemma. Fix x ∈ X. Let S = {Wm · x : m ∈ ω}. Then S is countable. We claim that for any x ˆ ∈ [x]S∞ , f (ˆ x) ∈ S. In fact f (ˆ x) = Wm(ˆx) · x ˆ. Let g ∈ S∞ be such that xˆ = g · x. Then by the invariance property of the base {Wm }, there is some m ∈ ω such that Wm(ˆx) g = Wm . Then f (ˆ x) = Wm(ˆx) g · x = Wm · x ∈ S. This proves Lemma 12.5.6(i). To show Lemma 12.5.6(ii) let x, y ∈ X and assume f (x) = f (y). Then [x]S∞ is comeager in f (x) and [y]S∞ is comeager in f (y), and it follows that [x]S∞ and [y]S∞ have a nonempty intersection, and therefore they coincide. X Now by Lemma 12.5.6 EG is essentially countable. The methods introduced in this section can be applied to investigate arbitrary potential classes of S∞ -orbit equivalence relations. In this manner Hjorth, Kechris, and Louveau have completely determined the possible potential classes of all Borel isomorphism relations. Exercise 12.5.1 Prove Lemma 12.5.2.

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Exercise 12.5.2 For any tree T on ω × ω let p[T ] = {x ∈ ω ω : ∃y ∈ ω ω (x, y) ∈ [T ]}. Note that T is an element of X = 2(ω×ω) on ω × ω such that p[T ] is uncountable.

<ω

. Let P be the set of all trees T

(1) Show that p[T ] is uncountable iff there is a subtree S of T such that the following property holds for S: for every (s, u) ∈ S there are (t, v), (r, w) ∈ S with t⊥r and (s, u) ⊆ (t, v), (r, w). (2) Show that the set P is Σ11 .

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Chapter 13 Natural Classes of Countable Models

In the previous two chapters we have developed the theory of countable models in some depth from the perspectives of both mathematical logic and descriptive set theory. For invariant Borel classes with Borel isomorphism relations we have obtained satisfactory characterizations of the models. These also led to powerful results about Borel S∞ -orbit equivalence relations. However, when natural classes of countable models are considered the isomorphism relations turn out to be mostly non-Borel, and often correspond to the universal S∞ -orbit equivalence relation. In this chapter we consider various natural classes of countable models and substantiate this observation. Historically these natural invariant Borel classes are not only examples to interpret the theoretical results, but also an integral part of the countable model theory and source of inspirations for further theoretical tools. In this chapter we will focus on countable graphs, trees, linear orderings, and groups. Similar work has been done for countable lattices, fields, and Boolean algebras.

13.1

Countable graphs

In previous discussions of countable model theory our general setup has always contained an arbitrary countable relational language L. There were few results that rely on assumptions of the composition of L. However, different languages L give rise to very different isomorphism relations ∼ =L . In Section 3.6 we proved that Mod(L) is a universal Borel S∞ -space iff L contains relation symbols of arbitrarily high arity. On the other extreme, it is easy to see that if L contains only unary relation symbols then the isomorphism relation on Mod(L) is Borel (and in fact Borel reducible to =+ ). In this section we will show that if L contains at least one relation symbol of arity ≥ 2 then the isomorphism relation on Mod(L) is Borel bireducible with the universal S∞ -orbit equivalence relation. This completely classifies the languages L in terms of the complexity of the isomorphism relation ∼ =L . Γ Throughout this section we fix a language L = {R} with one binary relation symbol R. Let γ be the LΓ -sentence ∀x ∀y [ ¬R(x, x) ∧ ( R(x, y) ↔ R(y, x) ) ].

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Then every model M ∈ Mod(γ) is essentially a countable graph (V, E) with the vertex set V = ω and the edge set E = {{a, b} : M |= R[a, b]}. Conversely every countable graph is obviously represented by an element of Mod(γ). In view of this we will refer to the elements of Mod(γ) as countable graphs and the isomorphism relation ∼ =γ the graph isomorphism. We will use the usual graph theoretic terminology when discussing the ingredients of a countable graph, such as vertex (or node), edge, degree, cycle, subgraph, and so on. If a, b are nodes in a graph, we denote the edge {a, b} by ab or ba for brevity. We will also use the following terminology. Definition 13.1.1 Let C be an invariant Borel class. (a) We say that C (or ∼ = C) is Borel complete if ∼ = C is Borel bireducible with the universal S∞ -orbit equivalence relation. ∼ C) is faithfully Borel complete if every S∞ -orbit (b) We say that C (or = equivalence relation is faithfully Borel reducible to ∼ = C. (c) The topological Vaught conjecture for C, denoted TVC(C), is the statement TVC(C, ∼ = C). Clearly faithful Borel completeness implies Borel completeness. By Theorem 9.5.5 if C is faithfully Borel complete then TVC(C) implies TVC(S∞ ), the Lω1 ω -Vaught conjecture. Our objective is to show that the class of all countable graphs is faithfully Borel complete. Theorem 13.1.2 The class of all countable graphs is faithfully Borel complete. Proof. Let L = {Rn }n≥2 be a countable relational language with each Rn an n-ary relation symbol. By Theorem 3.6.1 Mod(L) is a universal Borel S∞ -space. Hence by Proposition 5.2.2 Mod(L) is faithfully Borel complete. It suffices to construct a faithful Borel reduction from Mod(L) to the class of all countable graphs Mod(γ). So for each L-structure M ∈ Mod(L), we need to associate a countable graph Γ(M ). For n ≥ 1 an n-tag is a graph Tn with the vertex set {a1 , . . . , an } ∪ {b1,1 , b2,1 , b2,2 , . . . , bn,1 , . . . , bn,n , c, d1 , d2 , d3 , f } = A ∪ B, where the two displayed sets A and B are disjoint and the demonstrated elements of B are distinct, and with the following set of edges: { a1 b1,1 , b1,1 c, a2 b2,1 , b2,1 b2,2 , b2,2 c, ...... an bn,1 , bn,1 bn,2 , . . . , bn,n−1 bn,n , bn,n c, cd1 , d1 d2 , d2 d3 , d3 d1 , f d2 }.

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Figure 13.1 illustrates an n-tag. It is important that the n-tags have no f r r d2 b b br d3 " " d1 r" c r c rbn,n  c cp pp r p b1,1 r   b2,2   c rbn,2 r  b2,1 · · · cc  rb   cn,1 r r ······ cr  a1 a2 an Figure 13.1 An n-tag Tn (a1 , . . . , an ). symmetry; each vertex in an n-tag is uniquely determined by its properties. To be specific, note that in such a graph f has degree 1, d1 , d2 , d3 form a 3-cycle, c has degree n + 1, and each other vertex in B has degree 2. Such an n-tag will be used to code the tuple (a1 , . . . , an ), and we denote such an n-tag Tn (a1 , . . . , an ). We call the vertex c the center of the n-tag. If a1 , . . . , an are (not necessarily distinct) vertices in some graph Γ0 , then by adding Tn (a1 , . . . , an ) to Γ0 we mean to add fresh elements from B and the edges of Tn to form a new graph. Now given an L-structure M we first let Γ0 be the graph on |M | = ω with a 1-tag added for each a ∈ |M |. Then for each Rn , n ≥ 2, and tuple (a1 , . . . , an ), add an n-tag Tn (a1 , . . . , an ) iff M |= Rn [a1 , . . . , an ]. The resulting graph is denoted Γ(M ). Note that in Γ(M ) each 3-cycle is created only by the addition of an n-tag. From this the centers of n-tags can be identified because of their adjacency to the 3-cycles. Let C(M ) denote the set of centers of all n-tags for n ≥ 1. Use the abbreviation ∃= y1 . . . yk \ x1 . . . xl ϕ for the formula ⎤ ⎡ ⎢ ∃y1 . . . ∃yk ⎢ ⎣

'

1≤i
⎥ ( ¬yi = yi ∧ ¬yi = xj ) ∧ ϕ ⎥ ⎦.

Let θ(x) be the LΓωω -formula ∃= y1 , y2 , y3 , z \ x [ R(x, y1 ) ∧ R(x, z) ∧ R(y1 , y2 ) ∧ R(y2 , y3 ) ∧ R(y3 , y1 ) ]. Then Γ(M ) |= θ[c] iff c ∈ C(M ).

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For i ≥ 1, let ηi (x) be the LΓωω -formula ∃= y1 . . . yi

'

'

R(x, yj ) ∧ ¬∃= y1 . . . yi+1

1≤j≤i

R(x, yj ).

1≤j≤i+1

Then Γ(M ) |= ηi [c] iff c has degree i. Also for l ≥ 1 let λl (x, y) be the LΓωω -formula ∃= z1 . . . zl \ x, y [ ¬x = y ∧ R(x, z1 ) ∧ R(zl , y) ∧ % 1≤j≤l

η2 (zj ) ∧

% 1≤j
R(zj , zj+1 ) ].

Then Γ(M ) |= λl [a, c] iff there is a path between a and c with l elements of degree 2 in between. Now note that for each c ∈ C(M ), if the degree of c is 2 then the tag it is a center of is a 1-tag. It follows that a vertex a of Γ(M ) is in |M | iff there is a path of length 2 from a to a center of a 1-tag. Let δ(x) be the LΓωω -formula ∃y ( λ1 (x, y) ∧ θ(y) ∧ η2 (y) ). Then Γ(M ) |= δ[a] iff a ∈ |M |. In general if c ∈ C(M ) then the degree of c is exactly one more than the arity of the tuple it codes. Thus if c has degree n + 1, then c codes a unique n-tuple (a1 , . . . , an ). For n ≥ 2, let ρn (x1 , . . . , xn ) be the LΓωω -formula ' ∃y ( δ(xi ) ∧ θ(y) ∧ ηn+1 (y) ∧ λi (xi , y) ). 1≤i≤n

Then Γ(M ) |= ρn [a1 , . . . , an ] iff there is c ∈ C(M ) and c codes the tuple (a1 , . . . , an ) in the sense that the unique n-tag Tn (a1 , . . . , an ) has center c. It is now clear that Γ(M ) |= ρn [a1 , . . . , an ] iff M |= Rn [a1 , . . . , an ]. We are ready to show that for M, N ∈ Mod(L), M ∼ = N iff Γ(M ) ∼ = Γ(N ). In fact, the invariance of the map Γ is obvious. We only check the converse. For this suppose π : Γ(M ) ∼ = Γ(N ). Then for all a ∈ Γ(M ), a ∈ |M | iff Γ(M ) |= δ[a] iff Γ(N ) |= δ[π(a)] iff π(a) ∈ |N |. Thus π restricted on |M | is a bijection between |M | and |N |. Now for any n ≥ 2, a1 , . . . , an ∈ |M |, M |= Rn [a1 , . . . , an ] iff Γ(M ) |= ρn [a1 , . . . , an ] iff Γ(N ) |= ρn [π(a1 ), . . . , π(an )] iff N |= Rn [π(a1 ), . . . , π(an )]. Thus π  |M | is an isomorphism between M and N . Before proceeding to show that Γ is a faithful Borel reduction, we note the following further properties of the graphs Γ(M ): (i) There is an LΓωω -formula μ1 (x1 , y1,1 , z, u1 , u2 , u3 ) such that for any countable graph Γ, Γ |= μ1 [a1 , b1,1 , c, d1 , d2 , d3 ] iff the subgraph of Γ with the vertex set {a1 , b1,1 , c, d1 , d2 , d3 } forms a 1-tag T1 (a1 ) as demonstrated in the definition of 1-tags;

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(ii) More generally for each n ≥ 2 there is a formula μn (x1 , . . . , xn , . . . ) such that for any countable graph Γ, Γ |= μn [a1 , . . . , an , . . . ] iff the subgraph of Γ with the vertex set {a1 , . . . , an , . . . } forms an n-tag as demonstrated in the definition of n-tags; (iii) Furthermore, for each n ≥ 1 and 1 ≤ p ≤ N (n) =def (n2 + 3n + 8)/2, there is a formula ξn,p (x) such that for any Γ, Γ |= ξn,p [a] iff a is contained in an n-tag and is the p-th element in the demonstration in the definition of n-tags. Let χ be the conjunction of the following LΓω1 ω -sentences: ∀x ∀x ∀x

&

n≥1,p ξn,p (x),

% n,p>n

& n,p≤n



ξn,p (x) →

%

 ¬ξ (x) , m,q (m,q)=(n,p)

( ξn,p (x) → ξ1,1 (x) ) .

By now it is easy to verify that any countable graph satisfying χ is isomorphic to Γ(M ) for some M ∈ Mod(L). We are now ready to show that Γ is a faithful Borel reduction from Mod(L) to Mod(γ). In the above we have shown that Γ : Mod(L) → Mod(γ) is a reduction. With any reasonable coding of Γ(M ) by a genuine element of Mod(γ) Γ can be easily seen to be a Borel map. We check finally that Γ is faithful. For this let C be an invariant Borel subset of Mod(L). Then there is an Lω1 ω -sentence ψ such that C = Mod(ψ). We can obtain an LΓω1 ω -sentence ψ Γ by replacing, for all n ≥ 2, all occurrences of Rn in ψ by ρn . It is clear that whenever M |= ψ we have Γ(M ) |= ψ Γ . We claim that Mod(γ ∧ χ ∧ ψ Γ ) is the smallest invariant Borel class containing Γ(C). In fact, if Γ ∈ Mod(γ ∧χ∧ψ Γ ), then there is M ∈ Mod(L) such that Γ ∼ = Γ(M ). Now it must be the case that M |= ψ, since otherwise M |= ¬ψ, and by our construction Γ(M ) |= ¬ψ Γ , a contradiction. In model theory the reduction constructed in the above proof is called an interpretation because it has a much stronger property than we need for reducibility of the isomorphism relations. As shown in the proof the countable graph used to code the countable L-structure preserves all the Lω1 ω -properties in a uniform way. Also, since the formulas ρn are first-order, the coding formulas stay in the same countable fragment generated by the original formulas. This implies that the structure Γ(M ) has approximately the same Scott rank as the original L-structure M . More examples of interpretations can be found in Reference [84] Chapter 5. We remark that the Lω1 ω -Vaught conjecture, or equivalently TVC(S∞ ), is still a major open problem. By the above theorem it is equivalent to the topological Vaught conjecture for countable graphs.

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We state the following corollary of the theorem and leave the details of the proof to the reader. Corollary 13.1.3 Let L be a countable relational language with at least one relation symbol of arity ≥ 2. Then Mod(L) is faithfully Borel complete. Exercise 13.1.1 Prove Corollary 13.1.3. Exercise 13.1.2 Give the details of the formulas μn and ξn,p , n ≥ 1 and 1 ≤ p ≤ N (n), in the proof of Theorem 13.1.2. Exercise 13.1.3 Let Γ be the reduction defined in the proof of Theorem 13.1.2. Give a formal definition of Γ(M ) as an element of Mod(γ). Exercise 13.1.4 Let Γ be the reduction defined in the proof of Theorem 13.1.2. Show that if L is a finite relational language then there is an LΓωω sentence χ such that any countable graph satisfying χ is isomorphic to Γ(M ) for some M ∈ Mod(L).

13.2

Countable trees

Recall that in descriptive set theory a tree T on ω is a subset of ω <ω closed under initial segments, that is, if s ⊆ t and t ∈ T then s ∈ T . By definition, every tree on ω contains the empty sequence ∅. For trees S, T on ω, an isomorphism is a bijection π : S → T preserving initial segments, that is, for all s1 , s2 ∈ S, s1 ⊆ s2 iff π(s1 ) ⊆ π(s2 ). For any s ∈ S, lh(π(s)) = lh(s). In <ω particular, π(∅) = ∅. Every tree on ω is apparently an element of 2(ω ) . Let Tr be the set of all trees on ω. Then Tr is a closed subset of the Polish space <ω 2(ω ) , hence is itself Polish. In graph theory, a tree is an acyclic connected graph, that is, a graph with no cycles and in which there is a path between any pair of vertices. Here a tree on ω would be called a rooted tree, meaning a tree with a distinguished element known as the root. Conversely, a graph theoretic rooted tree can clearly be coded by a tree on ω. By this correspondence the class of all trees on ω becomes an invariant Borel class. Theorem 13.2.1 (Friedman–Stanley) The class of all countable trees on ω is Borel complete.

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Proof. We define a Borel reduction from the class of all countable graphs to that of trees on ω. For each countable graph Γ we associate a tree T (Γ) on ω. For this fix a countable graph Γ with the underlying set ω and the edge relation R. Let T0 be the full tree of nonrepeating finite sequences in ω <ω . T (Γ) will be obtained by adding at most one new, terminal, immediate successor to each node in T0 . For m, n ∈ ω, let m, n = 2m 3n . Then, for all m, n > 0, if s = (x1 , . . . , xm,n ) ∈ T0 , then add s x1 to T0 iff R(xm , xn ). The resulting tree is denoted T (Γ). Clearly the map T : Γ → T (Γ) is continuous. We check that it is a reduction. First let π : Γ ∼ = Γ . Let π ∗ : ω <ω → ω <ω be the automorphism induced by π: π ∗ (x1 , . . . , xk ) = (π(x1 ), . . . , π(xk )). Then π ∗ (T0 ) = T0 . Now (x1 , . . . , xk , x1 ) ∈ T (Γ) ⇐⇒ ∃m, n > 0 ( k = m, n ∧ RΓ (xm , xn )  ⇐⇒ ∃m, n > 0 ( k = m, n ∧ RΓ (π(xm ), π(xn )) ⇐⇒ (π(x1 ), . . . , π(xk ), π(x1 )) ∈ T (Γ ). Thus π ∗ (T (Γ)) = T (Γ ) and T (Γ) ∼ = T (Γ ). ∼ Conversely, suppose σ : T (Γ) = T (Γ ). By a back-and-forth argument we find two permutations π and π  of ω such that for all l ∈ ω, σ(π(0), . . . , π(l)) = (π  (0), . . . , π  (l)). This is done by induction on l. For the base case, let π(0) = 0, s0 = (π(0)), and define π  (0) so that σ(s0 ) = (π  (0)). Next let π  (1) be the least element of ω − {π  (0)}, and t1 = (π  (0), π  (1)). Define π(1) so that σ −1 (t1 ) = (π(0), π(1)). Since t1 is not a terminal node of T (Γ ), neither is σ −1 (t1 ), and hence π(1) = π(0). In general suppose distinct π(0), . . . , π(l) and distinct π  (0), . . . , π  (l) have been defined. Suppose l is odd. Let π(l + 1) be the least element of ω − {π(0), . . . , π(l)} and sl+1 = (π(0), . . . , π(l + 1)). Then σ(sl+1 ) = (π  (0), . . . , π  (l), y) for some y ∈ {π  (0), . . . , π  (l)} since sl+1 and σ(sl+1 ) are not terminal nodes. Define π  (l + 1) = y and continue the construction. If l is even then the definition is similar to the case l = 0. Now we claim that π  ◦ π −1 is an isomorphism between Γ and Γ . To see this suppose RΓ (a, b). Let m = π −1 (a) − 1, n = π −1 (b) − 1 and k = m, n. Then the node (π(0), . . . , π(k − 1), π(0)) is a terminal node of T (Γ). It follows that σ(π(0), . . . , π(k − 1), π(0)) is a terminal node of T (Γ ). Hence σ(π(0), . . . , π(k −1), π(0)) = (π  (0), . . . , π  (k −1), π  (0)) ∈ T (Γ ). This implies   that RΓ (π  (m), π  (n)) or RΓ (π  ◦ π −1 (a), π  ◦ π −1 (b)). By symmetry we have Γ Γ  −1 R (a, b) iff R (π ◦ π (a), π  ◦ π −1 (b)) for any a, b ∈ ω.

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There are some important differences between the above reduction and the one to countable graphs in Theorem 13.1.2. One cannot recover the original graph by the graph theoretic properties of the rooted tree constructed, and the reduction defined is not faithful. Gao [52] has shown that the class of trees on ω is not faithfully Borel complete, hence the topological Vaught conjecture for this class (which is known by results of Marcus [117], A. Miller [122], and Steel [150]) does not imply the Lω1 ω -Vaught conjecture. As we remarked above the theorem immediately implies the Borel completeness of all countable graph theoretic rooted trees. The proof of the theorem also has the following corollary about the other classes of countable graphs related to trees. Corollary 13.2.2 The classes of all countable trees and of all countable acyclic graphs are both Borel complete. Proof. In the proof of Theorem 13.2.1 the tree T (Γ) constructed has the graph theoretic property that for every vertex v of T (Γ) there is at most one vertex u of T (Γ) of degree 1 and uv ∈ T (Γ). Now let S(Γ) be the graph theoretic tree obtained from T (Γ) by adding two new vertices v0 , v1 and edges between each of them and the root of T (Γ). Then Γ ∼ = Γ iff S(Γ) ∼ = S(Γ )  since any isomorphism between S(Γ) and S(Γ ) gives rise to an isomorphism between T (Γ) and T (Γ ). The class of all finite splitting trees on ω is a Borel subset of Tr, hence is a standard Borel space. If we restrict our attention to finite splitting trees on ω then the isomorphism relation is much simpler, as the following theorem shows. Theorem 13.2.3 The isomorphism relation for finite splitting trees on ω is smooth. Proof. Let F denote the countable set of all finite trees on ω. Let F0 ⊆ F contain a tree of each isomorphism type. Let θ : F → F0 be such that for any T ∈ F , θ(T ) ∼ = T . Let S be a finite splitting tree on ω. For each n ∈ ω let Sn = {s ∈ S : lh(s) ≤ n}. Then define f (S) = (θ(Sn ))n∈ω . Now f is a Borel function from the class of all finite splitting trees to F0ω . We claim that S ∼ = S  iff f (S) = f (S  ). The invariance of f is obvious, since every isomorphism between S and S  is also an isomorphism between Sn and Sn for all n ∈ ω. For the converse, suppose f (S) = f (S  ). Then Sn ∼ = Sn for all n ∈ ω. Let  σn be an isomorphism between Sn and Sn . We note that for each s ∈ S, for all n ≥ lh(s), lh(σn (s)) = lh(s). Since there are only finitely many nodes of length lh(s) in S  , there is a subset Ns of ω such that for all n, m ∈ Ns ,

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σn (s) = σm (s), that is, σn (s) is constant. The same observation applies to s ∈ S  and σ −1 . Now using this observation, and by a back-and-forth construction, we may obtain a subset N of ω such that (i) for all s ∈ S, for all but finitely many n ∈ N , the value σn (s) is a constant; and (ii) for all s ∈ S  , for all but finitely many n ∈ N , the value σn−1 (s) is constant. Define σ(s) to be the eventual constant value of σn (s) for sufficiently large n ∈ N . By (ii) σ is a bijection between S and S  . We check that σ is an isomorphism between S and S  . To see this let s, t ∈ S and assume s ⊆ t. Then for sufficiently large n ∈ N , σ(s) = σn (s) ⊆ σn (t) = σ(t). By symmetry s ⊆ t iff σ(s) ⊆ σ(t). It is easy to see that there are uncountably many pairwise nonisomorphic finite splitting trees on ω. Thus the isomorphism relation for them is Borel bireducible with id(2ω ). A graph theoretic tree is locally finite if every vertex has finite degree. The following result shows a subtle difference between the descriptive set theoretic concept and the graph theoretic one. It also shows that there is no way to identify the root as we did in the proof of Corollary 13.2.2. Theorem 13.2.4 (Jackson–Kechris–Louveau) The isomorphism relation for countable locally finite trees is Borel bireducible with E∞ . Proof. To see that this isomorphism relation is essentially countable, we use Kechris’ Lemma 12.5.6. For each locally finite tree T and any vertex t in T , let St be the finite splitting tree on ω corresponding to the rooted tree with root t. Let F0 and f be given by the proof of Theorem 13.2.3. Then for each t ∈ T , f (St ) is an element of F0ω . Now given a locally finite tree T (with the underlying set a subset of ω) let t ∈ T be a canonically selected element (for instance the least element of T ⊆ ω). Then define g(T ) = f (St ). We check that g satisfies the hypotheses of Lemma 12.5.6. Fix a locally finite tree T , let A(T ) = {f (St ) : t ∈ T }. ∼ T }. This is Then A(T ) is countable. We note that A(T ) ⊇ {g(T  ) : T  = because, if π : T  ∼ = T and t ∈ T  is chosen, then f (St ) = f (Sπ(t ) ) since St ∼ = Sπ(t ) . This shows that the range of g on every isomorphism class is countable. On the other hand, if g(T  ) = g(T ) then there is t ∈ T  and t ∈ T such that f (St ) = f (St ). This implies that St ∼ = St and hence in particular

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T ∼ = T . Thus Lemma 12.5.6 is applicable, and it follows that the isomorphism of locally finite countable trees is essentially countable. For the converse we code the shift equivalence of 2F2 by the isomorphism of locally finite trees. For each subset A of F2 we associate a locally finite tree T (A) as follows. Let a, b be the generators of F2 . Let K be a labeled directed tree with vertex set F2 and two edge relations Ra and Rb defined by Ra (x, y) ⇐⇒ xa = y, and Rb (x, y) ⇐⇒ xb = y. If Ra (x, y) we say that there is a directed edge xy with label a, and similarly for Rb (x, y). Then we let T0 be a locally finite tree obtained by the following operations illustrated by Figure 13.2: r v2 r x

a -

r y

−→ r x

r u1

r v1

r u

r v

r y

r v3 r v2 r x

b -

r y

−→ r x

r u1

r v1

r u

r v

r y

Figure 13.2 Coding Ra (x, y) by Ta (x, y) and Rb (x, y) by Tb (x, y). (i) For each directed edge xy with label a in K, we replace the edge by a tree Ta (x, y) with vertex set {x, y} ∪ {u, u1, v, v1 , v2 } and edge set {xu, uv, vy, uu1, vv1 , v1 v2 }. (ii) For each directed edge xy with label b in K, we replace the edge by a tree Tb (x, y) with vertex set {x, y} ∪ {u, u1 , v, v1 , v2 , v3 }

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and edge set {xu, uv, vy, uu1, vv1 , v1 v2 , v2 v3 }. Note that T0 is a tree, and in T0 every element of F2 has degree 4, whereas every other element of T0 has degree at most 3. Now we obtain T (A) by adding to T0 a new vertex x∗ for each element x of F2 and a new edge xx∗ . In T (A) a vertex v has degree ≥ 4 iff v ∈ F2 , and v has degree 5 iff v ∈ A. We claim that T : A → T (A) is a Borel reduction from E∞ to the isomorphism. First suppose A, A ⊆ F2 and g ∈ F2 such that gA = A . Consider σg : F2 → F2 defined by σg (x) = gx. Then it is easy to see that σg is an automorphism of the labeled directed tree K. It follows that σg induces an automorphism σg∗ of T0 . Moreover, σg∗ can be extended to an isomorphism of T (A) onto T (A ), since for each additional edge xx∗ in T (A), x ∈ A, and so gx ∈ A and (gx)(gx)∗ ∈ T (A ); thus we may let σg∗ (x∗ ) = (gx)∗ . Finally suppose A, A ⊆ F2 and σ : T (A) ∼ = T (A ). Then by the degree properties we have that σ(F2 ) = F2 and σ(A) = A . Now because of the difference between the constructions of (i) and (ii), as well as the asymmetry involved, we must have that σ induces an automorphism of K with σ(A) = A . Let now g = σ(1F2 ). Then for any x ∈ F2 , we must have σ(x) = gx. This is because we can write x as x11 . . . xnn , where x1 , . . . , xn ∈ {a, b} and 1 , . . . , n ∈ {+1, −1}. Then there is a directed path from 1F2 to x with labels given by the sequence of xi i . By isomorphism there is a directed path from g = σ(1F2 ) to σ(x) with the same sequence of labels. This implies that σ(x) = gx. We thus have that gA = A and the proof is complete. In the remainder of this section we consider well-founded trees on ω. Let WF be the class of all well-founded trees on ω. Then WF is a Π11 , non-Borel subset of Tr. However, for each α < ω1 , if we let WFα be the class of all well-founded trees on ω of rank ≤ α, then each WFα is a Borel subset of Tr, and hence is a standard Borel space. For each α < ω1 , let ∼ =α denote the isomorphism relation on WFα . The following theorem shows that the collection of equivalence relations is essentially the same as the Friedman– Stanley tower. These isomorphism relations are in fact the original tower of equivalence relations considered by Friedman and Stanley. Theorem 13.2.5 For each α < ω1 , ∼ =3+α is Borel bireducible with =α+ . Proof. The proof is by induction on α < ω1 . For the base case we need to show that ∼ =3 and id(2ω ) are Borel bireducible. For this note that a tree T on ω of rank ≤ 3 has depth ≤ 3, that is, for any s ∈ T , lh(s) ≤ 2. Let T1 be the set of elements in T of length 1. Then it is easy to see that the isomorphism type of T is determined by the cardinality of T1 as well as the set of number of immediate successors for each element of T1 . Thus it is determined essentially

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by a subset of ω, or an element of 2ω . Conversely, it is also easy to see that every subset of ω can be coded by a tree of depth ≤ 3 in the same sense. For the successor inductive step, without loss of generality assume T has rank 3 + α + 1. Again let T1 be the set of nodes in T of length 1. For each t ∈ T1 let Tt be the tree on ω given by s ∈ Tt ⇐⇒ t s ∈ T. If lh(t) = 1 and t ∈ T1 , let Tt be the empty tree. Then T ∼ = T  iff there is a   ∼ bijection π between T1 and T1 such that for all t ∈ T1 , Tt = Tπ(t) . Let fα be a α+ Borel reduction form ∼ =3+α to = . Define fα+1 (T ) = (fα (Tt ))lh(t)=1 . Then T ∼ = T  iff fα+1 (T )=(α+1)+ fα+1 (T  ). The converse reduction is similar. Finally, if T has rank α, where α is a limit, then for any t ∈ T1 , Tt has rank β(t) < α. By similar reductions as defined for the successor case, we obtain Borel bireducibility between ∼ =3+α and =α+ from the inductive hypotheses. Exercise 13.2.1 Let LT = {R1 , R2 }, where R1 is unary and R2 is binary. Give an LTω1 ω -sentence ϕ such that for any M ∈ Mod(ϕ), R1M is a distinguished element of M and (M, R2M ) is a tree. Exercise 13.2.2 Show that the isomorphism relation for all pruned trees on ω is Borel complete. Deduce that the class of all countable trees with no vertices of degree 1 is Borel complete. Exercise 13.2.3 Show that the isomorphism relation for countable locally finite acyclic graphs is Borel bireducible with =+ . Exercise 13.2.4 Show that reduction defined in the proof of Theorem 13.2.4 is a faithful Borel reduction. Exercise 13.2.5 Give an invariant Borel subclass of countable locally finite trees so that the isomorphism relation is Borel bireducible with E0 .

13.3

Countable linear orderings

In this section we prove the Borel completeness of the class of all countable linear orderings of ω. Let L = {<} be the language with one binary relation symbol. Let ρ be the Lωω sentence that is the conjunction of the axioms of linear orders: ∀x ¬(x < x), ∀x ∀y ( x < y ∨ x = y ∨ y < x ), ∀x ∀y ∀z [ (x < y ∧ y < z) → x < z ].

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Then every element of Mod(ρ) is a linear ordering of ω. The proof that the class of all countable linear orderings is Borel complete will have some resemblance to the proof of Theorem 13.2.1, in the sense that some base linear order will be defined first and then the linear orders coding other structures will be obtained by a uniform operation on the base linear order. Before we start the proof of the theorem let us first describe the base linear order we will use. We will be working with dense linear orders without endpoints. Of course, a standard back-and-forth argument shows that there is only one such countable order up to isomorphism. The natural linear order (Q, <) on the set of all rational numbers is such an order. If P = {Pm : m ∈ ω} is a partition of Q, we say that P is mutually dense if for any p < q ∈ Q and any m ∈ ω, there is r ∈ Pm with p < r < q. In particular, if P is a mutually dense partition of Q, then every (Pm , <) is a dense linear order without endpoints. Lemma 13.3.1 There exists a mutually dense partition of (Q, <). Proof. Enumerate Q as a nonrepeating sequence q0 , q1 , . . . . We define a function f : Q → ω so that the partition P = {f −1 (m) : m ∈ ω} is mutually dense. For any i, j, m ∈ ω, let i, j, m = 2i 3j 5m . Then ·, ·, · is an injection from ω 3 into ω. By induction on n ∈ ω we define a finite set Dn and f (q) for each q ∈ Dn so that the following properties hold: (i) qn ∈ Dn and Dn ⊆ Dn+1 for all n ∈ ω; (ii) if n = i, j, m for some i, j, m ∈ ω with i = j, then there is r ∈ Dn with f (r) = m and either qi < r < qj or qj < r < qi . For the base step of the induction let D0 = {q0 } and f (q0 ) = 0. For the inductive step n > 0 assume Dn−1 has been defined and f (q) have been defined for all q ∈ Dn−1 . If n = i, j, m for i, j, m ∈ ω and i = j, we have either qi < qj or qj < qi . In either case by the density of Q there is r ∈ Dn−1 such that qi < r < qj or qj < r < qi . Let k be the least so that qk has this property. Let Dn = Dn−1 ∪ {qn , qk }. Let f (qk ) = m. If qn ∈ Dn−1 ∪ {qk } then f (qn ) is already defined, otherwise let f (qn ) = 0. We have that (i) and (ii) are satisfied in this case. Now if n ∈ {i, j, m : i = j, m ∈ ω}, then let Dn = Dn−1 ∪ {qn }. If qn ∈ Dn−1 then f (qn ) is already defined; otherwise let f (qn ) = 0. This finishes the inductive definition.

Now by (i) we have that n Dn = ω, hence f is defined on the entire Q. To see that f has the required property, let p < q ∈ Q and m ∈ ω. For some unique i, j we have p = qi and q = qj . Also i = j. Then for n = i, j, m, by (ii) we have some r ∈ Dn with f (r) = m with p = qi < r < qj = q, as required.

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A standard back-and-forth argument shows that any two mutually dense partitions of Q are isomorphic, that is, if P1 = {Pm,1 : m ∈ ω} and P2 = {Pm,2 : m ∈ ω} are mutually dense partitions of Q, then there is an orderpreserving bijection π : Q → Q such that for any m ∈ ω and q ∈ Q, q ∈ Pm,1 iff π(q) ∈ Pm,2 . Fix a mutually dense partition P = {Pm : m ∈ ω} for Q. We define a labeled linear order Q<ω with labels in ω <ω . Q<ω will be the union of a sequence of inductively defined linear orders Qn with labels in ω ≤n so that Qn ⊆ Qn+1 for all n ∈ ω. The labeling function will be denoted λ : Q<ω → ω <ω . We also define l : Q<ω → ω by l(x) = lh(λ(x)). Thus l(x) represents the level of λ(x), and l(x) = n iff λ(x) ∈ ω n . We will say that x ∈ Q<ω is of level n if l(x) = n. To begin the inductive definition, let Q0 = (Q, <), and for every x ∈ Q0 , define λ(x) = ∅ and l(x) = 0. Suppose Qn has been defined and λ and l have been defined for elements of Qn . Let Qn+1 be the linear order obtained from adding a copy of (Q, <) to the immediate right of each element of Qn . Formally, Qn+1 = Qn × ({−∞} ∪ Q) with the lexicographic order, where ({−∞} ∪ Q, <) is an extension of (Q, <) with −∞ < q for all q ∈ Q. In this formal definition we identify each x ∈ Qn with (x, −∞) ∈ Qn+1 , thus maintaining Qn ⊆ Qn+1 . Note that the new elements of Qn+1 form the product set Qn × Q. Thus for each x ∈ Qn and q ∈ Q, we define l(x, q) = n+ 1 and λ(x, q) = λ(x) m, where m is the unique number such that q ∈ Pm ∈ P. Then λ on Qn × Q has the properties: (i) for each x ∈ Qn and q ∈ Q, λ(x, q) ⊇ λ(x) and λ(x, q) ∈ ω n+1 ; and (ii) for each x ∈ Qn , the partition {λ−1 (λ(x) m) : m ∈ ω} is mutually dense in {x} × Q. This finishes the inductive definition of Qn , and also of Q<ω . Theorem 13.3.2 (Friedman–Stanley) The class of all countable linear orderings is Borel complete. Proof. It suffices to define a Borel reduction from any Borel complete class to the class of countable linear orderings. For notational convenience we will use the class of all binary relations on ω. Note that it is Borel complete since the class of all countable graphs is a subclass. Let LR = {R} where R is a binary relation symbol. Since the language is finite, for each n ∈ ω the set a : M ∈ Mod(LR ), a ∈ ω n } Φn = {ϕM, 0 a of formulas with n free variables v0 , . . . , vn−1 is finite. Recall that ϕM, is the 0 atomic type of  a over M defined in the Scott analysis in Section 12.1. Let

Φ = n Φn . Fix a bijection c : Φ → ω so that for all ϕ, ψ ∈ Φ if ϕ ∈ Φn ,

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ψ ∈ Φm and n < m, then c(ϕ) < c(ψ). Thus the function c gives a coding of atomic types of tuples by natural numbers. For each n ∈ ω define a linear order (Bn , <) by Bn = D1 ∪ Fn ∪ D2 where D1 and D2 are dense linear orders without endpoints, Fn contains n+ 2 elements, and for any p ∈ D1 , r ∈ Fn and q ∈ D2 , p < r < q. Note that there is an Lωω -formula θn (u, v) for any n ∈ ω such that for any linear ordering N ∈ Mod(ρ) and a, b ∈ |N |, N |= θn [a, b] iff a < b and the linear order {x ∈ N : a < x < b} is isomorphic to Bn . We let ψn (u) be the formula ∃v θn (u, v) ∨ ∃v θn (v, u). We are finally ready to define for each M ∈ Mod(LR ) a countable linear order Q(M ). As before we describe Q(M ) informally and leave the exact details of defining Q(M ) as an element of Mod(L) as an exercise. Note that M,λ(x) ). each x ∈ Q<ω gives rise to a tuple λ(x), which in turn is coded by c(ϕ0 M,λ(x) ). Now Q(M ) is obtained from Q<ω In view of this define cx = c(ϕ0 by replacing each element x of Q<ω by a copy of the linear order Bcx . This finishes the definition of the map Q : M → Q(M ). The proof for the Borelness of the map Q is routine. We check that Q is a reduction. First suppose π : M → M  , where M, M  ∈ Mod(LR ). Since π is a bijection from ω onto ω, it induces an automorphism π ∗ of the tree ω <ω , where π ∗ (a1 , . . . , an ) = (π(a1 ), . . . , π(an )). Furthermore, by an easy induction π ∗ induces an automorphism π n∗ of Qn as labeled linear ∗ ∗ ∗  Qn = πn∗ . Let π<ω = n πn∗ . Then π<ω is an orders such that πn+1 ∗ ∗ automorphism of Q<ω . Now if x ∈ Q<ω then π (λ(x)) = λ(π<ω (x)) and ∗ (x) . Thus for any x ∈ Q<ω , the copy of Bc cx = cπ<ω x in Q(M ) replacing x ∗ (x). This shows is isomorphic to the copy of Bcπ∗ (x) in Q(M  ) replacing π<ω <ω  ∼ that Q(M ) = Q(M ). Conversely, assume M, M  ∈ Mod(LR ) and σ : Q(M ) ∼ = Q(M  ). Note that for any a ∈ Q(M ) there is some n ∈ ω such that Q(M ) |= ψn [a], which implies that Q(M  ) |= ψn [σ(a)]. It follows that σ induces an order-preserving bijection σ  from the copy of Q<ω in the construction of Q(M ) to the copy of Q<ω in the construction of Q(M  ). Moreover, for each x ∈ Q<ω , cx = cσ (x) , M,λ(x)

M  ,λ(σ (x))

and hence ϕ0 = ϕ0 . Also by our construction, if x, y ∈ Q<ω then λ(x) ⊆ λ(y) iff x < y and for all z with x < z < y we have l(x) < l(z). Since this property is preserved by σ  , we have that for all x, y ∈ Q<ω , λ(x) ⊆ λ(y) iff λ(σ  (x)) ⊆ λ(σ  (y)). By these and using a back-and-forth argument similar to the proof of Theorem 13.2.1 we can construct two permutations π and π  of ω such that for all n ∈ ω, M,(π(0),...,π(n))

ϕ0

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M  ,(π  (0),...,π  (n))

= ϕ0

.

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Then π  ◦ π −1 is an isomorphism from M to M  just as in the proof of Theorem 13.2.1. Gao [52] has shown that the class of countable linear orderings is not faithfully Borel complete. Rubin has proved the Vaught conjecture for countable linear orderings (see Reference [150]). Steel [150] has considered a concept of generalized trees defined as a partial order < such that the initial segment below any element is linearly ordered, that is, satisfying the formula ∀x ∀y ∀z[ (y < x ∧ z < x) → (y < z ∨ y = z ∨ z < y) ]. This apparently defines a larger class than both that of countable trees and of countable linear orderings. He proved the topological Vaught conjecture for this class. Exercise 13.3.1 Show that any two mutually dense partitions of Q are isomorphic. Exercise 13.3.2 Let (Bn , <) be the linear order defined in the proof of Theorem 13.3.2. Let P = {Pm : m ∈ ω} be a mutually dense partition of Q. For each A ⊆ ω let Q(A) be obtained from Q by replacing each element of Pm by a copy of Bm iff m ∈ A. Show that A = A iff Q(A) ∼ = Q(A ). Exercise 13.3.3 Give an alternative proof of Theorem 13.3.2 by defining a linear order Q(T ) for each countable tree T on ω so that T ∼ = = T  iff Q(T ) ∼  Q(T ).

13.4

Countable groups

Countable groups already played an important role in invariant descriptive set theory because of their connections to countable Borel equivalence relations (see Chapter 7). In this section we study the isomorphism relation for various classes of countable groups. Apparently there is a vast subfield of group theory which concentrates exclusively on the classification of countable groups. Moreover, there has already been a lot of interactions between model theory and group theory. Thus it is both unnecessary and impossible to develop the entire subject in this short section. Instead we will focus on several group theoretic results with interaction with descriptive set theory. In fact, the results we present here have become standard knowledge in countable model theory. Most of their proofs are lengthy; we will omit them since they can be found in standard group theory textbooks. We need to interpret various classes of countable groups as invariant Borel classes for a relational language. Of course, the standard language of group

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theory involves a multiplication operation (and very often an inverse operation as well). But as remarked before, the multiplication can be represented by a 3-ary relation P (x, y, z) with the intended meaning x · y = z. Thus we fix a language LG = {P } with a 3-ary relation symbol P . The axioms of groups can then be expressed by an LG ωω -sentence ξ. The class of countable groups is therefore Mod(ξ). In our discussions below we will freely use conventional notation and terminology of group theory; for instance we refer to multiplication and inverse without bothering with the relation P . The following theorem of Mekler [120] confirms the common wisdom that the isomorphism for countable groups is complicated. Theorem 13.4.1 (Mekler) The class of all countable groups is faithfully Borel complete. The proof is complicated, and we omit it here. However, we remark that in the proof Mekler constructed countable nilpotent groups of rank 2 to code the isomorphism types of countable graphs. Moreover, the coding is in fact a two-way interpretation in the sense that properties of the graphs correspond to properties of the groups (see Reference [84] Sections 5.5 and A.3). As we remarked before, this is stronger than the faithfulness of the Borel reduction and has other model theoretic consequences. Now recall that a group G is rank 2 nilpotent iff G/Z(G) is abelian, where Z(G) = {x ∈ G : ∀y ∈ G x · y = y · x} is the center of G. Thus rank 2 nilpotent groups are characterized by the following sentence, ∀g, h, x [ ζ(x) → ∃y ( ζ(y) ∧ g · h · x = h · g · y ) ], where ζ(x) is the formula characterizing the center, that is, Z(G) = {x ∈ G : ζ(x)}. It follows that the class of all countable nilpotent groups of rank 2 is invariant Borel. Theorem 13.4.2 (Mekler) The class of all countable nilpotent groups of rank 2 is faithfully Borel complete. The class of all countable abelian groups is obviously an invariant Borel class. However, it is an open problem if it is Borel complete. In the rest of this section we discuss countable abelian groups, and in doing this we follow the convention of group theory to denote the group additive rather than multiplicative. Note that every countable abelian group G can be uniquely expressed as the direct sum of a torsion abelian group (its torsion part) and a torsion-free abelian group G∼ = GT ⊕ GF . Recall that an abelian group A is torsion-free if for all g ∈ A, g = 0 = 1A , and integer n > 0, ng = 0. For an abelian group G the torsion subgroup GT

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of G is defined by GT = {g ∈ G : ng = 0 for some integer n > 0}. An abelian group A is torsion if AT = A. If p is a prime number, we may define further the p-torsion subgroup GT,p of G by GT,p = {g ∈ G : pn g = 0 for some integer n > 0}. An abelian group A is p-torsion if A = AT,p . We have in general GT =

/

GT,p .

p

The definitions make it clear that the subclasses of countable abelian groups consisting of p-torsion, torsion, or torsion-free groups respectively are all invariant Borel classes. Thus it makes sense to think of their isomorphism relations as S∞ -orbit equivalence relations. Ulm gave a complete classification of countable torsion abelian groups up to isomorphism. His classification motivated the concept of Ulm classifiability, introduced in Section 9.2. We describe the original Ulm invariants below. Let p be a prime and A a countable abelian group. For α < ω1 define a subgroup pα A by induction as follows: p0 A = A, pα+1 A = p(pα A) = {pg : g ∈ pα A}, pλ A =

α<λ

pα A, if λ is a limit.

It is easy to see by induction that each pα A is a subgroup and pα A ⊆ pβ A if α ≥ β. Since A is countable there exists α0 < ω1 such that pα0 A = pα0 +1 A. The least such α0 is called the p-length of A, and we denote it by lp (A). Now suppose A is p-torsion. Let A[p] = {g ∈ A : pg = 0}. Then it is easy to see that / A[p] ∼ (Z/pZ) = i∈I

for an index set I that is at most countable. The cardinality of I is called the dimension of 0 A[p]. Note that (pA)[p] = A[p] ∩ pA and there is I  ⊆ I such ∼ that (pA)[p] = i∈I  (Z/pZ). Thus it makes sense to speak of the dimensions for both (pA)[p] and A[p]/(pA)[p], to be respectively the cardinalities of I  and I − I  . Now for each α < ω1 , let fα (A) be the dimension of (pα A)[p]/(pα+1 A)[p]; fα (A) is called the α-th Ulm invariant of A. Note that fα (A) = 0 for all α ≥ α0 . In general, fα (A) is an element of ω ∪ {∞}.

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Theorem 13.4.3 (Ulm) Let p be a prime and G, H be countable p-torsion abelian groups. Then G∼ = H iff for all α < ω1 , fα (G) = fα (H). For general torsion abelian groups A we obtain a transfinite sequence of Ulm invariants fαp (A) for each prime p, and the mixed sequence (fαp (A))p,α is a complete invariant for the group A. In Section 9.2 we defined the abstract Ulm invariants to be elements of the space 2<ω1 . This can be seen to be equivalent to the original Ulm invariants via a Δ12 isomorphism. Moreover, the Ulm classification of countable p-torsion abelian groups provides a proof of Δ12 bireducibility between the isomorphism relation and the equality of Ulm invariants. By metamathematical methods it can be shown that this isomorphism relation is nonsmooth but E0 is not Borel reducible to it. Thus the Glimm–Effros dichotomy fails for it. For the remainder of this section we turn to countable torsion-free abelian groups. Around the same time Ulm classified countable torsion abelian groups, Baer investigated torsion-free abelian groups in a somewhat similar manner. Let A be a countable torsion-free abelian group and p a prime. For each g ∈ A, the p-height of g is defined by  sup{n ∈ ω : g ∈ pn A − pn+1 A}, if g ∈ pω A, hp (g) = ∞, otherwise. Let P be the set of all prime numbers. Let h(g) = (hp (g))p∈P . Then h : A → (ω ∪ {∞})P . Call two elements g, h ∈ A linearly dependent if there are integers m, n = 0 such that mg + nh = 0. Lemma 13.4.4 Let A be a countable torsion-free abelian group and g, h ∈ A are linearly dependent. Then (i) for all p ∈ P , hp (g) = ∞ iff hp (h) = ∞, and (ii) for all but finitely many p ∈ P , hp (g) = hp (h). Proof. Suppose mg + nh = 0 for integers m, n = 0. Since A is torsion-free we may assume (m, n) = 1. Suppose hp (g) = ∞, that is, g ∈ pω A. It follows that mg ∈ pω A and hence nh ∈ pω A. Now assume n = pi q with (q, p) = 1. Then for any k > 0, we have that nh ∈ pk+i A, that is, there is x ∈ A with pi qh = pk+i x. Again by torsion-freeness we obtain that qh = pk x. Since (pk , q) = 1 there are integers r, s with rpk + qs = 1. Let y = sx + rh. Then pk y = pk sx + pk rh = qsh + rpk h = h. This shows that h ∈ pk A for all integers k > 0. Hence h ∈ pω A and hp (h) = ∞. For (ii) let p be a prime with (p, m) = (p, n) = 1. Suppose hp (g) = k, that

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is, g ∈ pk A − pk+1 A. Then for some x ∈ A, g = pk x and so nh = pk (−mx). If k = 0 then h ∈ pk A. If k > 0 then a similar argument as above shows that h ∈ pk A as well. By symmetry we have both (i) and (ii). In view of the above lemma, define an equivalence relation ∼ on (ω ∪{∞})P by f = (fp )p∈P ∼ f  = (fp )p∈P iff for all p ∈ P , fp = ∞ iff fp = ∞, and for all but finitely many p ∈ P , fp = fp . Lemma 13.4.5 Let A be a countable torsion-free abelian group and x ∈ A. If f ∈ (ω ∪ {∞})P is such that f ∼ h(x), then there is y ∈ A such that f = h(y). Proof. Let F = {p ∈ P : fp = hp (x)}. Then F is finite and for all p ∈ F , fp , hp (x) < ∞. If F = ∅ there is nothing to prove. Otherwise enumerate F as p1 < · · · < pn . Let ni = hpi (x) for 1 ≤ i ≤ n. We define x1 , . . . , xn ∈ A successively as follows. If n1 = 0 then let x1 = x. Otherwise, x ∈ pn1 A, so there is x1 with x = pn1 x1 . By the argument in the preceding proof we have that hp1 (x1 ) = 0 and hp (x1 ) = hp (x) for all p = p1 . In a similar manner successively define x2 , . . . , xn such that for each 2 ≤ i ≤ n, hpi (xi ) = 0 and hpi (xi ) = hp (xi−1 ) for all p = pi . It follows that hpi (xn ) = 0 for all 1 ≤ i ≤ n and hp (xn ) = hp (x) for all p ∈ F . Now let y = pf11 . . . pfnn xn . Then it is easy to check that hpi (y) = fi for all 1 ≤ i ≤ n and hp (y) = hp (x) for all p ∈ F . Thus y is as required. A countable torsion-free group A is of rank 1 if every two nonzero elements of A are linearly dependent. It is easy to see that every subgroup of (Q, +) is a torsion-free group of rank 1. The following theorem of Baer completely classifies countable torsion-free groups of rank 1. Theorem 13.4.6 (Baer) If G, G be countable torsion-free abelian groups of rank 1, then G ∼ = G iff    there is g ∈ G and g ∈ G such that h(g) ∼ h(g ). Proof. The implication (⇒) is obvious. For (⇐) let g ∈ G and g  ∈ G be such that h(g) ∼ h(g  ). Let N = {p ∈ P : hp (g) = ∞}. Then N = {p ∈ P : hp (g  ) = ∞}. We define an isomorphism π from G onto G with π(g) = g  . For this consider an arbitrary element h of G. Let (m, n) = 1 be integers such that mh + ng = 0. Write m = mF mN where (mF , mN ) = 1, mF contains no prime factor in N , and mN contains only prime factors in N , unless mN = 1. Similarly write n = nF nN . It is easy to see that hp (nN g) = hp (g) and hp (mN h) = hp (h) for all p ∈ P . By Lemma 13.4.4 h(mN h) ∼ h(nN g). We have h(nN g  ) = h(g  ) ∼ h(mN h). By the proof of Lemma 13.4.5 there is x ∈ G such that mF x+nF (nN g  ) = 0. Such x is unique. Since hp (x) ∼ hp (h) for all p ∈ P , there is a unique h such that x = nN h . We thus obtain a unique

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h such that mh + ng  = 0. Define π(h) = h . By symmetry π is a bijection. It is now easy to check that π is an isomorphism. Corollary 13.4.7 The isomorphism for countable torsion-free abelian groups of rank 1 is Borel bireducible with E0 . Proof. Given an arbitrary countable torsion-free abelian group G we let g0 ∈ G be a canonically distinguished element (for instance when G has underlying set ω we may let g0 = 0). The let NG = {p ∈ P : hp (g) = ∞} and FG = P − NG . Define hG : FG → ω by letting hG (p) = hp (g0 ) for all p ∈ FG . By Theorem 13.4.6 and Lemmas 13.4.4 and 13.4.5, we have that G∼ = G iff NG = NG and for all but finitely many p ∈ FG , hG (p) = hG (p). The function G → (NG , hG ) is easily seen to be Borel, and is a reduction from the isomorphism relation to E = id(2ω ) × E0 . But E is hyperfinite, hence E ≤B E0 . Conversely we show that E0 on ω ω is Borel reducible to the isomorphism of subgroups of Q. Let P be enumerated as p0 , p1 , . . . . For x ∈ ω ω , let Gx be the subgroup of Q generated by −x(i)

{1, pi

: i ∈ ω}.

Then Gx is torsion-free and hpi (1) = x(i) by our construction. Now for x, y ∈ ω, xE0 y iff hGx (1) ∼ hGy (1). It follows from Theorem 13.4.6 and Lemmas 13.4.4 and 13.4.5 that xE0 y iff Gx ∼ = Gy . In general elements g1 , . . . , gn in a torsion-free abelian group G are said to be linearly independent if for any integers m1 , . . . , mn , if m1 g1 +· · ·+mn gn = 0 then m1 = · · · = mn = 0. A torsion-free abelian group G is of rank n if there is a linearly independent set with n elements but there are no linearly independent sets with n + 1 elements. If G is of rank n for some 1 ≤ n < ∞, then G is said to be of finite rank. It is not hard to see that any torsionfree abelian group of rank ≤ n is isomorphic to a subgroup of Qn . Thus the standard Borel space S(Qn ) of all subgroups of Qn , as a Borel subspace of n the Polish space 2Q is another representation of the invariant Borel class of countable torsion-free abelian groups of rank ≤ n. Lemma 13.4.8 For each integer n ≥ 1, the isomorphism relation for countable torsion-free abelian groups of rank ≤ n is essentially countable. Proof. It suffices to show that the isomorphism relation on S(Qn ) is countable. For this fix a countable subgroup G of Qn . We verify that there are only countably many subgroups of Qn isomorphic to G. Let g1 , . . . , gm be a linearly

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independent set in G, where m ≤ n is the rank of G. Then every element of G is uniquely expressed as a rational linear combination of g1 , . . . , gm . It follows that if G ∈ S(Qn ) is isomorphic to G, then there are linearly independent  ∈ G such that for all rationals q1 , . . . , qm ∈ Q, g1 , . . . , gm  q1 g1 + · · · + qm gm ∈ G ⇐⇒ q1 g1 + · · · + qm gm ∈ G .

In this sense the group G is completely determined by the set of linearly  . Since there are only countably many such independent elements g1 , . . . , gm sets, the number of subgroups of Qn isomorphic to G is countable. Using deep results in Zimmer’s superrigidity theory, Thomas [153] has shown that the isomorphism relation for countable torsion-free abelian groups of rank ≤ n form a strictly increasing chain of essentially countable equivalence relation in the Borel reducibility hierarchy. Such results are significant not only for group theory but also for descriptive set theory of equivalence relations. Countable torsion-free abelian groups that are not of finite rank are said to have infinite rank. It is an open problem whether the class of all countable torsion-free abelian groups of infinite rank is Borel complete. Hjorth [71] has shown that all equivalence relations in the Friedman–Stanley tower are Borel reducible to the isomorphism relation of this class. Exercise 13.4.1 Show that for any n ≥ 2 the class of all countable nilpotent groups of rank n is an invariant Borel class. Deduce that the class of all countable nilpotent groups is an invariant Borel class. Exercise 13.4.2 Show that for any n ≥ 2 the class of all countable solvable groups of rank n is an invariant Borel class. Deduce that the class of all countable solvable groups is an invariant Borel class. Exercise 13.4.3 Show that each countable torsion-free abelian group of rank n is isomorphic to a subgroup of Qn . Exercise 13.4.4 Let GL(Qn ) be the group of all n×n matrices with nonzero determinant. Show that two subgroups G, H of Qn iff there is μ ∈ GL(Qn ) such that μG = H.

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Part IV

Applications to Classification Problems

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Chapter 14 Classification by Example: Polish Metric Spaces

In this final part of the book we focus on applications of the theory of equivalence relations to classification problems in mathematics. In these applications classification problems in mathematics are first identified as equivalence relations on standard Borel spaces. Then the exact complexity of each equivalence relation is determined by a comparison to some benchmark equivalence relation in the Borel reducibility hierarchy. In fact in Chapter 13 we already discussed isomorphic classification problems for invariant Borel classes of countable structures. Precise measurement of the complexity of some classification problems has been achieved, but in the process knowledge is needed from both descriptive set theory and the particular field (combinatorics and algebra, to be more specific). In this chapter we continue to consider classification problems for uncountable mathematical structures, thus expanding the applicability of invariant descriptive set theory to other mathematical fields (such as topology, geometry, and analysis). As such classification/nonclassification results are numerous and are still mushrooming as we speak, it is impossible to give a comprehensive account of applications in many directions. To make this book more useful, we will instead focus on the fundamentals, or the framework of these applications. In this chapter we will concentrate on one subject, the classification of Polish metric spaces, and illustrate how results about equivalence relations are relevant to the classification problems. When the reader attempts to consider classification problems for structures of a different nature, he will find the framework flexible enough to be used again.

14.1

Standard Borel structures on hyperspaces

When considering isomorphic classification of a family of countable structures, we have been setting up the problem by identifying the family as an invariant Borel class. Thus the isomorphism relation for the class of structures becomes an equivalence relation on a standard Borel space. And then

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it becomes mathematically sound to apply the Borel reducibility notion and results about the Borel reducibility hierarchy. If the structures considered are uncountable, then in theory the above method is no longer valid and the applicability of invariant descriptive set theory is limited. However, in practice, a large number of classes of mathematical structures can be turned into standard Borel spaces. Examples of such classes are ubiquitous in topology, geometry, and analysis. In this section we take only the example of Polish metric spaces, that is, separable complete metric spaces. Let X = the hyperspace of all Polish metric spaces. We show that, even if in general Polish metric spaces are uncountable, X can be turned into a standard Borel space by a suitable coding. In fact we give two coding methods to do this, and prove that they are equivalent. The first method, used first by Vershik [163], is based on the following simple observation. Every Polish metric space contains a countable dense subset, and the metric structure on the whole space is completely determined by and can be fully recovered from the metric structure on any dense subset by the operation of completion. Thus a Polish metric space can be coded by a canonical enumeration of a canonical countable dense subset. In practice, when a space is presented to us a canonical countable dense subset together with a canonical enumeration is often immediately recognizable. For instance, if the space R is presented to us we immediately recognize a canonical countable dense subset Q and a canonical enumeration of its elements. In view of this we define X to be the subspace of Rω×ω consisting of elements (ri,j )i,j∈ω such that (1) for all i, j ∈ ω, ri,j ≥ 0, and ri,j = 0 iff i = j; (2) for all i, j ∈ ω, ri,j = rj,i ; (3) for all i, j, k ∈ ω, ri,j ≤ ri,k + rk,j . Since Rω×ω is a Polish space with the product topology, and X is a Gδ subset of Rω×ω , X is Polish. Now if a Polish metric space (X, d) is given, as we required, together with an enumeration (xi )i∈ω of a canonical countable dense set DX , we let rX = (ri,j )i,j∈ω = (d(i, j))i,j∈ω . Note that rX depends also on the sequence (xi ), but as our assumption goes it is a part of the presentation of the space X, and hence we suppress mentioning (xi ) for notational simplicity. Conversely, for any r ∈ X we define metric spaces Dr = (ω, dω ) with dω (i, j) = ri,j and Xr to be the completion of Dr . It is clear that rX contains the full information of the metric structure on X, since the map ϕ : ω → D with ϕ(i) = xi is an isometry and is uniquely extended to an isometry ϕ∗ from Xr onto X.

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In this sense we have established a correspondence between elements of X with those of X, and this correspondence induces a standard Borel structure (in fact a Polish topological structure) on X . We now turn to the second method of endowing X with a standard Borel structure following the approach of Gao and Kechris [58]. This method is based on the following more sophisticated observations. There is a universal Polish metric space, for instance, the universal Urysohn space U, in which every Polish metric space is isometrically embedded into (see Section 1.2); the Effros Borel space F (U) is standard Borel (see Section 1.4). Thus in an informal sense we may let elements of F (U) represent the elements of X . However, to pin down the standard Borel structure on X exactly, and to allow a proof that this Borel structure is equivalent to the one defined via X, we need to well define the correspondence between X and F (U). This turns out to be highly nontrivial, as the following discussion shows. In the following we need the reader to be familiar with Section 1.2. Let a Polish metric space (X, d) be given, together with an enumeration (xi )i∈ω of a canonical countable dense set DX . We need to associate with X a particular closed subset FX of U. Recall from Section 1.2 that for each separable metric space (Y, δ), a sequence (Yn , δn )n≤ω of consecutive extensions of (Y, δ) is defined: (Y0 , δ0 ) = (Y, δ), (Yn+1 , δn+1 ) = E(Yn , ω), (Yω , δω ) =

n∈ω (Yn , δn ),

with the completion of (Yω , δω ) isometric to U. In particular, U is the completion of (Rω , dω ) where d is the usual metric on R. Thus given X above we may follow the constructions of (Xn )n≤ω to obtain an isometric embedding from X into U. We claim that every step of the construction can be carried out canonically given the enumeration of the countable dense set DX . To see this, we need to verify that a canonical countable dense subset can be obtained for each step of the construction, and that a canonical isometry between Xω and U can be obtained with a canonical countable dense subset of Xω . For the first part, it suffices to note that if a separable metric space (Y, δ) and a countable dense subset D are given, then the set E(Y, D, ω) =def {f ∈ E(Y ) : the support of f is a finite subset of D } is a countable dense subset of E(Y, ω). Moreover under the canonical isometric embedding x → fx from Y to E(Y, ω), D is a subset of E(Y, D, ω). Thus E(Y, D, ω) is a canonical countable dense subset of E(Y, ω). Since every element of E(Y, D, ω) is completely determined by its support, we also obtain a canonical enumeration of E(Y, D, ω) by canonically enumerating all finite sequences in D. Applying this observation for all Xn , we obtain a canonical

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enumeration of a canonical countable dense subset of Xω , which is also dense in the completion of Xω . In the same fashion, we also obtain a canonical enumeration of a countable dense subset of Rω that is also dense in U. For the second part of the claim, that a canonical isometry between the completion of Xω and U can be obtained from canonical countable dense subsets of Xω and Rω respectively, simply note that the proof of Theorem 1.2.5 relies only on countable dense subsets of two complete spaces with the Urysohn property. Therefore we have associated a closed subset of U to every element of X presented to us. We note two properties of this associated map, that it is one-to-one, and that it is not onto F (U). To see that it is one-to-one, note that if two distinct elements X and X  of X are given, then the two canonical enumerations (xi ) and (xi ) are distinct, and there are i, j ∈ ω such that d(xi , xj ) = d(xi , xj ). By our construction above, the canonical countable dense subsets of Xω and Xω contain the given ones respectively, hence Xω and Xω continue to be distinct elements of X . Finally the proof of Theorem 1.2.5 is carried out such that if the two countable dense subsets of Xω and Xω respectively are distinct then so are the resulting isometries between the respective spaces and U. To see that the associated map is not onto, note that Xω contains X as a proper subset, and therefore U itself is not in the image. At this moment we are still short of a satisfactory correspondence between X and F (U). But for the record we have given all the essential ingredients for a proof of the following theorem. Theorem 14.1.1 There is a Borel embedding J from X into F (U) such that for any r ∈ X, Xr is isometric with J(r). Proof. From the above discussions we obtain the following two Borel functions. Corresponding to the process of obtaining Xω from X we have a function E : X → X such that for any r ∈ X, DE(r) is the canonical countable dense subset of (Xr )ω . In fact there is a fixed injection e : ω → ω such that for any r ∈ X and i, j ∈ ω, ri,j = E(r)e(i),e(j) . This extension function E is clearly Borel since each of its entries is defined from the entries of r using finitely many supremum operations (as it is a value of some dn in Xn ). Corresponding to the next step of the construction, namely obtaining an isometry between the completion of Xω and U by the proof of Theorem 1.2.5, we have a function I : X → Uω such that for any r ∈ X, if Dr has the Urysohn property, then I(r) enumerates a countable dense subset of Rω . Here I(r)(n) is in fact the element of U corresponding to the n-th element in Dr under the resulting isometry. Since this function is obtained by following the backand-forth argument in the proof of Theorem 1.2.5, we have that for any n, I(r)(n) is the limit of a sequence (xm ), where each xm is an element of the

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canonical countable dense subset of Rω . Furthermore the requirement that each xm satisfies is a Borel condition involving the entries of r. It follows that each I(r)(n) is a Borel function from X to U, and hence I is Borel. Now the embedding J : X → F (U) is defined, for each r ∈ X, by J(r) = the closure of {IE(r)(n) : n ∈ e(ω)} in U. It only remains to check that J is Borel. For this let U ⊆ U be an open set. We have r ∈ J −1 ({F ∈ F (U) : F ∩ U = ∅}) ⇐⇒ J(r) ∩ U = ∅ ⇐⇒ ∃n ∈ e(ω) IE(r)(n) ∈ U, which is Borel since both I and E are Borel. It follows from the above theorem that J(X) is a Borel subset of F (U) and therefore a standard Borel space. Also from the definition of J and earlier discussions we may correspond the elements of X with those of J(X) and thus obtain a standard Borel structure on X . This standard Borel structure is certainly equivalent to the one induced by X by the above theorem. Now the following theorems finally establish a satisfactory correspondence between X and F (U). Theorem 14.1.2 There is a Borel embedding j from F (U) into J(X) such that for any F ∈ F (U), F is isometric with j(F ). Proof. We let X = U be given with the canonical countable dense subset of Rω , and obtain r ∈ X to code X. By the constructions of Theorem 14.1.1, we obtain the extension Xω and an isometry between the completion of Xω with U again. Let j(U) be the image of X in U following the construction. In symbols, j(U) = J(r). For each F ∈ F (U), the construction yields a closed subset of J(r) in a natural sense, and we let j(F ) be this closed subset of J(r). It is clear that the map j is a Borel embedding. By definition, j(F ) is isometric with F for any F ∈ F (U). Theorem 14.1.3 There is a Borel isomorphism Θ between J(X) and F (U) such that for any r ∈ X, Xr is isometric with Θ(r). Proof. Let j be the Borel embedding in Theorem 14.1.2. The identity map i from J(X) into F (U) obviously satisfies that i(F ) is isomorphic to F for all F ∈ J(X). By the usual proof of the Cantor–Bernstein Theorem 1.3.3, a

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Borel isomorphism Θ can be obtained from the Borel embeddings i and j so that the isometry types of the elements are preserved. Now the correspondence between X and J(X), composed with the Borel isomorphism Θ, finally gives a correspondence between X and F (U). Of course it follows immediately from Theorem 14.1.3 that Θ ◦ J is a Borel isomorphism between X and F (U) preserving the isometry types of elements. The existence of such an isomorphism means that the two approaches to equip X with a standard Borel structure are equivalent. Note that it is mathematically nontrivial to prove this equivalence. However, in practice, whenever we have different approaches to equip standard Borel structures on hyperspaces such as X , they always end up to be equivalent despite the difficulty of the proof. We summarize this phenomenon and generalize it philosophically in the following statement: For any hyperspace H of mathematical structures, if B1 and B2 are two natural standard Borel structures on H, then there is a Borel isomorphism Ψ between (H, B1 ) and (H, B2 ) such that for any space X in H, Ψ(X) and X are isomorphic as mathematical structures. This is apparently a philosophical statement rather than a mathematical one, since there is no mathematical way to define naturalness of a standard Borel structure. For readers familiar with computability theory, the statement is similar to the Church–Turing Thesis, which states that any two natural ways to define computability are equivalent. Such statements can be refuted mathematically, that is, if two different definitions are given and are proven inequivalent, and it is accepted that both definitions are natural. While there is no way to confirm such a statement, there might be mathematical theorems which can attest to its plausibility. One reason that the Church–Turing Thesis is widely accepted now is the practical need to reduce the work done and presented to establish its concrete instances again and again. To a large extent such work is mathematically insignificant and irrelevant to the understanding of the main problem. For our purpose the objective is to understand the complexity of various classification problems for, say, Polish metric spaces. While the theorems in this section are nontrivial and their proofs contain important ideas that can be used later to tackle the classification problems, the statements of the theorems themselves are not helping in our understanding of the complexity of the classification problems. Exercise 14.1.1 Let Xcpt be the set of all r ∈ X such that Xr is compact. Show that Xcpt is a Borel subset of X, and therefore a standard Borel space. Exercise 14.1.2 Let K(U) be the set of all F ∈ F (U) such that F is compact. Show that K(U) is a Borel subset of F (U), and therefore a standard Borel space.

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The next three exercise problems discuss a natural map from F (U) into X preserving isometry types. Exercise 14.1.3 Let X be a Polish space and {Un }n∈ω a countable base for X. Show that for each n ∈ ω, the map F → F ∩ Un from F (X) to F (X) is Borel. (Note that in general the intersection operation on F (X) is not Borel.) Exercise 14.1.4 Let {Un }n∈ω be a countable base for U. Let s : F ∗ (U) → U be a selection function given by Theorem 1.4.6, where F ∗ (U) = {F ∈ F (U) : F = ∅}. For each n ∈ ω define pn : F (U → U by  s(F ∩ Un ), if F ∩ Un = ∅, pn (F ) = s(F ), otherwise. Show that each pn is a Borel function and for any F ∈ F ∗ (U), {pn (F )}n∈ω is dense in F . Exercise 14.1.5 Let F ∞ (U) = {F ∈ F (U) : F is infinite}. (a) Show that F ∞ (U) is a Borel subset of F (U). (b) Using Exercise 14.1.4 define a Borel map k : F ∞ (U) → X such that for any F ∈ F ∞ (U), Xk(F ) is isometric with F . (c) Show that k is not an embedding.

14.2

Classification versus nonclassification

In mathematics, a classification problem is associated with a hyperspace of mathematical structures and a notion of equivalence. As we did in the preceding section, if the hyperspace can be turned into a standard Borel space X, then the equivalence notion becomes an equivalence relation E on the standard Borel space X. In our example, the space X of all Polish metric spaces have been equipped with a standard Borel structure. The isometric classification of Polish metric spaces is thus an equivalence relation on X , which we denote by ∼ =i . In this section we discuss classification and nonclassification results in mathematics. We first give an example of a complete and satisfactory solution to a classification problem. The problem is the isometric classification of compact metric spaces. Let Xcpt be the hyperspace of all compact metric spaces. Since Xcpt is a subspace of X , to see that it is a standard Borel space it suffices to show that it is a Borel subset of X , which was done in Exercises 14.1.1 and 14.1.2 for both approaches to equip standard Borel structures on X . In particular,

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recall that for any Polish space X, the space K(X) of all compact subsets of X form a standard Borel space with the Effros Borel structure. We will use this fact in the proof of the following theorem. Theorem 14.2.1 (Gromov) The isometric classification problem for compact metric spaces is smooth.

Proof. Given a compact metric space X, for any n ∈ ω we define a map 2 Mn : X n+1 → R(n+1) by Mn (x0 , . . . , xn ) = (d(xi , xj ))0≤i,j≤n . 2

Then Mn is a continuous function from X n+1 into R(n+1) . Since X is com2 pact, so is X n+1 and its image Dn (X n+1 ) as a subset of R(n+1) . We let M (X) be the sequence (Mn (X n+1 ))n∈ω . Then M is a map from Xcpt into the product space  2 Y = K(R(n+1) ). n∈ω

We note that M is a Borel map. To see this, we use the correspondence between X and X. For any X ∈ Xcpt let DX , a canonical countable dense subset of X, and an enumeration (xi ) of DX be given. Then for any n ∈ ω, n+1 n+1 is dense in X n+1 , and therefore Mn (X n+1 ) is the closure of Mn (DX ) DX 2 2 (n+1) (n+1) . Now for any open U ⊆ R , in R n+1 ) ∩ U = ∅ Mn (X n+1 ) ∩ U = ∅ ⇐⇒ Mn (DX

⇐⇒ ∃i0 , . . . , in ∈ ω Mn (xi0 , . . . , xin ) ∈ U. This shows that M is Borel as a function from Xcpt into Y . To prove the theorem we show that for any X, X  ∈ Xcpt , X ∼ =i X  iff  M (X) = M (X ). By the definition of M it is clear that X ∼ =i X  implies    M (X) = M (X ). For the converse, assume (X, d), (X , d ) are compact metric spaces with M (X) = M (X  ). We first define an isometric embedding ϕ from X into X  . Let D ⊆ X be countable dense and (xi )i∈ω be an enumeration of D. For each n ∈ ω, n+1 Mn (x0 , . . . , xn ) ∈ Mn (X n+1 ) = Mn (X  ), hence there are y0n , . . . , ynn ∈ X  such that Mn (x0 , . . . , xn ) = Mn (y0n , . . . , ynn ). Now consider the sequence (y0n )n∈ω . Since X  is compact there is a convergent subsequence of (y0n ); this means that there is a subset A0 ⊆ ω and x0 ∈ X  such that limn∈A0 y0n = x0 . Without loss of generality we may assume 0 ∈ A0 . By similar arguments we obtain an infinite sequence A0  A1  A2  . . .

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of subsets of ω and x0 , x1 , . . . of elements of X  such that inf Am ∈ Am+1 n and limn∈Am ym = xm for all m ∈ ω. Let A = {inf Am : m ∈ ω}. Then n n A − Am is finite for each m ∈ ω, and so limn∈A ym = limn∈Am ym = xm .    We claim that d (xi , xj ) = d(xi , xj ) for all i, j ∈ ω. For this fix i, j ∈ ω and let m = max{i, j}. Note that for any n ≥ m, by Mn (x0 , . . . , xn ) = Mn (y0n , . . . , ynn ), we have that d(xi , xj ) = d (yin , yjn ). By continuity we get that d(xi , xj ) = limn∈A d (yin , yjn ) = d (xi , xj ). Thus we may define ϕ(xi ) = xi . Then ϕ is an isometric embedding from D into X  , which uniquely extends to an isometric embedding from X into X  , still denoted by ϕ. By symmetry we can also define an isometric embedding ψ : X  → X. Thus ψ ◦ ϕ : X → X is an isometric embedding of X into itself. It follows that ψ ◦ ϕ must be onto. To see this, let θ = ψ ◦ ϕ and assume x ∈ X − θ(X). Let r = d(x, θ(X)). Since θ(X) is compact, r > 0. Now consider the sequence x, θ(x), θ2 (x), . . . . Note that d(x, θn (x)) ≥ r for all n ≥ 1, and by isometry d(θm (x), θn (x)) ≥ r for all n > m. This implies that the sequence does not contain any Cauchy subsequences. But being a sequence in a compact space it contains a convergent subsequence, and therefore a Cauchy one, a contradiction. We thus have that ψ ◦ ϕ is onto, which implies that both ψ and ϕ are onto. Hence ϕ is an isometry from X onto X  , as required. It is trivial to see that there are perfectly many pairwise nonisometric compact metric spaces (even with two points). Thus as an equivalence relation ∼ =i for compact metric spaces is Borel bireducible with id(2ω ). In the intuitive sense a complete classification only requires the determination of complete invariants for the objects in question, and therefore corresponds to a Borel reduction to a known equivalence relation. A satisfactory classification, on the other hand, should leave no room for significant improvement, and therefore can be interpreted as a natural equivalence relation that is Borel bireducible with the classification problem. When both these are achieved, we refer to the result as a classification theorem. Next we turn to the full isometric classification problem for all Polish metric spaces. Before we obtain a classification theorem for this problem we first show that it is essentially more complex than the isometric classification of compact metric spaces. Thus not only the Gromov invariants for compact metric spaces are no longer complete invariants for general Polish metric spaces, but also there is no way to expand the Gromov invariants to such complete invariants. It has to be that the nature of the complete invariants for general Polish metric spaces is different than that of the Gromov invariants. We therefore refer to a result such as the following one as a nonclassification theorem. Theorem 14.2.2 The graph isomorphism is Borel reducible to ∼ =i . In particular, the isometric classification of Polish metric spaces is not smooth.

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Proof. By Theorem 13.2.1 the graph isomorphism is Borel bireducible with the isomorphism relation for countable connected graphs. Thus it suffices to define a Borel reduction from the latter to ∼ =i . For any countable connected graph Γ, we define a metric space S(Γ) on Γ with the metric given by the geodesic distance: dS (x, y) = the length of the shortest path from x to y. S(Γ) is countable and discrete, hence it is Polish and dS is complete. Γ can be recovered from S(Γ) since xy ∈ Γ iff dS (x, y) = 1. It is easy to see that Γ → S(Γ) is a Borel reduction from ∼ = to ∼ =i . With the background of theory of equivalence relations, this simple theorem now has far-reaching consequences. For instance, it implies that ∼ =i is nonBorel and is above the Friedman–Stanley tower. It now becomes a simple abstract argument to see that, for instance, a countable set of compact metric spaces can not be coded by a simple compact metric space, but can be coded by a Polish metric space. Invariant descriptive set theory makes it possible to prove nonclassification theorems by proving reductions. In the usual practice of mathematics, nonclassification is usually suggested but seldom proved; here, as we saw above, nonclassification theorems are precise mathematical theorems. In the rest of this section we give a complete classification for Polish metric spaces up to isometry. The equivalence relation we use as complete invariants is an Iso(U)-orbit equivalence relation. More precisely, we consider the natural Iso(U) action on the standard Borel space F (U), and denote the induced orbit equivalence relation by EI . Theorem 14.2.3 (Gao–Kechris) ∼ =i ≤B EI . That is, the isometric classification of Polish metric spaces is Borel reducible to the Iso(U)-orbit equivalence relation on F (U). Proof. The reduction function is the map J defined in Theorem 14.1.1. We showed that it is a Borel embedding from X into F (U). Here we check that for Polish metric spaces X and X  , X ∼ =i X  iff J(rX )EI J(rX  ). First assume J(rX )EI J(rX  ), that is, for some ϕ ∈ Iso(U), ϕ(J(rX )) = J(rX  ). Then in particular, J(rX ) ∼ =i J(rX  ). By Theorem 14.1.1, X ∼ =i J(rX ) and X ∼ =i X  . For the other direction, =i J(rX  ). Thus by transitivity we have X ∼ assume X ∼ =i X  . Then from the definition of J, we have isometries ψ between the completion of Xω and U and ψ  between the completion of Xω and U. Now if π is an isometry between X and X  , by the construction of Xω and Xω there is an extension π ∗ between Xω and Xω such that π ∗  X = π. π ∗ can be uniquely extended to an isometry between the completions of Xω and Xω , which we continue to denote by π ∗ . Then ψ  ◦ π ∗ ◦ ψ −1 is an element of

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Iso(U) so that ψ  ◦ π ∗ ◦ ψ −1 (J(rX  )) = ψ  ◦ π ∗ (X) = ψ  (X  ) = J(rX  ). This shows that J(rX )EI J(rX  ). In the next section we will show in a precise sense that there is no room to improve this classification significantly. Thus, regardless of the reader’s familiarity with the equivalence relation used as complete invariants, the theorem cannot be materially improved with the use of another kind of invariant. Exercise 14.2.1 Show that the isometric biembeddability between compact metric spaces is a smooth equivalence relation. Exercise 14.2.2 Give a direct coding of countable sets of compact metric spaces by single Polish metric spaces, that is, associate with each countable set {Xn }n∈ω of pairwise nonisometric compact metric spaces a Polish metric  space X so that X ∼ =i Xπ(n) =i X  iff there is a permutation π of ω with Xn ∼ for all n ∈ ω. The following three problems give an alternative proof of Gromov’s Theorem 14.2.1. Exercise 14.2.3 Let (X, dX ), (Y, dY ) be compact metric spaces. Suppose that for any  > 0, n ∈ ω, and x0 , . . . , xn ∈ X, there are y0 , . . . , yn ∈ Y such that |d(xi , xj ) − d(yi , yj )| < . Show that there is an isometric embedding from X into Y . Exercise 14.2.4 Let K(U) be the space of all compact subsets of U with the Vietoris topology (see Section 1.1 and Exercise 1.4.4). Show that every ∼ =i -equivalence class is closed. Exercise 14.2.5 For compact metric spaces X, Y , the Gromov–Hausdorff metric dG (X, Y ) is defined as the infimum of dZ (ϕ(X), ψ(Y )), where Z is a metric space, ϕ : X → Z and ψ : Y → Z are isometric embeddings, and dZ is the Hausdorff metric on K(Z). Show that for X, Y ∈ K(U), dG (X, Y ) = inf{dH (X  , Y  ) : X  ∼ =i X, Y  ∼ =i Y }, where dH is the Hausdorff metric on K(U).

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14.3

Invariant Descriptive Set Theory

Measurement of complexity

At first sight the equivalence relation EI considered in Theorem 14.2.3 is far from canonical as complete invariants for Polish metric spaces. However, in this section we will show that EI is universal for all orbit equivalence relations induced by Borel actions of Polish groups. Moreover, we will show that the classification problem has the same complexity as that of EI , that is, that they are Borel bireducible to each other. Putting these results together, we obtain a complete classification for Polish metric spaces up to isometry that cannot be improved; and the complete invariants we use have a characteristic property making them in a sense canonical. We now start to present a proof that any orbit equivalence relation is Borel reducible to ∼ =i . This recent theorem was proved by Gao and Kechris, and also independently by Clemens (see References [22] and [58]). We present Clemens’ proof here since it is more elementary. Let G be a Polish group and X a Borel G-space. We will associate with X y iff Mx ∼ each element x ∈ X a Polish metric space Mx so that xEG =i My . First, in view of Exercises 3.3.6 and 3.1.9 we may assume X is a compact Polish G-space. We will need a special compatible metric on G given by the following lemma. Lemma 14.3.1 Let G be a Polish group and X a compact Polish G-space. Let dX ≤ 1 be a compatible metric on X. Then there is a left-invariant compatible metric dG ≤ 1 on G such that, for any x ∈ X and g, h ∈ G, 1 dG (g, h) ≥ dX (g −1 · x, h−1 · x). 2 Let d0 ≤ 1 be a compatible left-invariant metric on G. Define 1 1 dG (g, h) = d0 (g, h) + sup{dX (g −1 · x, h−1 · x) : x ∈ X}. 2 2 It is easy to see that dG ≤ 1 is a left-invariant metric on G satisfying dG (g, h) ≥ 1 −1 · x, h−1 · x) for all x ∈ X. It only remains to check that dG is a 2 dX (g compatible metric on G. Since dG (g, h) ≥ 12 d0 (g, h) for all g, h ∈ G, the topology induced by dG is finer than that induced by d0 . For the converse let  > 0. By left-invariance of d0 and dG it suffices to find δ > 0 such that, whenever d0 (g, 1G ) < δ, we have dG (g, 1G ) < . Since the action is continuous as a function from G × X to X, we have that for any x ∈ X, there is 0 < δ < /2 such that for all g ∈ G and y ∈ X with d0 (g, 1G ) < δx and dX (y, x) < δx , we have dX (g −1 · y, x) < /2. It follows that for such g and y,   dX (g −1 · y, y) ≤ dX (g −1 · y, x) + dX (y, x) < + = . 2 2 Proof.

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By compactness of X there are finitely many points x0 , . . . , xn ∈ X such that for all y ∈ X, there is i ≤ n with dX (y, xi ) < δxi . Let δ = inf{δx0 , . . . , δxn }. Then for all g ∈ G with d0 (g, 1G ) < δ and all y ∈ X, dX (g −1 · y, y) < ; hence dG (g, 1G ) =

1 1 1 1 d0 (g, 1G ) + sup{dX (g −1 · y, y) : y ∈ X} <  +  = . 2 2 2 2

Now fix a compatible metric dX ≤ 1 on X and a compatible left-invariant metric dG ≤ 1 on G given by Lemma 14.3.1. Without loss of generality we may assume that there are elements x, y ∈ X with dX (x, y) = 1. Also we may assume that sup{dG (g, h) : g, h ∈ G} = 1. By left-invariance this implies that for any g ∈ G, sup{dG (g, h) : h ∈ G} = 1. Fix a countable dense subset D of X, and let (xn )n∈Z be an enumeration of D. We are now ready to define the Polish metric spaces Mx for each x ∈ X. n = −1

n=0

n=1

i=1:

···

···

i=0:

···

···

Figure 14.1 The space H = G × Z × {0, 1}. Let H = G × Z × {0, 1}. Let π : Z → ω be the bijection defined as  2n, if n ≥ 0, π(n) = −2n − 1, otherwise. Define a metric dx on H by letting, for (g1 , n1 , i1 ), (g2 , n2 , i2 ) ∈ H, dx ((g1 , n1 , i1 ), (g2 , n2 , i2 )) ⎧ dG (g1 , g2 ), ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 3 ⎪ ⎪ ⎨ + 4−|n1 −n2 | (1 + dG (g1 , g2 )), 2 = ⎪ ⎪ ⎪ 1 + 4−π(n1 −n2 )−1 (1 + dX (xn −n , g −1 · x)), ⎪ 1 2 ⎪ 2 ⎪ ⎪ ⎪ ⎪ ⎩ 1 + 4−π(n2 −n1 )−1 (1 + dX (xn2 −n1 , g1−1 · x)),

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if n1 = n2 and i1 = i2 , if n1 = n2 and i1 = i2 , if i1 = 0 and i2 = 1, if i1 = 1 and i2 = 0.

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It is straightforward to check that dx is a metric (see Exercise 14.3.1). With this definition each H can be viewed as the union of countably many copies of G indexed by elements of Z × {0, 1}, as illustrated in Figure 14.1. Note that for any two elements in different copies of G, their distance is greater than 1, while the metric dx on each copy of G coincides with dG ≤ 1. ˆ be the completion of G with dG , and continue to denote the metric Let G by dG . Define Mx to be the completion of H with dx , and we continue to denote the metric on Mx by dx . Then Mx has the underlying set ˆ =G ˆ × Z × {0, 1}, H ˆ indexed by elements of Z × which is a union of countably many copies of G ˆ coincides with the {0, 1}, while the completed metric dx on each copy of G completed metric dG . We note that the map x → Mx is a Borel function from X into X . To see this just note that a canonical countable dense subset Dx of Mx can be obtained from any canonical countable dense subset of G, and the distance between two points in Dx is explicitly defined in the definition of dx , which is clearly Borel. Lemma 14.3.2 X y then Mx ∼ If xEG =i My . Proof. Suppose y = h · x. We check that the map (g, n, i) → (hg, n, i) is an isometry from (H, dx ) onto (H, dy ). Since this isometry extends uniquely to an isometry from Mx onto My , we have Mx ∼ =i My . The map is obviously a permutation of H. We only need to verify that it preserves the metric. Among the four cases of the definition of dx , the first two cases follow easily by left-invariance of dG . By the symmetry of the remaining cases, we only show one of them. So consider two points (g1 , n1 , i1 ), (g2 , n2 , i2 ) ∈ H with i1 = 0 and i2 = 1. Then dx ((g1 , n1 , i1 ), (g2 , n2 , i2 )) = 1 + 4−π(n1 −n2 )−1 (1 + dX (xn1 −n2 , g2−1 · x)) = 1 + 4−π(n1 −n2 )−1 (1 + dX (xn1 −n2 , (hg2 )−1 · y)) = dy ((hg1 , n1 , i1 ), (hg2 , n2 , i2 )).

Lemma 14.3.3 X If Mx ∼ y. =i My then xEG

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Proof. Suppose ϕ : Mx ∼ =i My . Since Mx contains countably many copies ˆ with the diameter of each copy at most 1 and distances between points of G, from different copies greater than 1, it follows that ϕ must send each copy of ˆ in Mx onto a copy of G ˆ in My . This means that ϕ induces a bijection f : G ˆ ϕ(h, n, i) ∈ G×{f ˆ Z×{0, 1} → Z×{0, 1} such that for all (h, n, i) ∈ H, (n, i)}. For convenience we will use the following terminology to address subsets ˆ Call G ˆ × {(n, i)} the (n, i)-copy of G, ˆ the collection G ˆ × Z × {0} the of H. ˆ and G ˆ × Z × {1} the 1-chain of copies of G. ˆ 0-chain of copies of G, ˆ in the 0-chain have Now note further that points from different copies of G distance greater than 3/2, whereas points from different chains have distance ˆ in the same chain in Mx to at most 3/2. It follows that ϕ sends copies of G those in the same chain in My . Furthermore, for i ∈ {0, 1} and n1 = n2 ∈ Z, ˆ = 3 + 2 · 4−|n1 −n2 | , sup{dx ((h1 , n1 , i), (h2 , n2 , i)) : h1 , h2 ∈ G} 2 ˆ = 1. This implies the following rigidity since sup{dG (h1 , h2 ) : h1 , h2 ∈ G} property of ϕ: if f (0, i) = (n0 , j), then either f (m, i) = (n0 + m, j) for all m ∈ Z or f (m, i) = (n0 − m, j) for all m ∈ Z. We now consider two cases. Case 1: f (0, 0) = (n0 , 0) for some n0 ∈ Z. It follows that f (0, 1) = (m0 , 1) for some m0 ∈ Z. However, if n0 = m0 , then for g1 , g2 ∈ G, dx ((g1 , 0, 0), (g2 , 0, 1)) = 1 + 4−1 (1 + dX (x0 , g2−1 · x)) ≥

5 4

but dy ((g1 , n0 , 0), (g2 , m0 , 1)) = 1 + 4−π(n0 −m0 )−1 (1 + dX (xn0 −m0 , g2−1 · y)) ≤ 1 + 2 · 4−π(n0 −m0 )−1 ≤

9 . 8

Thus we must have m0 = n0 . Also for g1 , g2 ∈ G and any m ∈ Z, dx ((g1 , m, 0), (g2 , 0, 1)) ∈ [1 + 4−π(m)−1 , 1 + 2 · 4−π(m)−1 ] dy ((g1 , n0 + m, 0), (g2 , n0 , 1)) ∈ [1 + 4−π(m)−1 , 1 + 2 · 4−π(m)−1 ] dy ((g1 , n0 − m, 0), (g2 , n0 , 1)) ∈ [1 + 4−π(−m)−1 , 1 + 2 · 4−π(−m)−1 ] The first two intervals being the same and disjoint from the third, it follows ˆ it follows that f (m, 0) = (n0 + m, 0) for all m ∈ ω. Since G is comeager in G, that there is a comeager subset C of G such that for all g ∈ C, and for all m ∈ ω, ϕ(g, m, 0) ∈ H and ϕ(g, 0, 1) ∈ H. Let g1 , g2 ∈ C. We have for all m ∈ ω, dx ((g1 , m, 0), (g2 , 0, 1)) = 1 + 4−π(m)−1 (1 + dX (xm , g2−1 · x)).

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Let h1,m ∈ G be such that ϕ(g1 , m, 0) = (h1,m , n0 + m, 0), and let h2 ∈ G be such that ϕ(g2 , 0, 1) = (h2 , n0 , 1). Then dx ((g1 , m, 0), (g2 , 0, 1)) = dy (ϕ(g1 , m, 0), ϕ(g2 , 0, 1)) = dy ((h1,m , n0 + m, 0), (h2 , n0 , 1)) = 1 + 4−π(m)−1 (1 + dX (xm , h−1 2 · y)). It follows that for all m ∈ ω, dX (xm , g2−1 · x) = dX (xm , h−1 2 · y), and hence −1 X g2−1 · x = h−1 · y. Thus y = h g · x and yE x. 2 2 2 G Case 2: f (0, 0) = (n0 , 1) for some n0 ∈ Z. Then a similar argument shows that f (0, 1) = (n0 , 0) and f (m, 1) = (n0 − m, 0) for all m ∈ Z. Again there are g1 , h1,m , g2 , h2 ∈ G such that ϕ(g1 , m, 1) = (h1,m , n0 − m, 0) and ϕ(g2 , 0, 1) = (h2 , n0 , 0). Thus dx ((g1 , m, 1), (g2 , 0, 0)) = dy (ϕ(g1 , m, ), ϕ(g2 , 0, 0)) = dy ((h1,m , n0 − m, 0), (h2 , n0 , 1)) = 1 + 4−π(m)−1 (1 + dX (xm , h−1 2 · y)), X x. from which it follows as before that yEG

We have thus proved the following theorem. Theorem 14.3.4 (Clemens–Gao–Kechris) X ≤B ∼ Let G be a Polish group and X a Borel G-space. Then EG =i . X Recall that a universal orbit equivalence relation is one of the form EG , where G is a Polish group and X a Borel G-space, such that for all Polish Y X groups H and Borel H-space Y , EH ≤B EG . The following corollary is now immediate.

Corollary 14.3.5 (Gao–Kechris) EI is a universal orbit equivalence relation. Proof. Note that EI is an orbit equivalence relation. By Theorems 14.2.3 and 14.3.4, every orbit equivalence relation is Borel reducible to EI . Corollary 14.3.6 (Gao–Kechris) The isometric classification for Polish metric spaces is Borel bireducible to the universal orbit equivalence relation. Proof.

It follows from Theorem 14.3.4 that EI ≤B ∼ =i ∼B EI . =i , and thus ∼

Our result is therefore a proper classification theorem for the isometric problem of Polish metric spaces, and the universality property of the equivalence relation EI makes it a benchmark equivalence relation.

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Exercise 14.3.1 Show that dx defined on the set H is a metric. Exercise 14.3.2 Let G be a cli Polish group, that is, a Polish group with a compatible complete left-invariant metric and X a Polish G-space. Show that X is Borel reducible to the isometric classification of Polish metric groups. EG Exercise 14.3.3 Let G be an abelian Polish group and X a Polish G-space. X Show that EG is Borel reducible to the isometric classification of abelian Polish metric groups. Exercise 14.3.4 Let G be Iso(U) with a compatible left-invariant metric d ≤ 1. Show that the G-orbit equivalence relation on L(G, d) is a universal orbit equivalence relation.

14.4

Classification notions

In this section we discuss some different classification notions for Polish metric spaces. The isometric classification is the finest notion of equivalence for metric spaces. In mathematics we are often interested in other classification notions for the same class of mathematical structures. In the case of metric spaces, we recall the following list of notion of equivalence, all considered natural and helpful in our understanding of the properties of Polish metric spaces. Definition 14.4.1 Let (X, dX ) and (Y, dY ) be metric spaces. (1) We say that X and Y are homeomorphic if there is a bijection f : X → Y such that both f and f −1 are continuous. (2) We say that X and Y are uniformly homeomorphic if there is a bijection f : X → Y such that both f and f −1 are uniformly continuous. (3) We say that X and Y are Lipschitz isomorphic if there is a bijection f : X → Y such that both f and f −1 are Lipschitz. (4) We say that X and Y are isometrically biembeddable if there are isometric embeddings f : X → Y and g : Y → X. This definition does not exhaust all natural notions of equivalence for metric spaces. Note that uniform homeomorphism, Lipschitz isomorphism, and isometric biembeddability are still metric notions, but homeomorphism is a topological notion.

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Each of the above classification notions corresponds to an equivalence relation on the standard Borel space X of all Polish metric spaces. One can check that the metric notions are Σ11 (see Exercise 14.4.1). Here we show that the homeomorphism problem is Σ12 , as noted by Ferenczi, Louveau, and Rosendal [42]. Similar claims have been shown in Reference [21]. The proof below is due to Kechris. Proposition 14.4.2 The homeomorphic classification for all Polish metric spaces is Σ12 . Proof. We consider the homeomorphism relation ≈ on F (U). We claim that, for closed subsets X, Y of U, X ≈ Y iff there are countable dense subsets DX , DY of X, Y , respectively, and a homeomorphism f between DX and DY , such that for any sequence (xk )k∈ω in DX , (xk ) is Cauchy iff (f (xk )) is Cauchy. To see this, suppose X ≈ Y and ϕ : X → Y is a homeomorphism. Let DX be any countable dense subset of X. Then ϕ(DX ) is dense in Y . Thus if we let DY = ϕ(DX ) and f = ϕ  DX , ϕ is an extension of f to the completion of DX , which is X, as required. Conversely, suppose the displayed property holds for X, Y ∈ F (U). Then define ϕ : X → Y as follows. For any x ∈ X let (xk ) be a Cauchy sequence in DX with limk xk = x and define ϕ(x) = limk f (xk ). The definition of ϕ does not depend on the choice of the Cauchy sequence (xk ), since if (xk ) is another Cauchy sequence in DX converging to x, we may define a new Cauchy sequence (xk ) by letting x2k = xk and x2k+1 = xk for all k ∈ ω; then by the assumption (f (xk )) is Cauchy and hence limk f (xk ) = limk f (xk ). A similar argument also shows that ϕ is a bijection between X and Y , and that ϕ is a homeomorphism. This finishes the proof of the claim. We next show that the displayed property is Σ12 . Fix a countable base {Un }n∈ω for U. Note that the relation x ∈ X for x ∈ U and X ∈ F (U) is Borel since x ∈ X ⇐⇒ ∀n ( x ∈ Un → X ∩ Un = ∅ ). Moreover, if (xk )k∈ω ∈ Uω and X ∈ F (U), then {xk : k ∈ ω} is a dense subset of X iff ∀k ∈ ω xk ∈ X ∧ ∀n ∈ ω ∃k ∈ ω ( X ∩ Un = ∅ → xk ∈ Un ), and hence is Borel as well. Let d be the metric on U. For (xk )k∈ω ∈ Uω , (xk ) is d-Cauchy ⇐⇒ ∀n ∈ ω ∃m ∈ ω ∀p, q ≥ m d(xp , xq ) < 2−n is a Borel relation. If (xk )k∈ω , (yk )k∈ω ∈ Uω , define f : {xk : k ∈ ω} → {yk : k ∈ ω} by letting f (xk ) = yk ; then f is continuous iff ∀k, n ∈ ω ∃m ∈ ω ∀l ∈ ω ( d(xl , xk ) < 2−m → d(yl , yk ) < 2−n ).

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It follows that the relation that xk → yk is a homeomorphism between {xk : k ∈ ω} and {yk : k ∈ ω} is Borel. Finally for X, Y ∈ F (U), we have that X ≈ Y iff ∃(xk )k∈ω , (yk )k∈ω ∈ Uω [{xk : k ∈ ω} is a dense subset of X ∧ {yk : k ∈ ω} is a subset of Y ∧ xk → yk is a homeomorphism between {xk : k ∈ ω} and {yk : k ∈ ω} 1 ∧ ∀z ∈ ω ω ( (xz(k) )k∈ω is d-Cauchy ↔ (yz(k) )k∈ω is d-Cauchy ) This shows that ≈ is Σ12 . It is not known that the statement of the proposition is optimal, but this computation of descriptive complexity suggests that the homeomorphism relation is much more complicated than all the metric notions. For the rest of this section we consider the homeomorphic classification of compact metric spaces. We denote this homeomorphism equivalence relation by ≈c . As a contrast to Proposition 14.4.2 we note that ≈c is Σ11 . Proposition 14.4.3 The homeomorphic classification for compact metric spaces is Σ11 . Proof. The main difference here is that continuous functions between compact metric spaces have much nicer properties. First, they are closed maps, which implies that any continuous bijection is a homeomorphism. Second, they are uniformly continuous, and therefore completely determined by restrictions on countable dense subsets. To be more specific, let X and Y be compact metric spaces and D a countable dense subset of X. Then a function f : X → Y is continuous iff f  D is a uniformly continuous function from D into Y , and for a continuous function f : X → Y , f is onto iff f (D) is dense in Y . It follows that f is a homeomorphism from X onto Y iff f  D is one-to-one, f (D) is dense, and f  D is uniformly continuous. These are now Borel conditions on Xcpt . In fact, ≈c is Borel reducible to the universal orbit equivalence relation. But we will not prove it here. In contrast to the Gromov Theorem 14.2.1 we show that ≈c is nonsmooth. Proposition 14.4.4 The equivalence relation =+ is Borel reducible to ≈c . In particular, the homeomorphic classification of compact metric spaces is not smooth.

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Proof. First we define a reduction of certain finite splitting trees to compact metric spaces. Let T0 be the set of all finite splitting trees on ω such that the following conditions hold: (i) ∅ has exactly three immediate successors, (ii) every node in T has at least three immediate successors. For each tree T in T0 we define a compact subset ST of R2 as follows. First let (0, 1) ∈ ST . This point corresponds to ∅ ∈ T . Then corresponding to the three nodes t1 , t2 , t3 in T of length 1, we let p1 = (−2−1 , 2−1 ), p2 = (0, 2−1 ), p3 = (2−1 , 2−1 ) ∈ ST together with the line segments linking each of them with (0, 1). Next if t1 has n immediate successors s1 , . . . , sn then we find n points q1 , . . . , qn with y-coordinate 2−2 and x-coordinates in [−5/8, −3/8], add q1 , . . . , qn to ST , together with the line segment linking each of them with p1 . The construction up to this point is illustrated in Figure 14.2. Repeat

p1 r  \ L  r r r L\ r\r ···

r1 @ @ @ @ @r p3 r p2 T D L r r L r r r DrT r ··· 0

Figure 14.2 The compact metric space ST . this process for the rest of the nodes in T , and we obtain a subset of R2 . The construction should be carried out that the set contains all of its accumulation points with nonzero y-coordinates. Finally let ST be the closure of the resulting set, that is, by adding to the resulting set all of its accumulation points (and note that they all have zero y-coordinates). It is clear that ST is path-connected. In a natural sense we may regard T as a subset of ST . Then the degree of nodes in T can be topologically characterized: each node of degree d ≥ 3 becomes a point x in ST with the property that ST − {x} has d many pathconnected components. It is now easy to see that T ∼ = T  iff ST ≈c ST  . By Theorem 13.2.3 the isomorphism on T0 is Borel bireducible to id(2ω ). This implies that there is a Borel map x → Sx from 2ω into Xcpt such that Sx ≈c Sy for x = y. Now if A is a countable subset of 2ω , then we let SA be the countable union of a copy of Sx for each x ∈ A. For each x ∈ A, Sx is

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homeomorphic to a connected component of SA . It then follows easily that A = A iff SA ≈ SA . In fact Proposition 14.4.3 can be proved by many different constructions. We present this particular proof here to give the flavor of the subject and to show the power of invariant descriptive set theory in gaining new insight. The homeomorphic classification problem for compact metric spaces has been studied extensively in topology and descriptive set theory. However, its exact complexity is still unknown. Exercise 14.4.1 Show that uniform homeomorphism, Lipschitz isomorphism, and isometric biembeddability are all Σ11 equivalence relations on X . Exercise 14.4.2 Show that =+ is Borel reducible to the homeomorphism classification of connected compact metric spaces. Exercise 14.4.3 Give a Borel reduction from id(2ω ) to the homeomorphism classification of 0-dimensional compact metric spaces.

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Chapter 15 Summary of Benchmark Equivalence Relations

We have given in this book an introduction of definable equivalence relations. In decreasing order of comprehensiveness, we have treated the topics of orbit equivalence relations, general Borel equivalence relations, general Σ11 equivalence relations, and general Π11 equivalence relations. While these classes do not exhaust all definable equivalence relations, they are most relevant to many other areas of mathematics and results about them often do not go beyond the usual axioms of mathematics and set theory. Many equivalence relations we have considered have characteristic properties which give them distinct places in the hierarchy of Borel reducibility. This kind of canonicity makes them benchmark equivalence relations suitable for use in gauging the complexity of other equivalence relations and classification problems arising in mathematics. In this chapter we summarize these equivalence relations and mention classification problems with identical complexity.

15.1

Classification problems up to essential countability

On a very crude scale there are four benchmark equivalence relations up to the universal countable Borel equivalence relation, and they are id(ω), id(2ω ), E0 , E∞ . We recall the basic facts we proved about the Borel reducibility hierarchy related to these equivalence relations. Below id(ω) are equivalence relations with finitely many equivalence classes, which are obviously determined by the cardinality of the quotient space. By the Silver dichotomy (Theorem 5.3.5) every Π11 equivalence relation is either at most id(ω) or at least id(2ω ) in the Borel reducibility quasiorder. This is no longer true for Σ11 equivalence relations (Section 9.1). The statement that it holds for orbit equivalence relations is the topological Vaught conjecture and is one of the major open problems of the entire field. The Glimm–Effros dichotomy proved by Harrington–Kechris–Louveau (Theorem 6.3.1) states that every Borel equivalence relation is either at most id(2ω ) or else at least E0 .

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Smooth equivalence relations are the ones up to id(2ω ) in the Borel reducibility hierarchy. Up to Borel bireducibility they can be listed as id(1), id(2), . . . , id(ω), id(2ω ). Essentially hyperfinite equivalence relations are either smooth or Borel bireducible with E0 , thus up to Borel bireducibility they are listed as id(1), id(2), . . . , id(ω), id(2ω ), E0 . Equivalence relations beyond E0 cannot be listed sequentially any more. E∞ is a universal countable Borel equivalence relation. It is known that the Borel reducibility hierarchy between E0 and E∞ is very complicated, but we did not get into the details in this book, and interested readers can find further results in References [1] and [86]. Many classification problems in mathematics have been identified to have the same complexity as one of the four equivalence relations. The following are some examples. Example 15.1.1 In topology, a compact orientable surface is completely classified up to homeomorphism by its genus. Since the genus is a natural number, and its computation involves only finite triangulations and is obviously Borel, we obtain that the homeomorphic classification problem for closed orientable surfaces is Borel bireducible with id(ω). Example 15.1.2 In algebra, a finitely generated abelian group can be uniquely expressed, up to isomorphism, as a direct sum Z/pr11 Z ⊕ · · · ⊕ Z/prnn Z ⊕ Z ⊕ · · · ⊕ Z 34 5 2 m

where m, n ∈ ω, p1 , . . . , pn are prime numbers (not necessarily distinct) and r1 , . . . , rn ≥ 1. It follows that the isomorphic classification problem for finitely generated abelian groups is Borel bireducible with id(ω). Example 15.1.3 In linear algebra, the similarity type of a complex square matrix is completely determined by its Jordan normal form. Recall that two n×n complex matrixes A and B are similar if there is an invertible matrix P such that P −1 AP = B. Every square matrix is similar to one in Jordan normal form, which is determined by the eigenvalues of the matrix as well as their multiplicities. These finitely many eigenvalues, together with their multiplicities, can be coded by a single real number. It follows that the similarity classification problem for complex square matrices is smooth.

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Example 15.1.4 In dynamical systems, a Bernoulli shift is a triple (X, μ, T ), where for some n ≥ 1, X = {1, 2, . . . , n}Z with the standard Borel structure given by the product topology, μ is an product measure on X given by nonnegative real numbers p1 , . . . , pn with i=1 pi = 1, and T is the shift: for any x = (xn )n∈Z ∈ X, (T x)n+1 = xn for all n ∈ Z. Given a Bernoulli shift as above, its entropy is defined as E=−

n 

pi log pi .

i=1

Two Bernoulli shifts (X1 , μ1 , T1 ) and (X2 , μ2 , T2 ) are isomorphic if there is a map ϕ : (X1 , μ1 ) → (X2 , μ2 ) that is measure-preserving, that is, μ2 (ϕ(A)) = μ1 (A) for any Borel set A ⊆ X1 , and such that for all x ∈ X1 , ϕ(T1 x) = T2 ϕ(x). The following celebrated theorem of Ornstein classifies Bernoulli shifts up to isomorphism by their entropies. Since the entropy is a real number, it essentially shows that the isomorphic classification problem for Bernoulli shifts is smooth. Theorem 15.1.5 (Ornstein) Two Bernoulli shifts are isomorphic iff they have the same entropy. Somewhat related is the shift action of Z on 2Z . The induced orbit equivalence relation is Borel bireducible with E0 . This is also useful in the study of dynamical systems. As another example in algebra, we have seen that E0 comes up in Baer’s classification of torsion-free abelian groups of rank 1. Example 15.1.6 A metric space (X, d) is Heine–Borel if every closed bounded subset is compact. A Heine–Borel metric space is separable and complete. Consider the isometric classification problem for all Heine–Borel ultrametric spaces. It is a theorem of Gao and Kechris [58] that it is Borel bireducible with E0 . The proof is left as exercise. The universal countable Borel equivalence relation E∞ is less well known. But Kechris’ theorem (Corollary 7.5.3) that any locally compact Polish group action induces essentially countable orbit equivalence relations gives a sweeping classification for a wide variety of objects considered in algebra, dynamical systems, geometry, and analysis. We mention two results that employ E∞ as the exact complexity of classification problems (from References [58] and [79] respectively). Theorem 15.1.7 (Hjorth) The isometric classification of all Heine–Borel Polish metric spaces is Borel bireducible with E∞ .

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Theorem 15.1.8 (Hjorth–Kechris) The conformal equivalence for Riemann surfaces is Borel bireducible with E∞ . There are many essentially countable equivalence relations strictly in between E0 and E∞ (see References [1] and [153]). Many results involving them require the deep theory of superrigidity developed by Zimmer and others in dynamical systems. However, so far none has been recognized as a benchmark as in the cases of E0 and E∞ . Exercise 15.1.1 Show that any Heine–Borel metric space is separable and complete. Exercise 15.1.2 Let (X, d) be a Heine–Borel ultrametric space. For any x ∈ X and r > 0, let Brc (x) = {y ∈ X : d(x, y) ≤ r}. (a) Show that for any x, y ∈ X and r > d(x, y), Brc (x) = Brc (y). (b) Show that two Heine–Borel ultrametric spaces (X, dX ) and (Y, dY ) are isometric iff for any x ∈ X and y ∈ Y there is n ∈ ω such that for all c c (x) ∼ (y). m ≥ n, Bm =i Bm (c) Deduce from (b) that the isometric classification of Heine–Borel ultrametric spaces is Borel reducible to E0 . (d) Give a Borel reduction from E0 to the isometry of countable Heine–Borel ultrametric spaces.

15.2

A roadmap of Borel equivalence relations

In this section we identify benchmark Borel equivalence relations. Recall that any Borel equivalence relation E is strictly below its Friedman–Stanley jump E + (Theorem 8.3.6). Also we may define finite or countable products of Borel equivalence relations; in particular, for any Borel equivalence relation E we have E ≤B E ω . These operations allow us to generate a complicated list of equivalence relations even starting from a relatively short list of basic equivalence relations. We will first focus on giving a list of Π03 equivalence relations. In Chapter 8 we have introduced the following Π03 equivalence relations: E0 , E∞ , p (1 ≤ p < ∞), E1 , ∞ , E0ω , c0 , =+ , ≡m . Dougherty and Hjorth showed that for 1 ≤ p, q < ∞, p < q iff p < q (Theorem 8.4.3). It is a theorem of Kechris and Louveau [102] that E1 is the unique nonhyperfinite hypersmooth equivalence relation up to Borel bireducibility.

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They also showed that E1 is not below any orbit equivalence relation (Theorem 10.6.1). Rosendal’s Theorem 8.4.2 identifies ∞ as a universal Kσ equivalence relation; this positions it above every equivalence relation that comes before it in the above list. It is known that E0ω is below c0 and =+ , and that 2 and =+ are below the measure equivalence ≡m (Section 8.5). In fact =+ is the universal Π03 S∞ -orbit equivalence relation (Theorem 12.5.5). And by the theory of turbulence, neither 1 nor c0 is below =+ (Section 10.5). This implies that none of p (1 ≤ p ≤ ∞), c0 , or ≡m is below =+ . We summarize these reductions in the following figure. + rH ∞ HH HH HHr +  E1  r ≡m    B   B  r c0 =+ r B L B L B L B L B L ω rE1ω r rE∞ ∞  B L  H r H Br  L

AA H HH    B L
A  B r  2 A HHrE  ωL
r r r E0  1    1 E ∞    AA

   A
  A r 
 E0 r id(2ω ) r id(ω)

Figure 15.1 Borel reductions among Π03 equivalence relations. In Figure 15.1 each line represents a known Borel reduction from the horizontally lower equivalence relation to the higher one. Other Borel reductions can be obtained by compositions. For pairs of equivalence relations for which no Borel reduction is implied by the figure, some nonreducibility results are known, such as the ones mentioned in the preceding paragraph, but more often the question is an open problem. For instance, it is not known if p ≤B ≡m iff p ≤ 2, or whether c0 ≤B ≡m .

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Among Π03 equivalence relations ∞ and ≡m show up the most often in classification problems. The measure equivalence ≡m is known to be Borel bireducible with the unitary equivalence of bounded normal operators (or unitary operators, or bounded self-adjoint operators) by the Spectral Theory of functional analysis (see, for example, Reference [104]). By Rosendal’s theorem ∞ is known to be bireducible with a number of classification problems involving biembeddability. For instance, the isometric biembeddability problem for compact metric spaces is known to have this complexity [132]. This is in contrast with Gromov’s theorem that the isometric classification of compact metric spaces is smooth (Theorem 14.2.1). There are very few benchmarks of Borel equivalence relations beyond Π03 . The Friedman–Stanley tower is a cofinal ω1 -sequence of Borel S∞ -orbit equivalence relations (Section 12.2) (we postpone their further discussion to the next section). Rosendal [132] recently constructed a cofinal ω1 -sequence of Borel equivalence relations, but their connection with classification problems has not been well studied. Exercise 15.2.1 Show that E ω ≤B (E × id(2ω ))+ . Thus if E is a Borel equivalence relation such that E × id(2ω ) ≤B E, then E ω ≤B E + . Exercise 15.2.2 Verify all the reducibility claims in Figure 15.1.

15.3

Orbit equivalence relations

In Figure 15.1 most equivalence relations are induced by Polish group actions. The exceptions are all the equivalence relations from E1 and above. It is an open problem if nonreducibility from E1 characterizes orbit equivalence relations. Among the orbit equivalence relations the best understood are S∞ -orbit equivalence relations. In Figure 15.1 these include all the equivalence relations from =+ and below. The rest of the Friedman–Stanley tower include =α+ for all 2 ≤ α < ω1 , which form a strictly increasing ω1 -sequence of Borel equivalence relations. By the Scott analysis the Friedman–Stanley tower is cofinal in all Borel S∞ -orbit equivalence relations. There are two distinguished nonBorel isomorphism relations which are frequently used as benchmarks to gauge the complexity of other equivalence relations and classification problems. The first is the isomorphism relation for countable torsion abelian groups. Because of its classification by the Ulm invariants (Sections 9.2 and 13.4), we denote this equivalence relation id(2<ω1 ). One can form a tower similar to the relativized Friedman–Stanley tower above id(2<ω1 ), but it has not been very useful for classification problems (mostly isomorphism problems for countable structures) considered so far. The other distinguished isomorphism relation

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is the universal S∞ -orbit equivalence relation, sometimes denoted by ES∞∞ . By our results in Chapter 13 it has many realizations, but we use the graph isomorphism most often. Figure 15.2 summarizes S∞ -orbit equivalence relations. ∞ rES∞ or graph isomorphism B B p B p p B p B p B B r=α+ B B p B p B p B r =2+ B B B r= + B ω B rE∞ B @ @rE∞ E0ω r B @ BBr <ω1 @r ) E0  id(2   r id(2ω ) 

r id(ω) Figure 15.2 S∞ -orbit equivalence relations. Many classification problems allow countable structures as complete invariants, and therefore regardless of the nature of the objects their complexity becomes that of one of the S∞ -orbit equivalence relations. For instance, Camerlo and Gao [17] showed that the isomorphism relation of commutative almost finite-dimensional C ∗ -algebras is Borel complete, that is, Borel bireducible with the graph isomorphism. Gao and Kechris [58] showed that the isometric classification of Polish ultrametric spaces is also Borel complete. Among other orbit equivalence relations the one with an obvious importance is the universal orbit equivalence relation. As we proved in the preceding chapter, it can be realized as a universal Iso(U)-orbit equivalence relation, ∞ , or as the isometric classification problem for all which we denote by EIso(U) Polish metric spaces, which is denoted ∼ =i . It has been shown by Melleray [121] and Weaver [165] that the linear isometric classification for separable Banach spaces is Borel bireducible with ∼ =i .

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∞ be a universal G-orbit equivExercise 15.3.1 For any Polish group G let EG ∞ alence relation. Give a diagram of Borel reducibility among EG where G varies over the following list:

Z, F2 , 2 , S∞ , Iso(U).

15.4

General Σ11 equivalence relations

By general Σ11 equivalence relations we mean Σ11 equivalence relations that are non-Borel and nonorbit equivalence relations. As we discussed in Chapter 9 the definable reducibility notion suitable for general Σ11 equivalence relations seems to be that of Δ12 reductions. There are more absolute notions such as C-measurable reducibility and absolutely Δ12 reducibility. For equivalence relations and classification problems arising naturally, very often one can show Borel reductions or even continuous reductions. There are two kinds of general Σ11 equivalence relations that come up often in practice. One kind includes equivalence relations with ω1 many equivalence classes. We saw in Chapter 9 some examples of such equivalence relations. One can show that they are Δ12 bireducible with one another, but probably not Borel bireducible. To be consistent with the notation we have been using, we denote any representative of this kind of equivalence relations by id(ω1 ). The Burgess trichotomy theorem (Theorem 9.1.5) is an analog of the Silver dichotomy for Σ11 equivalence relations. It states that any Σ11 equivalence relation has either countably many, ω1 many, or perfectly many equivalence classes. id(ω1 ) is above id(ω) but Δ12 reducible to id(2<ω1 ). By the Silver dichotomy no Π11 equivalence relation is Δ12 bireducible with id(ω1 ). But whether it is Δ12 bireducible with an orbit equivalence relation is equivalent to the topological Vaught conjecture and is therefore open. No natural classification problems have been identified with this equivalence relation. Hjorth and Kechris [76] have proved an analog of the Glimm–Effros dichotomy for Σ11 equivalence relations. At the top is the universal Σ11 equivalence relation, which we denote by EΣ11 . Louveau and Rosendal [111] have recently obtained combinatorial realizations of this equivalence relation. They showed that the biembeddability of trees on ω are universal Σ11 . More recent theorems of Ferenczi, Louveau, and Rosendal [42] related this equivalence relation with classification problems. Among the classification problems proved to be universal Σ11 are the isomorphism problem of separable Banach spaces and the uniform homeomorphic classification of Polish metric spaces.

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15.5

353

Beyond analyticity

Silver dichotomy is formulated for Π11 equivalence relations. Hjorth [67] has shown that there is a universal Π11 equivalence relation. However, in practice true Π11 equivalence relations appear far less frequently than Σ11 equivalence relations. There are examples of classification problems on even higher levels of the projective hierarchy. We have seen in the preceding chapter that the homeomorphism problem for general Polish metric spaces is Σ12 . There are no known classification problems beyond analytic ones whose exact complexity has been determined. However, we should probably close by remarking that the objective of invariant descriptive set theory is not to reach equivalence relations ever higher in the reducibility hierarchy, but to obtain ever deeper understandings of the ones that are most relevant to mathematics and its applications.

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Appendix A Proofs about the Gandy–Harrington Topology

In this appendix we give proofs for the theorems mentioned in Section 1.8. These theorems are harder to find in the literature and not always included in standard textbooks. However, the techniques and some of the statements can be found in References [126] and [133].

A.1

The Gandy basis theorem

In this section we give a proof of the Gandy basis theorem and deduce some consequences toward a characterization of low elements. We will use Kleene’s Theorem 1.7.5 in our proofs. This is not always necessary, but it allows us to reach the proofs faster. Lemma A.1.1 CK(x) CK(y) ≤ ω1 } is Σ11 . The set {(x, y) : ω1 CK(x)

Proof. By Spector’s Theorem 1.6.12 the condition ω1 alent to

CK(y)

≤ ω1

is equiv-

∀u ∈ Δ11 (x) [ u ∈ WO → (∃z)(∃f )( z is computable in y and f is an order-preserving bijection of (ω,
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Recall that for any tree T on ω the Kleene–Brouwer ordering of T is defined by s
Σ11 set contains an element x such that ω1

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= ω1CK .

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Now suppose A ∈ Σ11 . Consider the set

Proof.

{(x, y) : x ∈ A and y ∈ Δ11 (x)}. It follows from Kleene’s Theorem 1.7.5 that the set is Σ11 . If A = ∅ then the above set is nonempty, since there are only countably many reals in Δ11 (x). By the Kleene basis theorem above there are x, y such that (x, y) is computable in O, x ∈ A, and y ∈ Δ11 (x). Thus x ∈ Δ11 (O) and y ∈ Δ11 (O) but y ∈ Δ11 (x), and thus O ∈ Δ11 (x). CK(x) We verify that ω1 = ω1CK . Toward a contradiction assume this is not CK(x) CK , and it follows that there is a u computable the case. Then ω1 < ω1 in x such that ot(
(i) ω1

= ω1CK .

(ii) ∀A ∈ Σ11 [x ∈ A → ∃B ∈ Σ11 (x ∈ B ∧ A ∩ B = ∅)]. (iii) ∀A ∈ Σ11 [x ∈ A → ∃B ∈ Δ11 (x ∈ B ∧ A ∩ B = ∅)]. Proof. That (iii)⇒(ii) is obvious. The implication (ii)⇒(i) is an immediate corollary of the Gandy basis theorem. In fact, assume that it fails for x. Let CK(y) A = {y : ω1 = ω1CK }, which is Σ11 by Lemma A.1.1. Then x ∈ A, and by the assumed property there is a Σ11 set B with x ∈ B and A ∩ B = ∅. Since B contains x, it is nonempty, hence must meet A by the Gandy basis theorem. This contradicts B ∩ A = ∅. CK(x) = ω1CK . Let A ∈ Σ11 with It remains to show (i)⇒(iii). For this let ω1 x ∈ A and T be a computable tree such that y ∈ A ↔ ∃z ∀n ( y  n, z  n ) ∈ T. Now the tree Tx = {s : (x  lh(s), s) ∈ T } is computable in x and is wellfounded because x ∈  A. Let γ be the order type of the Kleene–Brouwer

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x . Then γ < ω1CK = ω1 ordering
since Tx is computable in x. Consider T

y B = {y : the order type of the Kleene–Brouwer ordering
Since WO0γ is Δ11 , so is B. Now we have x ∈ B. But for any y ∈ A, Ty is ill-founded, and hence y ∈ B. So A ∩ B = ∅.

A.2

The Gandy–Harrington topology on Xlow

Let CK(x)

Xlow = {x : ω1

= ω1CK }.

Elements of Xlow are called low. We essentially showed in the preceding section that Xlow is Σ11 but not Π11 , and it meets every nonempty Σ11 set. However, for the record we note the following fact. Lemma A.2.1 Xlow is Borel. CK(x)

Proof. Note that if x ∈ Xlow then ω1 > ω1CK , and thus there is e ∈ ω x CK such that {e} ∈ WO and ot(<{e}x ) = ω1 . We thus have that 

x ∈ Xlow ⇐⇒ ∃e ∈ ω {e}x ∈ WOωCK − 1

WOα .

α<ω CK 1

Since WOα is Borel for all α < ω1 , it follows that Xlow is a countable intersection of Borel sets, and hence is Borel. Recall that the Gandy–Harrington topology on ω ω is the topology generated by all Σ11 sets. Being a Σ11 set, Xlow is basic open in the Gandy–Harrington topology. We consider the Gandy–Harrington topology restricted on Xlow . Theorem A.1.6 implies that every Σ11 set restricted to Xlow is clopen. Thus the Gandy–Harrington topology on Xlow is 0-dimensional, and in particular regular. Since it is also second countable, it thus is metrizable by the Urysohn metrization theorem. We next show that it is completely metrizable, and hence is Polish. Theorem A.2.2 Xlow with the Gandy–Harrington topology is Polish.

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Proof. We give an indirect proof using strong Choquet spaces. By Theorem 4.1.5 the Baire space ω ω with the Gandy–Harrington topology is strong Choquet. Since Xlow is an open subset of ω ω with the Gandy–Harrington topology, by Theorem 4.1.2 (a) the Gandy–Harrington topology on Xlow is strong Choquet. Now note that the Gandy–Harrington topology on Xlow is second countable and regular. It is finer than the usual topology, and hence is Hausdorff, and furthermore T3 . Hence this topology is second countable, T3 , and strong Choquet. By Choquet’s Theorem 4.1.4 it is Polish. It is possible to prove directly that the Gandy–Harrington topology on Xlow is completely metrizable by constructing a homeomorphism between this space and a Gδ subset of the Baire space ω ω with the usual topology.

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References

[1] S. Adams and A. S. Kechris, Linear algebraic groups and countable Borel equivalence relations, Journal of the American Mathematical Society 13 (2000), no. 4, 909–943. [2] R. Baer, Abelian groups without elements of finite order, Duke Mathematical Journal 3 (1937), no. 1, 68–122. [3] J. Barwise, Admissible Sets and Structures: An Approach to Definability Theory. Perspectives in Mathematical Logic. Springer-Verlag, Berlin, New York, 1975. [4] H. Becker, The topological Vaught’s conjecture and minimal counterexamples, Journal of Symbolic Logic 59 (1994), no. 3, 757–784. [5] H. Becker, Polish group actions: dichotomies and generalized elementary embeddings, Journal of the American Mathematical Society 11 (1998), no. 2, 397–449. [6] H. Becker, Topics in invariant descriptive set theory, Annals of Pure and Applied Logic 111 (2001), no. 3, 145–184. [7] H. Becker and A. S. Kechris, Borel actions of Polish groups, Bulletin of the American Mathematical Society 28 (1993), no. 2, 334–341. [8] H. Becker and A. S. Kechris, The Descriptive Set Theory of Polish Group Actions. London Mathematical Society Lecture Notes Series 232. Cambridge University Press, Cambridge, 1996. [9] M. Benda, Remarks on countable models, Fundamenta Mathematicae 81 (1974), 107–119. [10] Logic and Its Applications. Papers from the Conference on Logic and its Applications in Algebra and Geometry held at the University of Michigan, Ann Arbor, MI, April 11–13, 2003 and the Workshop on Combinatorial Set Theory, Excellent Classes and Schanuel’s Conjecture held at the University of Michigan, Ann Arbor, MI, April 14–15, 2003. Edited by A. Blass and Y. Zhang. Contemporary Mathematics 380. American Mathematical Society, Providence, RI, 2005. [11] G. S. Boolos and R. C. Jeffrey, Computability and Logic. Third edition. Cambridge University Press, Cambridge, 1989.

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