SOME APPLICATIONS OF SUPERRIGIDITY TO BOREL EQUIVALENCE RELATIONS

SIMON THOMAS

Abstract. There exists a countable Borel equivalence relation E such that E <_{B} E ⊕ E and E × I(2) ⋢_{B} E.

1. Introduction

Recently Zimmer’s superrigidity theorem [13, Theorem 5.2.5] has been used to solve a number of well-known problems concerning countable Borel equivalence relations. (For example, see [2], [11], [12].) In particular, Adams [1] used superrigidity, together with Ratner’s measure classification theorem [8, Theorem 3], to construct the first known examples of pairs of countable Borel equivalence relations E ⊆ F such that E ≰_{B} F. In this paper, we shall point out that Adams’ examples can also be used to solve two other well-known problems concerning Borel equivalence relations.

Let X be a standard Borel space; i.e. a Polish space equipped with its associated σ-algebra of Borel subsets. Then a Borel equivalence relation on X is an equivalence relation E ⊆ X^{2} which is a Borel subset of X^{2}. The Borel equivalence relation E is said to be countable iff every E-equivalence class is countable. If E, F are Borel equivalence relations on the standard Borel spaces X, Y respectively, then we say that E is Borel reducible to F and write E ≤_{B} F if there exists a Borel map f : X → Y such that xEy iff f(x)F f(y). Such a map f is called a Borel reduction from E to F. We say that E and F are Borel bireducible and write E ∼_{B} F if both E ≤_{B} F and F ≤_{B} E. Finally we write E <_{B} F if both E ≤_{B} F and F ≰_{B} E. Most of the Borel equivalence relations that we shall consider in this paper arise from group actions as follows. Let G be a lcsc group; i.e. a locally compact second countable group. Then a standard Borel G-space is a standard Borel space X equipped with a Borel action (g, x) ↦ g · x of G on X. The

Research partially supported by NSF Grants.

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corresponding G-orbit equivalence relation on X, which we shall denote by E_{G}^{X}, is a Borel equivalence relation. In fact, by Kechris [6], E_{G}^{X} is Borel bireducible with a countable Borel equivalence relation. Conversely, by Feldman-Moore [5], if E is an arbitrary countable Borel equivalence relation on the standard Borel space X, then there exists a countable group G and a Borel action of G on X such that E = E_{G}^{X}.

We are now ready to discuss the two questions that will be answered in this paper.

Definition 1.1. Let E be a Borel equivalence relation on the standard Borel space X and let 1 ≤ n ≤ ω. Then nE is the Borel equivalence relation on X × n defined by

(⟨x, i⟩, ⟨y, j⟩) ∈ nE iff (x, y) ∈ E and i = j.

We also write nE = \underbrace{E ⊕ ⋯ ⊕ E}_{n}.

A longstanding open question asked whether there existed a (not necessarily countable) Borel equivalence relation E such that E <_{B} E ⊕ E. In Section 3, we shall show that each of Adams’ examples satisfies nE <_{B} (n+1)E for all 1 ≤ n < ω. Of course, this implies that these relations also satisfy nE <_{B} ωE for all 1 ≤ n < ω.

If E, F are Borel equivalence relations, then we say that E is Borel embeddable in F and write E ⊑_{B} F if there exists an injective Borel reduction from E to F. We write E ≈_{B} F if both E ⊑_{B} F and F ⊑_{B} E. A countable Borel equivalence relation E is said to be aperiodic if every E-equivalence class is infinite. For example, each of Adams’ examples has the form E_{Γ}^{X} for some free Borel action of a countably infinite group Γ and hence is aperiodic. (Here the action of the group Γ on the standard Borel space X is said to be free if γ · x ≠ x for all x ∈ X and 1 ≠ γ ∈ Γ.) In [4], Dougherty-Jackson-Kechris asked the following question and noted that it was equivalent to asking whether E ×I(ω) ⊑_{B} E for each aperiodic countable Borel equivalence relation E.

Question 1.2. Suppose that E, F are aperiodic countable Borel equivalence relations. If E ∼_{B} F, is it necessarily true that E ≈_{B} F?

Definition 1.3. Let E be a Borel equivalence relation on the standard Borel space X and let 1 ≤ n ≤ ω. Then E × I(n) is the Borel equivalence relations on X × n defined by

(⟨x, i⟩, ⟨y, j⟩) ∈ E × I(n) iff (x, y) ∈ E.

If E is any countable Borel equivalence relation, then clearly E ∼_{B} E × I(n) for all 1 ≤ n ≤ ω. In Section 3, we shall answer Question 1.2 negatively by showing that each of Adams’ examples satisfies E × I(2) ⋢_{B} E.

This paper is organized as follows. In Section 2, we shall recall some basic notions from ergodic theory and the theory of Lie groups. In Section 3, we shall describe Adams’ examples and prove our main theorem, modulo a technical lemma which will be proved in Section 4. Acknowledgements: I would like to thank Scot Adams and Alexander Kechris for very helpful discussions concerning the material in this paper.

2. Preliminaries

In this section, we shall recall some basic notions from ergodic theory and the theory of Lie groups. Let Γ be a countable group and let X be a standard Borel Γ-space. Throughout this paper, a probability measure on X will always mean a Borel probability measure; i.e. a measure which is defined on the collection of Borel subsets of X. The probability measure μ on X is Γ-invariant iff μ(γ(A)) = μ(A) for every γ ∈ Γ and Borel subset A ⊆ X. In the proof of Lemma 3.4, we shall make use of the following well-known result.

Proposition 2.1. Suppose that Γ is a countable group and that X is a standard Borel Γ-space with an invariant probability measure μ. If A is a Borel subset of X and g : A → X is a Borel map such that g ⊆ E_{Γ}^{X}, then μ(g(A)) ≤ μ(A).

If μ is Γ-invariant measure, then the action of Γ on X is said to be ergodic iff for every Γ-invariant Borel subset A ⊆ X, either μ(A) = 0 or μ(A) = 1. In this case, we shall also say that μ is an ergodic probability measure. The following characterization of ergodicity is well-known.

Theorem 2.2. If μ is a Γ-invariant probability measure on the standard Borel Γ-space X, then the following statements are equivalent.

(i) The action of Γ on X is ergodic.

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(ii) If Y is a standard Borel space and f : X → Y is a Γ-invariant Borel function, then there exists a Γ-invariant Borel subset M ⊆ X with μ(M) = 1 such that f ↾ M is a constant function.

The action of Γ on X is said to be uniquely ergodic iff there exists a unique Γ-invariant probability measure μ on X. In this case, it is well-known that μ must be ergodic. (For example, see Bekka-Mayer [3, Section I.3].)

Suppose that Y is a standard Borel space and that f : X → Y is a Borel function. Then for every probability measure ν on X, we can define a corresponding probability measure f_{∗}ν on Y by

(f_{∗}ν)(A) = ν(f^{−1}(A))

for every Borel subset A ⊆ Y. Now suppose that ν is Γ-invariant and that Y is also a standard Borel Γ-space. Then the Borel function f : X → Y is said to be almost Γ-equivariant iff for all γ ∈ Γ, γ · f(x) = f(γ · x) for ν-a.e. x ∈ X. In this case, it is easily checked that f_{∗}ν is also Γ-invariant.

Finally let G be a connected centerless simple Lie group and let ν be a fixed Haar measure on G. Suppose that Γ is a discrete subgroup of G and let G/Γ be the set of left cosets of Γ in G. By the Effros-Mackey cross section theorem [9, Theorem 5.4.2], there exists a Borel transversal T for G/Γ; i.e. a Borel subset T ⊆ G which meets every coset in a unique element. The covolume ν(G/Γ) of Γ in G is defined to be ν(T) and Γ is said to be a lattice if ν(G/Γ) < ∞. (It is an easy exercise to show that the covolume ν(G/Γ) is independent of the choice of the transversal T.)

3. The example

In this section, we shall describe Adams’ examples and prove our main theorem, modulo a technical lemma which will be proved in Section 4. Adams’ examples are based on the existence of groups G, Γ, K and a homomorphism ρ : Γ → K satisfying the following conditions.

Hypothesis 3.1. (a) G is a connected centerless simple Lie group such that

ℝ-rank(G) ≥ 2.

(b) Γ is a lattice in G.

(b) K is a compact connected simple Lie group.

SOME APPLICATIONS OF SUPERRIGIDITY 5

(c) ρ : Γ → K is an injective homomorphism such that ρ(Γ) is dense in K.

A well-known arithmetic construction yields many examples of G, Γ, K and ρ satisfying Hypothesis 3.1. (For example, see [7], [10].) For the sake of concreteness, we shall next describe one particular instance of this situation. Clear and detailed proofs of the following assertions can be found in Adams [1, Section 3]. Readers who are not interested in seeing an actual instance of Hypothesis 3.1 can simply skip this example.

Example 3.2. Let k = ℚ(\sqrt{2}) and let O = ℤ[\sqrt{2}] be the ring of algebraic integers in k. Let F be the quadratic form defined over O by

F (x_{1}, …, x_{5}) = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} − \sqrt{2}x_{4}^{2} − \sqrt{2}x_{5}^{2};

and let SO(F) be the group of 5 × 5 matrices with real entries which have determinant 1 and preserve the form F. Then SO(F) is a centerless Lie group of ℝ-rank 2 and the subgroup Λ consisting of the matrices with entries in the ring O is a lattice in SO(F).

Next let σ be the nontrivial element of the Galois group Gal(k/ℚ) and let F′ = F^{σ} be the quadratic form defined over O by

F′(x_{1}, …, x_{5}) = x_{1}^{2} + x_{2}^{2} + x_{3}^{2} + \sqrt{2}x_{4}^{2} + \sqrt{2}x_{5}^{2}.

Let K = SO(F′) be the group of 5 × 5 matrices with real entries which have determinant 1 and preserve the form F′. Then K is a compact connected simple Lie group. For each g ∈ Λ, let φ(g) be the result of applying the field automorphism σ to each entry of the matrix g. Then φ : Λ → K is an embedding such that φ(Λ) is dense in K.

Finally let G = SO(F)^{0} be the connected component of the identity in the Hausdorff topology, let Γ = Λ ∩ G and let ρ = φ ↾ Γ. Then G is a connected centerless simple Lie group of ℝ-rank 2 and [SO(F) : G] = [Λ : Γ] = 2. Since Λ is a lattice in SO(F), it follows that Γ is a lattice in G. Furthermore, ρ(Γ) is also dense in K. (To see that this last assertion holds, note that the closure H of ρ(Γ) in K satisfies

[K : H] ≤ [φ(Λ) : φ(Γ)] = 2

and that the connected group K has no proper subgroups of finite index.)

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For the remainder of this paper, fix G, Γ, K and ρ satisfying Hypothesis 3.1. Let X = K and let μ be the Haar probability measure on X. Define a free action of Γ on X by γ · x = ρ(γ)x and let E = E_{Γ}^{X} be the corresponding orbit equivalence relation. Then μ is clearly Γ-invariant. In fact, by Adams [1, Lemma 4.2], the action of Γ on X is uniquely ergodic.

Theorem 3.3. With the above hypotheses,

(a) nE <_{B} (n + 1)E for all n ≥ 1; and

(b) E × I(2) ⋢_{B} E.

Theorem 3.3 is an easy consequence of the following lemma, which will be proved in Section 4.

Lemma 3.4. Suppose that X_{0} is a Γ-invariant Borel subset of X with μ(X_{0}) = 1 and that f : X_{0} → X is a Borel map such that (x, y) ∈ E iff (f(x), f(y)) ∈ E. Then

(a) μ(Γ · f(X_{0})) = 1; and

(b) if f is injective, then μ(f(X_{0})) = 1.

Proof of Theorem 3.3. Suppose that (n+1)E ≤_{B} nE and let h : X×(n+1) → X×n be a Borel map such that

(⟨x, i⟩, ⟨y, j⟩) ∈ (n + 1)E iff (h(x, i), h(y, j) ∈ nE.

For each ℓ < n+1, let h_{ℓ} : X → X ×n be the Borel map defined by h_{ℓ}(x) = h(x, ℓ); and for each m < n, let

X_{ℓ,m} = {x ∈ X | h_{ℓ}(x) ∈ X × {m}}.

Then the ergodicity of μ implies that for each ℓ < n + 1, there exists a unique integer m(ℓ) < n such that μ(X_{ℓ,m(ℓ)}) = 1; and applying Lemma 3.4, we see that for each ℓ < n + 1,

μ(Γ · h_{ℓ}(X_{ℓ,m(ℓ)})) = 1.

(Here we are slightly abusing notation by also writing μ for the corresponding probability measure on X ×{m(ℓ)}.) Let k, ℓ < n+1 be distinct integers such that m(k) = m(ℓ) = m. Then we must have that

Γ · h(X × {k}) ∩ Γ · h(X × {ℓ}) ⊇ Γ · h_{k}(X_{k,m}) ∩ Γ · h_{ℓ}(X_{ℓ,m}) ≠ ∅,

which is a contradiction. A similar argument shows that E × I(2) ⋢_{B} E.

4. The proof of Lemma 3.4

In this section, we shall prove Lemma 3.4. So suppose that X_{0} is a Γ-invariant Borel subset of X with μ(X_{0}) = 1 and that f : X_{0} → X is a Borel map such that

Γ · x = Γ · y iff Γ · f(x) = Γ · f(y)

for all x, y ∈ X_{0}. Examining the proof of Adams [1, Lemma 4.6], we see that there exist

(i) a Lie group automorphism ψ ∈ Aut(G), (ii) a Borel function f̃: X_{0} → X, and (iii) a Γ-invariant Borel subset X_{1} ⊆ X_{0} such that the following conditions are satisfied:

(1) ψ(Γ) ⩽ Γ;

(2) μ(X_{1}) = 1;

(3) f̃(x) ∈ Γ · f(x) for all x ∈ X_{1}; and

(4) f̃(γ · x) = ψ(γ) · f̃(x) for all γ ∈ Γ and x ∈ X_{1}. (It should be mentioned that the proof of Adams [1, Lemma 4.6] makes use of both Zimmer’s superrigidity theorem [13, Theorem 5.2.5] and Ratner’s measure classification theorem [8, Theorem 3].)

First note that ψ(Γ) = Γ. To see this, fix a Haar measure ν on G and consider the covolume ν(G/Γ) of Γ in G. Since G is a connected centreless simple Lie group, it follows that the Lie group automorphism ψ ∈ Aut(G) preserves the Haar measure ν; and this implies that ν(G/Γ) = ν(G/ψ(Γ)). (For example, see Adams [1, Section 4].) Since ν(G/ψ(Γ)) = [Γ : ψ(Γ)]ν(G/Γ), it follows that ψ(Γ) = Γ.

Next note that f̃(X_{1}) ⊆ Γ·f(X_{0}). Also notice that if f is an injective map, then we can define a surjective Borel map g : f(X_{1}) → f̃(X_{1}) by g(f(x)) = f̃(x); and since g ⊆ E, it follows that μ(f(X_{1})) ≥ μ(f̃(X_{1})). Hence to complete the proof of Lemma 3.4, it is enough to show that μ(f̃(X_{1})) = 1. To see this, note that since the action of Γ on X preserves μ, it follows that the action of ψ(Γ) on X preserves f̃_{∗}μ. As ψ(Γ) = Γ and the action of Γ on X is uniquely ergodic, this implies that f̃_{∗}μ = μ and so μ(f̃(X_{1})) = (f̃_{∗}μ)(f̃(X_{1})) = 1.

References

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[7] G. A. Margulis, Some remarks on invariant means, Monatshefte fur Mathematik 90 (1980), 233-235.

[8] M. Ratner, On Raghunathan’s measure conjecture, Ann. Math. 134 (1991), 545-607.

[9] S. M. Srivastava, A Course on Borel Sets, Graduate Texts in Mathematics 180, Springer-Verlag, 1998.

[10] D. Sullivan, For n > 3, there is only one finitely additive rotationally invariant measure on the n-sphere on all Lebesgue measurable sets, Bull. Amer. Math. Soc. 1 (1981), 121-123.

[11] S. Thomas, On the complexity of the classification problem for torsion-free Abelian groups of rank two, to appear in Acta Math.

[12] S. Thomas, The classification problem for torsion-free Abelian groups of finite rank, preprint, 2000.

[13] R.Zimmer, Ergodic Theory and Semisimple Groups, Birkhäuser, 1984.

Mathematics Department, Rutgers University, 110 Frelinghuysen Road, Piscataway, New Jersey 08854-8019, USA

E-mail address: sthomas@math.rutgers.edu