PROPERTY (τ) AND COUNTABLE BOREL EQUIVALENCE RELATIONS

SIMON THOMAS

Abstract. We prove Borel superrigidity results for suitably chosen actions of groups of the form SL_{2}(ℤ[1/p_{1}, ⋯, 1/p_{t}]), where {p_{1}, …, p_{t}} is a finite nonempty set of primes, and present a number of applications to the theory of countable Borel equivalence relations. In particular, for each prime q, we prove that the orbit equivalence relations arising from the natural actions of SL_{2}(ℤ[1/q]) on the projective lines ℚ_{p} ∪ {∞}, p ≠ q, over the various p-adic fields are pairwise incomparable with respect to Borel reducibility.

1. Introduction

Let p be a prime and let P G(1, ℚ_{p}) = ℚ_{p} ∪ {∞} be the projective line over the field ℚ_{p} of p-adic numbers. Then GL_{2}(ℚ) has a natural Borel action on P G(1, ℚ_{p}) as a group of fractional linear transformations; i.e.

(\begin{matrix}a & b \\ c & d \end{matrix}) · x = \frac{ax + b}{cx + d}

for each x ∈ P G(1, ℚ_{p}). In this paper, we shall prove Borel superrigidity results for the corresponding actions of suitably chosen subgroups Γ ⩽ GL_{2}(ℚ). More specifically, for each finite nonempty set S = {p_{1}, ⋯, p_{t}} of primes, let

Γ_{S} = SL_{2}(ℤ[1/p_{1}, ⋯, 1/p_{t}])

and let E_{S}^{p} be the orbit equivalence relation induced by the action of Γ_{S} on P G(1, ℚ_{p}).

Theorem 1.1. Suppose that p, q are primes and that S, T are finite nonempty sets of primes such that p ∉ S and q ∉ T. If (p, S) ≠ (q, T), then E_{S}^{p} and E_{T}^{q} are incomparable with respect to Borel reducibility.

1991 Mathematics Subject Classification. Primary 03E15, 37A20.

Key words and phrases. Borel equivalence relation, Property (τ), Superrigidity.

Research partially supported by NSF Grants.

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In Thomas [30], an analogous result was proved for higher dimensional projective spaces; namely, if n > 2, then the orbit equivalence relations arising from the natural actions of SL_{n}(ℤ) on the (n − 1)-dimensional projective spaces P G(n − 1, ℚ_{p}) over the various p-adic fields are pairwise incomparable with respect to Borel reducibility. However, it remains an open problem whether the orbit equivalence relations arising from the actions of SL_{2}(ℤ) on the projective lines P G(1, ℚ_{p}) over the various p-adic fields are also pairwise incomparable with respect to Borel reducibility. (If true, this would show that there exist infinitely many treeable countable Borel equivalence relations up to Borel bireducibility.) In order to understand the essential differences between the cases of P G(1, ℚ_{p}) and P G(n − 1, ℚ_{p}) for n > 2, it will be helpful to review the main ingredients of the proof for higher dimensional spaces. These consist of:

(i) Zimmer’s measure theoretic superrigidity results [34] for higher rank simple Lie groups;

(ii) the techniques of Adams-Kechris [2] and Adams [1]; and (iii) Ratner’s measure classification theorem [24] for real Lie groups.

Unfortunately, the analogue of Zimmer’s measure theoretic superrigidity theorem fails for the low rank Lie group SL_{2}(ℝ) and hence there is no hope of applying the ideas of Thomas [30] to the action of the lattice SL_{2}(ℤ) on P G(1, ℚ_{p}). However, let S = {p_{1}, ⋯, p_{t}} be a finite nonempty set of primes. By Zimmer [34, 10.1.1] and [34, 10.1.6], if we identify Γ_{S} = SL_{2}(ℤ[1/p_{1}, ⋯, 1/p_{t}]) with its image under the diagonal embedding into

G = SL_{2}(ℝ) × SL_{2}(ℚ_{p_{1}}) × ⋯ × SL_{2}(ℚ_{p_{t}}),

then Γ_{S} is a lattice in G and the analogue of Zimmer’s measure theoretic superrigidity theorem holds for G. Furthermore, by Margulis-Tomanov [21], the analogue of Ratner’s measure classification theorem also holds for G. However, a serious problem arises when we attempt to adapt the ideas of Adams-Kechris [2] to our setting: many of their arguments make essential use of the fact that lattices in higher rank simple Lie groups have the Kazhdan property and it is well-known that Γ_{S} does not have the Kazhdan property. (In fact, by Zimmer [35], SL_{2}(ℂ) does not contain any finitely generated subgroups with the Kazhdan property.) Fortunately, Γ_{S} does

PROPERTY (τ) AND COUNTABLE BOREL EQUIVALENCE RELATIONS 3

have Property (τ), a weak form of the Kazhdan property, and this turns out to be enough to prove Theorem 1.1, as well as Γ_{S}-analogues of the other theorems of Thomas [30]. For example, for each (possibly infinite) nonempty set of primes J such that S ∩ J = ∅, let

K(J) = ∏_{p∈J} SL_{2}(ℤ_{p}),

where ℤ_{p} is the ring of p-adic integers. Then we can regard Γ_{S} as a subgroup of K(J) via the diagonal embedding. Let E_{S}^{J} be the orbit equivalence relation arising from the action of Γ_{S} on K(J) via left translations. Theorem 1.2. Suppose that S_{1}, S_{2} are finite nonempty sets of primes and that J_{1}, J_{2} are (possibly infinite) nonempty sets of primes such that S_{1} ∩ J_{1} = S_{2} ∩ J_{2} = ∅. If (J_{1}, S_{1}) ≠ (J_{2}, S_{2}), then E_{S_{1}}^{J_{1}} and E_{S_{2}}^{J_{2}} are incomparable with respect to Borel reducibility.

The basic idea of this paper can be summarized as follows. In Adams-Kechris [2], most applications of the Kazhdan property exploit the incompatibility of the actions of Kazhdan groups and amenable groups; and, to a large extent, this incompatibility is captured by the notion of E_{0}-ergodicity. In Section 5, we shall prove the E_{0}- ergodicity of orbit equivalence relations arising from dense embeddings of finitely generated groups with Property (τ) into profinite groups. Then, in Section 6, we shall see that the actions covered by Theorems 1.1 and 1.2 fit into this context and hence are E_{0}-ergodic. Combined with the fact that every proper algebraic subgroup of SL_{2}(ℚ_{p}) or SL_{2}(ℝ) is amenable, this will enable us to prove that various associated cocycles are Zariski dense in their target groups, and thus reduce our analysis to a situation where Zimmer’s measure theoretic superrigidity results [34] can be applied.

However, there are applications of the Kazhdan property which do not seem to follow directly from E_{0}-ergodicity. For example, suppose that G is a Kazhdan group and that X is a standard Borel G-space with an invariant ergodic probability measure. By Zimmer [36, 2.2], if H is a countable group, then for every cocycle α : G × X → H, there exists an equivalent cocycle β such that β(G × X) is contained in a finitely generated subgroup of H. In an earlier version of this paper, I conjectured that the following analogous result holds in the setting of this paper; and I pointed out that, combined with the methods of this paper, this would imply

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that if p ≠ q are distinct primes, then the classification problems for p-local and q-local torsion-free abelian groups of rank two are incomparable with respect to Borel reducibility. (The analogous result for the p-local groups of rank n ≥ 3 was proved in Thomas [31].)

Statement 1.3. Let p ≠ q be distinct primes and let H be a countable group. Then for every cocycle α : P SL_{2}(ℤ[1/q]) × P G(1, ℚ_{p}) → H, there exists an equivalent cocycle β such that β(P SL_{2}(ℤ[1/q])×P G(1, ℚ_{p})) is contained in a finitely generated subgroup of H.

Soon afterwards, Greg Hjorth found an elegant proof of Statement 1.3. Hjorth’s proof, together with the above application, can be found in Hjorth-Thomas [15].

This paper is organised as follows. In Section 2, we shall recall some basic notions and results concerning Borel equivalence relations, ergodic theory and irreducible lattices in products of real and p-adic algebraic groups. In Section 3, we shall formulate a Borel superrigidity theorem for a suitably chosen class of actions of the groups of the form P SL_{2}(ℤ[1/p_{1}, ⋯, 1/p_{t}]), where S = {p_{1}, ⋯, p_{t}} is a finite nonempty set of primes, and then present a number of easy applications, including Theorems 1.1 and 1.2. In Section 4, we shall prove that if E is an E_{0}-ergodic countable Borel equivalence relation and F is an orbit equivalence relation arising from a Borel action of an arbitrary locally compact second countable amenable group, then E is F -ergodic. In Section 5, we shall prove the E_{0}-ergodicity of the orbit equivalence relations arising from dense embeddings of finitely generated groups with Property (τ) into profinite groups. Then, in Section 6, we shall show that the countable Borel equivalence relations covered by our Borel superrigidity theorem fit into this context and hence are E_{0}-ergodic. In Section 7, we shall prove a measure classification theorem for lattices in products of real and p-adic algebraic groups, modulo a result of Witte Morris that will be proved in Appendix A. Finally, in Section 8, we shall present the proof of our Borel superrigidity theorem. Acknowledgements: I would like to thank Alexander Kechris and Dave Witte Morris for providing me with the proofs of Theorems 4.7 and 7.4 respectively. This paper would not have been possible without their invaluable assistance. I would also like to thank Greg Hjorth for some very helpful discussions concerning the material in this paper.

PROPERTY (τ) AND COUNTABLE BOREL EQUIVALENCE RELATIONS 5

2. Preliminaries

In this section, we shall recall some basic notions and results concerning Borel equivalence relations, ergodic theory and irreducible lattices in products of real and p-adic algebraic groups.

2.1. 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 [x]_{E} is countable. The subset A ⊆ X is said to be E-invariant iff A = ⋃{[x]_{E} | x ∈ A}.

If E, F are Borel equivalence relations on the standard Borel spaces X, Y respectively, then the Borel map f : X → Y is said to be a Borel homomorphism from E to F iff xEy implies f(x)F f(y). If f satisfies the stronger condition that xEy iff f(x)F f(y), then f is said to be a Borel reduction. In this case, we say that E is Borel reducible to F and write E ≤_{B} 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 corresponding G-orbit equivalence relation on X, which we shall denote by E_{G}^{X}, is a Borel equivalence relation. By Kechris [18], there exists a complete countable Borel section X_{0} for E_{G}^{X}; i.e. X_{0} ⊆ X is a Borel subset which intersects every G-orbit in a nonempty countable subset. Furthermore, E_{G}^{X} is Borel bireducible with the countable Borel equivalence relation E_{G}^{X} ↾ X_{0}. Conversely, by Feldman-Moore [8], 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}.

A detailed development of the general theory of countable Borel equivalence relations can be found in Jackson-Kechris-Louveau [16].

Throughout this paper, a probability measure on a standard Borel space X will always mean a Borel probability measure; i.e. a measure which is defined on the

6 SIMON THOMAS

collection of Borel subsets of X. Suppose that X, Y are a standard Borel spaces and that f : X → Y is a Borel function. If ν is a probability measure on X, then f_{∗}ν denotes the corresponding probability measure on Y defined by

(f_{∗}ν)(A) = ν(f^{−1}(A))

for every Borel subset A ⊆ Y.

If G is a lcsc group and X is a standard Borel G-space, then the probability measure μ on X is G-invariant iff μ(g(A)) = μ(A) for every g ∈ G and Borel subset A ⊆ X. If E is a countable Borel equivalence relation on the standard Borel space X and μ is a probability measure on X, then μ is said to be E-invariant iff μ is H-invariant for some (equivalently every) countable group H and Borel action of H on X such that E = E_{H}^{X}.

Suppose that E, F are Borel equivalence relations on the standard Borel spaces X, Y respectively and that μ is an E-invariant probability measure on X. Then (E, μ) is said to be F -ergodic iff for every Borel homomorphism f : X → Y from E to F, there exists an E-invariant Borel subset M ⊆ X with μ(M) = 1 such that f maps M into a single F -class. In this case, for simplicity, we shall often say that E is F -ergodic.

Throughout this paper, E_{0} will always denote the Vitali equivalence relation, defined on 2^{ℕ} by xE_{0}y iff x(n) = y(n) for almost all n.

2.2. Ergodic theory. Suppose that G is a lcsc group and that X is a standard Borel G-space. If μ is a G-invariant probability measure on X, then the action of G on (X, μ) is said to be ergodic iff for every G-invariant Borel subset A ⊆ X, either μ(A) = 0 or μ(A) = 1. The following characterization of ergodicity is wellknown. (In other words, letting Δ(Y) be the identity relation on Y, the action of G on (X, μ) is ergodic iff the corresponding orbit equivalence relation E_{G}^{X} is Δ(Y)-ergodic for every standard Borel space Y.)

Theorem 2.1. If μ is a G-invariant probability measure on the standard Borel G-space X, then the following statements are equivalent.

(i) The action of G on (X, μ) is ergodic.

PROPERTY (τ) AND COUNTABLE BOREL EQUIVALENCE RELATIONS 7

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

The action of G on X is said to be uniquely ergodic iff there exists a unique G-invariant probability measure μ on X. In this case, it is well-known that μ must be ergodic. (For example, see Bekka-Mayer [4, Section I.3].)

Suppose that Γ is a countable group and that X is a standard Borel Γ-space with an invariant ergodic probability measure μ. Let Λ ⩽ Γ be a subgroup such that [Γ : Λ] < ∞. Then a Λ-invariant Borel subset Z ⊆ X is said to be an ergodic component for the action of Λ on X iff

• μ(Z) > 0; and

• Λ acts ergodically on (Z, μ_{Z}), where μ_{Z} is the probability measure defined

on Z by μ_{Z}(A) = μ(A)/μ(Z). It is easily checked that there exists a partition Z_{1} ⊔ ⋯ ⊔ Z_{d} of X into finitely many ergodic components and that the collection of ergodic components is uniquely determined up to μ-null sets. Furthermore, if the action of Γ on X is uniquely ergodic, then the action of Λ on each ergodic component Z ⊆ X is also uniquely ergodic.

Finally suppose that K is a compact second countable group and that L is a closed subgroup. Let ν be the Haar probability measure on K and let π : K → K/L be the canonical surjection. Then μ = π_{∗}ν is called the Haar probability measure on K/L. It is well-known that μ is the unique K-invariant probability measure μ on the standard Borel K-space K/L. Throughout this paper, we shall make use of the following easy observations. Lemma 2.2. Let K be a compact second countable group, let L ⩽ K be a closed subgroup and let μ be the Haar probability measure on K/L. If Γ is a countable dense subgroup of K, then the following properties hold.

(a) The action of Γ on K/L is uniquely ergodic; i.e. μ is the unique Γ-invariant probability measure on K/L.

(b) Suppose that Λ ⩽ Γ is a subgroup such that [Γ : Λ] < ∞ and that H = \overline{Λ} is the closure of Λ in K. Then the ergodic components for the action of Λ on K/L are precisely the H-orbits {H · x | x ∈ K/L}.

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(c) Suppose that • Γ^{+} ⩽ K is a subgroup such that Γ ⩽ Γ^{+},

• Λ ⩽ Γ is a subgroup such that [Γ : Λ] < ∞ and Z is an ergodic

component for the action of Λ on K/L. If Δ = {g ∈ Γ^{+} | g(Z) = Z mod μ-null sets }, then [Γ^{+} : Δ] < ∞.

Proof. Clauses 2.2(a) and 2.2(b) are included in Thomas [30, 2.2]. To see that Clause 2.2(c) holds, let H = \overline{Λ} be the closure of Λ in K. Then [K : H] < ∞ and we can suppose that there exists x ∈ K/L such that Z = H · x. It follows that [Γ^{+} : Γ^{+} ∩ H] < ∞ and that Γ^{+} ∩ H ⩽ Δ. □

2.3. Lattices and algebraic groups. Throughout this paper, ℙ will denote the set of primes. For each prime p ∈ ℙ, the field of p-adic numbers will be denoted by ℚ_{p} and the ring of p-adic integers will be denoted by ℤ_{p}. We shall often use ℚ_{∞} to denote the field ℝ of real numbers.

Suppose that p ∈ ℙ ∪ {∞} and that G(ℚ_{p}) ⩽ GL_{n}(ℚ_{p}) is an algebraic ℚ_{p}- group. Then G(ℚ_{p}) is a lcsc group with respect to the Hausdorff topology; i.e. the topology obtained by restricting the natural topology on ℚ_{p}^{n^{2}} to G(ℚ_{p}). More generally, suppose that T is a finite subset of ℙ ∪ {∞} and that G_{p}(ℚ_{p}) is an algebraic ℚ_{p}-group for each p ∈ T. Then G = ∏_{p∈T} G_{p}(ℚ_{p}) is a lcsc group with respect to the product topology. Any topological notions concerning the groups of the form G = ∏_{p∈T} G_{p}(ℚ_{p}) will always refer to this topology.

Suppose that G is a lcsc group and that ν is a Haar measure on G. The discrete subgroup Γ ⩽ G is a said to be a lattice iff ν(G/Γ) is finite. Now suppose that G = ∏_{p∈T} G_{p}(ℚ_{p}), where T is a finite subset of ℙ ∪ {∞} and G_{p}(ℚ_{p}) is an almost ℚ_{p}-simple algebraic ℚ_{p}-group for each p ∈ T. (Recall that an algebraic ℚ_{p}-group is said to be almost ℚ_{p}-simple if every proper normal algebraic ℚ_{p}-subgroup is finite.) For each nonempty subset A ⊆ T, let G_{A} = ∏_{p∈A} G_{p}(ℚ_{p}). Then the lattice Γ in G is said to be irreducible iff for each nonempty subset A ⊆ T, Γ projects onto a dense subgroup of G/G_{A}. For example, let S = {p_{1}, ⋯, p_{t}} be a finite nonempty set of primes and let

Γ_{S} = SL_{2}(ℤ[1/p_{1}, ⋯, 1/p_{t}]).

PROPERTY (τ) AND COUNTABLE BOREL EQUIVALENCE RELATIONS 9

By Margulis [20, IX.1.7(iii)], if we identify Γ_{S} with its image under the diagonal embedding into

G = SL_{2}(ℝ) × SL_{2}(ℚ_{p_{1}}) × ⋯ × SL_{2}(ℚ_{p_{t}}),

then Γ_{S} is an irreducible lattice in G.

3. A Borel superrigidity theorem

In this section, we shall first formulate a Borel superrigidity theorem for a suitably chosen class of actions of the groups of the form

Λ_{S} = P SL_{2}(ℤ[1/p_{1}, ⋯, 1/p_{t}]),

where S = {p_{1}, ⋯, p_{t}} is a finite nonempty set of primes, and then we shall present a number of easy applications, including Theorems 1.1 and 1.2. The statement of our Borel superrigidity theorem involves the notion of a virtual isomorphism, which is defined as follows. (Our definition of a virtual isomorphism is slightly more stringent than that of Furman [10]. However, both definitions coincide in the special case when Γ_{1}, Γ_{2} have the form Λ_{S_{1}}, Λ_{S_{2}} for finite nonempty sets of primes S_{1}, S_{2}.) Definition 3.1. For each i = 1, 2, let Γ_{i} be a countable group and let X_{i} be a standard Borel Γ_{i}-space with an invariant ergodic probability measure μ_{i}.

(a) (X_{1}, Γ_{1}, μ_{1}) and (X_{2}, Γ_{2}, μ_{2}) are isomorphic iff there exist Γ_{i}-invariant Borel subsets Y_{i} ⊆ X_{i} with μ_{i}(Y_{i}) = 1, a Borel bijection f : Y_{1} → Y_{2} and an isomorphism φ : Γ_{1} → Γ_{2} such that

• f_{∗}μ_{1} = μ_{2}; and

• f(γ · x) = φ(γ) · f(x) for all γ ∈ Γ_{1} and x ∈ Y_{1}.

(b) (X_{1}, Γ_{1}, μ_{1}) and (X_{2}, Γ_{2}, μ_{2}) are virtually isomorphic iff there exist subgroups Δ_{i} ⩽ Γ_{i} with [Γ_{i} : Δ_{i}] < ∞ and ergodic components Z_{i} for the actions of Δ_{i} on X_{i} such that (Z_{1}, Δ_{1}, (μ_{1})_{Z_{1}}) and (Z_{2}, Δ_{2}, (μ_{2})_{Z_{2}}) are isomorphic.

We shall begin with a short discussion of the specific actions which are covered by the theorem. Suppose S = {p_{1}, ⋯, p_{t}} is a finite nonempty set of primes and

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that J is a (possibly infinite) nonempty set of primes such that S ∩ J = ∅. Let

K(J) = ∏_{p∈J} SL_{2}(ℤ_{p}),

where ℤ_{p} is the ring of p-adic integers. Then K(J) is a compact second countable group and we can regard Γ_{S} as a subgroup of K(J) via the diagonal embedding. Furthermore, by the Strong Approximation Theorem [22, 7.12], SL_{2}(ℤ) is a dense subgroup of K(J) and hence Γ_{S} is also a dense subgroup of K(J). Let L be a closed subgroup of K(J) and let μ_{J} be the Haar probability measure on K(J)/L. Since Γ_{S} is a dense subgroup of K(J), it follows that the action of Γ_{S} on (K(J)/L, μ_{J}) is uniquely ergodic.

In the statement of our theorem, we shall suppose that L has been chosen so that the following two hypotheses hold. Hypothesis 3.2. L contains the center Z(Γ_{S}) of Γ_{S}.

Of course, this means that Z(Γ_{S}) acts trivially on K(J)/L; and hence we obtain an induced action of Λ_{S} = P SL_{2}(ℤ[1/p_{1}, ⋯, 1/p_{t}]) on K(J)/L. Let

F_{S}(J, L) = {x ∈ K(J)/L | γ · x ≠ x for all 1 ≠ γ ∈ Λ_{S}}.

In other words, F_{S}(J, L) is the Borel subset of K(J)/L on which Λ_{S} acts freely. Hypothesis 3.3. μ_{J}(F_{S}(J, L)) = 1. Remark 3.4. Clearly if Hypothesis 3.3 holds, then L has infinite index in K(J). It seems conceivable that the converse is also true.

We are now ready to state our Borel superrigidity theorem. Theorem 3.5. Suppose that S_{1}, S_{2} are finite nonempty sets of primes and that J_{1}, J_{2} are (possibly infinite) nonempty sets of primes such that S_{1} ∩ J_{1} = S_{2} ∩ J_{2} = ∅. For i = 1, 2, suppose that

• L_{i} is a closed subgroup of K(J_{i}) satisfying Hypotheses 3.2 and 3.3;

• X_{i} ⊆ F_{S_{i}}(J_{i}, L_{i}) is a Λ_{S_{i}}-invariant Borel subset such that μ_{J_{i}}(X_{i}) = 1; and • E_{i} is the Λ_{S_{i}}-orbit equivalence relation on X_{i}.

Suppose that f : X_{1} → X_{2} is a Borel reduction from E_{1} to E_{2}. Then:

(a) S_{1} = S_{2};

(b) (K(J_{1})/L_{1}, Λ_{S_{1}}, μ_{J_{1}}) and (K(J_{2})/L_{2}, Λ_{S_{2}}, μ_{J_{2}}) are virtually isomorphic;

(c) μ_{J_{2}}(Λ_{S_{2}} · f(X_{1})) = 1.

PROPERTY (τ) AND COUNTABLE BOREL EQUIVALENCE RELATIONS 11

Remark 3.6. In general, there is no reason to suppose that J_{1} = J_{2}. For example, let S = S_{1} = S_{2} be any finite nonempty set of primes; and let J_{1}, J_{2} be any nonempty sets of primes such that S ∩ J_{1} = S ∩ J_{2} = ∅ and I = J_{1} ∩ J_{2} ≠ ∅. For i = 1, 2, let

L_{i} = ∏_{p∈J_{i}∖I} SL_{2}(ℤ_{p}) × ∏_{p∈I} Z(SL_{2}(ℤ_{p})).

Then

K(J_{1})/L_{1} ≅ ∏_{p∈I} P SL_{2}(ℤ_{p}) ≅ K(J_{2})/L_{2}

and the orbit equivalence relations arising from the free actions of Λ_{S} on K(J_{1})/L_{1}, K(J_{2})/L_{2} are Borel bireducible. Remark 3.7. The proof of Theorem 3.5 shows that if S_{1} ⊈ S_{2}, then E_{1} is E_{2}-ergodic. However, this does not follow generally from the assumption that S_{1} ≠ S_{2}. For example, let S_{1}, S_{2} be finite nonempty sets of primes such that S_{1} ⊊ S_{2}; and let J_{1} = J_{2} = {p}, where p ∉ S_{2}. Let

L_{1} = Z(SL_{2}(ℤ_{p})) = L_{2}.

Then K(J_{1})/L_{1} = K(J_{2})/L_{2} and the identity map is a nontrivial Borel homomorphism from E_{1} to E_{2}.

The proof of Theorem 3.5 will be given in Section 8. In the remainder of this section, we shall illustrate the use of Theorem 3.5 by presenting a number of applications, beginning with the proofs of Theorems 1.1 and 1.2.

Proof of 1.1. Throughout this proof, we shall regard P G(1, ℚ_{p}) as the space of 1dimensional subspaces of the 2-dimensional vector space V (2, ℚ_{p}) over the p-adics. It is easily checked that SL_{2}(ℤ_{p}) acts transitively on P G(1, ℚ_{p}). (For example, see Thomas [30, 6.1].) Fix some point x_{p} ∈ P G(1, ℚ_{p}) and let L_{p} be the stabilizer of x_{p} in SL_{2}(ℤ_{p}). Then L_{p} is a closed subgroup of SL_{2}(ℤ_{p}) and we can identify P G(1, ℚ_{p}) with the coset space SL_{2}(ℤ_{p})/L_{p}. Regarding Γ_{S} = SL(ℤ_{S}) as a subgroup of SL_{2}(ℤ_{p}), it is clear that Z(Γ_{S}) acts trivially on P G(1, ℚ_{p}) and so Z(Γ_{S}) is contained in L_{p}. Hence Hypothesis 3.2 holds. Let

P G^{∗}(1, ℚ_{p}) = {x ∈ P G(1, ℚ_{p}) | γ · x ≠ x for all γ ∈ Γ_{S} ∖ Z(Γ_{S})}.

In other words, P G^{∗}(1, ℚ_{p}) is the Borel subset of P G(1, ℚ_{p}) on which Λ_{S} = P SL_{2}(ℤ_{S}) acts freely.

Claim 3.8. P G(1, ℚ_{p}) ∖ P G^{∗}(1, ℚ_{p}) is a countable set.

Proof of Claim 3.8. If x ∈ P G(1, ℚ_{p}) ∖ P G^{∗}(1, ℚ_{p}), then there exists an element γ ∈ Γ_{S} ∖ Z(Γ_{S})} such that x is an eigenspace of γ. □

In particular, if μ_{p} is the Haar probability measure on P G(1, ℚ_{p}) = SL_{2}(ℤ_{p})/L_{p}, we have that μ_{p}(P G^{∗}(1, ℚ_{p})) = 1 and so Hypothesis 3.3 also holds.

Now suppose that f : P G(1, ℚ_{p}) → P G(1, ℚ_{q}) is a Borel reduction from E_{S}^{p} to E_{T}^{q}. Recall that E_{S}^{p}, E_{T}^{q} are the orbit equivalence relations arising from the actions of Γ_{S}, Γ_{T} on P G(1, ℚ_{p}), P G(1, ℚ_{q}) respectively. Of course, E^{p} and E_{T}^{q} are also the orbit equivalence relations arising from the induced actions of Λ_{S} and Λ_{T}. By Claim 3.8, there exists a Borel subset X ⊆ P G^{∗}(1, ℚ_{p}) with μ_{p}(X) = 1 such that f(X) ⊆ P G^{∗}(1, ℚ_{q}). Hence, applying Theorem 3.5, we obtain that

• S = T; and

• (P G(1, ℚ_{p}), Λ_{S}, μ_{p}) and (P G(1, ℚ_{q}), Λ_{S}, μ_{q}) are virtually isomorphic.

Arguing as in the proof of Thomas [30, 6.3], we see that (P G(1, ℚ_{p}), Λ_{S}, μ_{p}) and (P G(1, ℚ_{q}), Λ_{S}, μ_{q}) are virtually isomorphic iff p = q. This completes the proof of Theorem 1.1. □

Proof of 1.2. Suppose that E_{S_{1}}^{J_{1}} ≤_{B} E_{S_{2}}^{J_{2}}. For each i = 1, 2, let Z_{i} = Z(Γ_{S_{i}}) and let E_{i} be the orbit equivalence relation arising from the free action of Λ_{S_{i}} on K(J_{i})/Z_{i}. Clearly E_{1} ≤_{B} E_{2}. Hence, applying Theorem 3.5, we obtain that

• S_{1} = S_{2}; and

• (K(J_{1})/Z_{1}, Λ_{S_{1}}, μ_{J_{1}}) and (K(J_{2})/Z_{2}, Λ_{S_{2}}, μ_{J_{2}}) are virtually isomorphic. Arguing as in the proof of Thomas [30, 5.1], we see that K(J_{1})/Z_{1} and K(J_{2})/Z_{2} contain open subgroups which are isomorphic as topological groups. By Gefter-Golodets [11, A.6], it follows that J_{1} = J_{2}. □

We shall end this section with two applications that also require the use of Clause 3.5(c). Definition 3.9. Let E be a Borel equivalence relation on the standard Borel space X and let n ≥ 1. Then nE is the Borel equivalence relation on X × n defined by

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

PROPERTY (τ) AND COUNTABLE BOREL EQUIVALENCE RELATIONS 13

In Thomas [28], answering a longstanding open question, it was pointed out that each of Adams’ examples in [1] satisfies nE <_{B} (n + 1)E for all n ≥ 1. Further examples of countable Borel equivalence relations E with this property were later presented in Thomas [30, 4.9] and Hjorth-Kechris [14, 3.9]. In each case, the main point was to show that Borel reductions from E to E satisfied the analogue of Clause 3.5(c). Hence, arguing as in Thomas [28], we obtain the following result. Theorem 3.10. Suppose that S is a finite nonempty set of primes and that J is a (possibly infinite) nonempty set of primes such that S ∩ J = ∅. Suppose that

• L is a closed subgroup of K(J) satisfying Hypotheses 3.2 and 3.3;

• X ⊆ F_{S}(J, L) is a Λ_{S}-invariant Borel subset such that μ_{J}(X) = 1; and • E is the Λ_{S}-orbit equivalence relation on X.

Then nE <_{B} (n + 1)E for all n ≥ 1.

□

Finally recall that a Borel equivalence relation E is essentially countable iff there exists a countable Borel equivalence relation F such that E ≤_{B} F. Recently, answering a question of Jackson-Kechris-Louveau [16], Hjorth [13] has constructed an example of an essentially countable Borel equivalence relation E such that there does not exist a countable Borel equivalence relation F with E ∼_{B} F. Theorem 3.5 can be used to provide further examples of such Borel equivalence relations, as follows. Theorem 3.11. Let S = {p_{1}, ⋯, p_{t}} be a finite nonempty set of primes and let E_{S} be the orbit equivalence relation induced by the free action of SL_{2}(ℤ[1/p_{1}, ⋯, 1/p_{t}]) on the standard Borel space

X = ⋃_{∅≠J⊆ℙ∖S}{J} × K(J),

defined by γ · (J, k) = (J, γk). Then there exists a Borel equivalence relation E such that

• E ≤_{B} E_{S}, and

• there does not exist a countable Borel equivalence relation F with E ∼_{B} F.

Proof. By Theorem 3.5, the family {E_{S}^{J} | ∅ ≠ J ⊆ ℙ ∖ S} of countable Borel equivalence relations satisfies Fact 1.2 of Hjorth [13]. The result follows. □

14 SIMON THOMAS

4. E_{0}-ergodicity and actions of amenable groups

In this section, we shall prove that if E is an E_{0}-ergodic countable Borel equivalence relation and F is an orbit equivalence relation arising from a Borel action of an arbitrary (not necessarily countable) lcsc amenable group, then E is F -ergodic. This result will play a crucial role in the proof of Theorem 3.5, where it will be used to show that various associated cocycles are Zariski dense in their target groups. First we need to review some of the theory of hyperfinite and amenable countable Borel equivalence relations, as developed in Jackson-Kechris-Louveau [16]. Definition 4.1. Let F be a countable Borel equivalence relation on the standard Borel space X.

(a) F is said to be hyperfinite iff there exists an increasing sequence

F_{0} ⊆ F_{1} ⊆ ⋯ ⊆ F_{n} ⊆ ⋯

of finite Borel equivalence relations such that F = ⋃_{n∈ℕ} F_{n}. (Here an equivalence relation is said to be finite iff each of its equivalence classes is finite.)

(b) If ν is a (not necessarily F -invariant) probability measure on X, then F is ν-hyperfinite iff there exists an F -invariant Borel subset M ⊆ X with ν(M) = 1 such that F ↾ M is hyperfinite.

For example, the Vitali equivalence relation E_{0}, defined on 2^{ℕ} by

xE_{0}y iff x(n) = y(n) for almost all n,

is hyperfinite. Furthermore, by Dougherty-Jackson-Kechris [7], if F is an arbitrary hyperfinite relation, then F ≤_{B} E_{0}. Hence if E is a Borel equivalence relation, then E is E_{0}-ergodic iff E is F -ergodic for every hyperfinite equivalence relation F.

It is an interesting open question whether every orbit equivalence relation E_{H}^{X} arising from a Borel action of a countable (discrete) amenable group H on a standard Borel space X is hyperfinite. However, by Jackson-Kechris-Louveau [16], it is known that each such equivalence relation E_{H}^{X} is ν-hyperfinite for every probability measure ν on X. At the end of this section, we shall present an appropriate generalization of this result to the orbit equivalence relations arising from Borel

PROPERTY (τ) AND COUNTABLE BOREL EQUIVALENCE RELATIONS 15

actions of arbitrary lcsc amenable groups. The proof will make use of the following characterization of lcsc amenable groups. Definition 4.2. Let H be a lcsc group and let μ be a right-invariant Haar measure on H. Then H satisfies Day’s condition iff there exists a sequence

φ_{n} : H → ℝ_{+}, n ∈ ℕ,

of nonnegative μ-measurable functions such that the following conditions hold: (a) For each n ∈ ℕ, ∫_{H} φ_{n} dμ = 1.

(b) For each n ∈ ℕ and h ∈ H, let φ_{n}^{h} : H → ℝ_{+} be the right-translate of φ_{n}

defined by φ_{n}^{h}(g) = φ_{n}(gh). Then for each h ∈ H,
||φ_{n}^{h} − φ_{n}||_{1} = ∫_{H} |φ_{n}^{h} − φ_{n}| dμ → 0 as n → ∞.

Theorem 4.3 (Day [5]). If H is a lcsc group, then H is amenable iff H satisfies Day’s condition.

□

Motivated by Day’s Theorem, Jackson-Kechris-Louveau [16] introduced the following generalization of the notion of hyperfiniteness. Definition 4.4. Suppose that E is a countable Borel equivalence relation on the standard Borel space X. Then E is said to be 1-amenable iff there exists a sequence

⟨f_{n} : E → ℝ^{+} | n ∈ ℕ⟩

of non-negative Borel functions such that, letting f_{n}^{x}(y) = f_{n}(x, y), the following conditions hold for all x, y ∈ X:

(a) f_{n}^{x} ∈ ℓ_{1}([x]_{E}) and ||f_{n}^{x}||_{1} = 1;

(b) if xEy, then ||f_{n}^{x} − f_{n}^{y}||_{1} → 0 as n → ∞.

Lemma 4.5 (Jackson-Kechris-Louveau [16]). (a) If E is hyperfinite, then E

is 1-amenable.

(b) If H is a countable (discrete) amenable group and X is a standard Borel H-space, then E_{H}^{X} is 1-amenable.

(c) If E is a 1-amenable equivalence relation on the standard Borel space X, then E is ν-hyperfinite for every probability measure ν on X.

16 SIMON THOMAS

Proof. Clauses 4.5(a) and 4.5(b) are included in Jackson-Kechris-Louveau [16, 2.13]. Since the statement of Clause 4.5(c) is projective, it is enough to prove it under the assumption that CH holds. But then the result is an immediate consequence of Proposition 2.13(ii) [16] and Theorem 2.6 [16]. □

It remains open whether every 1-amenable equivalence relation is hyperfinite. However, because of the following easy observation, this issue will not cause any problems in this paper. Lemma 4.6. Let E, F be countable Borel equivalence relations on the standard Borel spaces X, Y respectively and let μ be an E-invariant probability measure on X. Suppose that:

(a) (E, μ) is E_{0}-ergodic; and

(b) F is 1-amenable. Then (E, μ) is F -ergodic.

Proof. Suppose that f : X → Y is a Borel homomorphism from E to F. Let ν be the probability measure on Y defined by ν = f_{∗}μ. By Lemma 4.5(c), there exists an F -invariant Borel subset Y_{0} ⊆ Y with ν(Y_{0}) = 1 such that F ↾ Y_{0} is hyperfinite. Let X_{0} = f^{−1}(Y_{0}). Since (E, μ) is E_{0}-ergodic, it follows that there exists an Einvariant Borel subset X_{1} ⊆ X_{0} with μ(X_{1}) = 1 such that f maps X_{1} to a single F -equivalence class. □

By Lemmas 4.5(b) and 4.6, if (E, μ) is E_{0}-ergodic and F is an orbit equivalence relation arising from a Borel action of a countable (discrete) amenable group H, then (E, μ) is F -ergodic. By the next result, which is due to Alexander Kechris, this is also true if H is an arbitrary lcsc amenable group. Theorem 4.7 (Kechris). Suppose that H is a lcsc amenable group and that X is a standard Borel H-space. Let X_{0} be a complete countable Borel section for the corresponding orbit equivalence relation E_{H}^{X}. Then the countable Borel equivalence relation E = E_{H}^{X} ↾ X_{0} is 1-amenable.

Proof. Let μ be a right-invariant Haar measure on H and let

φ_{n} : H → ℝ_{+}, n ∈ ℕ,

PROPERTY (τ) AND COUNTABLE BOREL EQUIVALENCE RELATIONS 17

be a sequence of nonnegative measurable functions which satisfies conditions 4.2(a) and 4.2(b). For each n ∈ ℕ, let ν_{n} be the probability measure on H defined by

ν_{n}(A) = ∫_{H} χ_{A}φ_{n} dμ,

where χ_{A} is the characteristic function of the Borel subset A ⊆ H. Next for each x ∈ X, let π_{x} : H → X be the Borel function defined by π_{x}(h) = h · x. Then for each x ∈ X and n ∈ ℕ,

m_{n}^{x} = (π_{x})_{∗}ν_{n}

is a probability measure on the H-orbit H · x containing x. Finally let b : X → X_{0} be a Borel function such that b(x) ∈ H · x ∩ X_{0} = [x]_{E} for all x ∈ X and let

p_{n}^{x} = b_{∗}m_{n}^{x}.

Then p_{n}^{x} is a probability measure on the countable set H · x ∩ X_{0} = [x]_{E}; and so the nonnegative Borel function

f_{n}^{x} : [x]_{E} → ℝ_{+}

defined by

f_{n}^{x}(y) = p_{n}^{x}({y})

satisfies f_{n}^{x} ∈ ℓ_{1}([x]_{E}) and ||f_{n}^{x}||_{1} = 1. We claim that the sequence of Borel functions

f_{n} : E → ℝ^{+}, n ∈ ℕ,

defined by f_{n}(x, y) = f_{n}^{x}(y) witnesses the 1-amenability of the countable Borel equivalence relation E on X_{0}. It only remains to check that if x, y ∈ X_{0} and xEy, then ||f_{n}^{x} − f_{n}^{y}||_{1} → 0 as n → ∞. Hence it is enough to verify the following identity.

Claim 4.8. Suppose that x, y ∈ X_{0}, h ∈ H and that h · y = x. Then for all n ∈ ℕ,

||f_{n}^{x} − f_{n}^{y}||_{1} = ||φ_{n} − φ_{n}^{h}||_{1}.

To see this, for each w, z ∈ X_{0} with wEz, let

A_{z}^{w} = {g ∈ H | b(g · w) = z}.

Then it is easily checked that

f_{n}^{w}(z) = ∫_{H} χ_{A_{z}^{w}}φ_{n} dμ.

Since h · y = x, it follows that for all z ∈ [x]_{E},

g ∈ A_{z}^{x} iff gh ∈ A_{z}^{y}.

Hence, using the right-invariance of μ, we obtain that

f_{n}^{y}(z) = ∫_{H} χ_{A_{z}^{y}}(g)φ_{n}(g) dμ(g)
= ∫_{H} χ_{A_{z}^{y}}(gh)φ_{n}(gh) dμ(g)
= ∫_{H} χ_{A_{z}^{x}}(g)φ_{n}^{h}(g) dμ(g)

and so

|f_{n}^{x}(z) − f_{n}^{y}(z)| = ∫_{H} χ_{A_{z}^{x}}|φ_{n} − φ_{n}^{h}| dμ.

Summing over all z ∈ [x]_{E}, we obtain that

||f_{n}^{x} − f_{n}^{y}||_{1} = ||φ_{n} − φ_{n}^{h}||_{1}.

This completes the proof of Claim 4.8 and hence also the proof of Theorem 4.7. □

5. E_{0}-ergodicity and Property (τ)

In this section, we shall prove the E_{0}-ergodicity of the orbit equivalence relations arising from dense embeddings of finitely generated groups with Property (τ) into profinite groups. Then, in Section 6, we shall show that the countable Borel equivalence relations covered by Theorem 3.5 fit into this context and hence are E_{0}- ergodic. First we need to recall the Jones-Schmidt [17] representation theoretical criterion for E_{0}-ergodicity. (Our account is based on Appendix A of Hjorth-Kechris [14].)

Suppose that G is a lcsc group and that X is a standard Borel G-space with an G-invariant probability measure μ. Then we can define an associated unitary representation of G on the Hilbert space L^{2}(X, μ) by

g · f(x) = f(g^{−1} · x).

Let L_{0}^{2}(X, μ) be the closed subspace of L^{2}(X, μ) defined by

L_{0}^{2}(X, μ) = {f ∈ L^{2}(X, μ) | ∫_{X} f dμ = 0}.

PROPERTY (τ) AND COUNTABLE BOREL EQUIVALENCE RELATIONS 19

Then L_{0}^{2}(X, μ) is the orthogonal complement of the space of constant functions and so L_{0}^{2}(X, μ) is G-invariant. The following well-known characterization of ergodicity is essentially just a reformulation of Theorem 2.1. Theorem 5.1. If μ is a G-invariant probability measure on the standard Borel G-space X, then the following statements are equivalent.

(i) The action of G on (X, μ) is ergodic.

(ii) The associated unitary action of G on L_{0}^{2}(X, μ) has no nonzero invariant

vectors.

□

If G is a countable (discrete) group, then the stronger notion of E_{0}-ergodicity is closely connected to the question of whether the associated unitary action admits nonzero almost invariant vectors. Definition 5.2. Let Γ be a countable group and let π : Γ → U(H) be a unitary representation on the Hilbert space H. Then π admits nonzero almost invariant vectors iff for every ϵ > 0 and every finite subset F ⊆ Γ, there exists 0 ≠ v ∈ H such that ||π(γ) · v − v|| < ϵ||v|| for all γ ∈ F. Theorem 5.3. Let Γ be a countable group and let X be a standard Borel Γ-space with Γ-invariant probability measure μ. If the associated unitary action of Γ on L_{0}^{2}(X, μ) does not admit nonzero almost invariant vectors, then the action of Γ on (X, μ) is E_{0}-ergodic.

Proof. This is an immediate consequence of the results of Schmidt [26] and Jones-Schmidt [17]. (For a more detailed discussion, see Appendix A of Hjorth-Kechris [14], especially A2.2, A3.1 and A3.4.) □

Recall that a countable group Γ has the Kazhdan property iff for every unitary representation π of Γ, if π admits nonzero almost invariant vectors, then there exists a nonzero Γ-invariant vector. (Hence if Γ has the Kazhdan property, then every ergodic action of Γ is actually E_{0}-ergodic.) Equivalently, the Kazhdan property can be defined as follows. Definition 5.4. Let Γ be a countable group. Then Γ has the Kazhdan property iff there exists an ϵ > 0 and a finite subset F ⊆ Γ such that for each nontrivial

20 SIMON THOMAS

irreducible unitary representation π : Γ → U(H) and each 0 ≠ v ∈ H, there exists an element γ ∈ F such that ||π(γ) · v − v|| ≥ ϵ||v||.

It is well-known that if Γ is a Kazhdan group, then Γ is finitely generated and F can be taken to be any finite set of generators. If we add the hypothesis that the group Γ is finitely generated and restrict our attention to the unitary representations which factor through finite quotients of Γ, then we obtain the weaker notion of Property (τ). Definition 5.5. Let Γ be a finitely generated group and let S be a finite set of generators. Then Γ has Property (τ) iff there exists an ϵ > 0 such that for each nontrivial irreducible unitary representation π : Γ → U(H) with [Γ : ker π] < ∞ and each 0 ≠ v ∈ H, there exists an element γ ∈ S such that ||π(γ) · v − v|| ≥ ϵ||v||. Remark 5.6. It is well-known that the above definition does not depend upon the choice of the finite generating set S. (For example, see Lubotzky [19, 3.2.5].)

We are now ready to state the main result of this section. (Recall that a group K is said to be profinite iff K is an inverse limit of a system of finite groups. Equivalently, K is a compact group whose open normal subgroups form a basis for the open neighborhoods of the identity.) Theorem 5.7. Let K be a compact second countable group, let L ⩽ K be a closed subgroup and let μ be the Haar probability measure on X = K/L. Let Γ be a finitely generated dense subgroup of K. Suppose that:

(a) K is a profinite group; and

(b) Γ has Property (τ). Then the action of Γ on (X, μ) is E_{0}-ergodic.

In the proof of Theorem 5.7, we shall make use of the following well-known result. Proposition 5.8 (Folklore). If K is a profinite group and π : K → U(H) is an irreducible unitary representation, then

(a) H is finite dimensional; and

(b) [K : ker π] < ∞.

Proof. Since K is compact, it follows that H is finite dimensional. (For example, see Folland [9, 5.2].) Hence U(H) is a real Lie group. By Helgason [12, II.B.5], there exists a neighborhood V of the identity e in U(H) such that V contains no

PROPERTY (τ) AND COUNTABLE BOREL EQUIVALENCE RELATIONS 21

nonidentity subgroups. Since π : K → U(H) is a continuous homomorphism, there exists a neighborhood W of the identity 1 in K such that π(W) ⊆ V. As the open normal subgroups of K form a basis for the open neighborhoods of 1, it follows that there exists a normal subgroup N ⊴ K such that [K : N] < ∞ and N ⊆ W. This implies that π(N) = {e} and so [K : ker π] < ∞. □

Proof of Theorem 5.7. By Theorem 5.3, it is enough to show that the associated unitary action of Γ on L_{0}^{2}(X, μ) does not admit almost invariant vectors. To see this, consider the action of the profinite group K on L_{0}^{2}(X, μ). Since K is compact, we can express

L_{0}^{2}(X, μ) = ⨁_{i∈I} H_{i}

as a direct sum of mutually orthogonal finite-dimensional irreducible K-invariant subspaces H_{i}. (For example, see Folland [9, 5.2].) For each i ∈ I, let

π_{i} : K → U(H_{i})

be the associated irreducible unitary representation. By Theorem 5.1, since K acts ergodically on X, each of the representations π_{i} is nontrivial. Also, since K is profinite, it follows that [K : ker π_{i}] < ∞ for each i ∈ I. Furthermore, since Γ is dense in K, it follows that each H_{i} is also irreducible under the action of Γ. Clearly [Γ : Γ ∩ ker π_{i}] < ∞.

Let S be a finite set of generators for Γ. Since Γ has Property (τ), there exists ϵ > 0 such that for every i ∈ I and every 0 ≠ w ∈ H_{i}, there exists γ ∈ S such that

||π_{i}(γ) · w − w|| ≥ ϵ||w||.

Now consider an arbitrary vector 0 ≠ v ∈ L_{0}^{2}(X, μ). Then we can express

v = ∑_{i∈I} v_{i},

where each v_{i} ∈ H_{i}. For each γ ∈ Γ, we have that

||γ · v − v||^{2} = ∑_{i∈I} ||π_{i}(γ) · v_{i} − v_{i}||^{2}.

It follows easily that there exists γ ∈ S such that

||γ · v − v|| ≥ \frac{ϵ}{\sqrt{|S|}}||v||.

22 SIMON THOMAS

Hence the action of Γ on L_{0}^{2}(X, μ) does not admit almost invariant vectors. This completes the proof of Theorem 5.7. □

Remark 5.9. The above argument actually proves a slightly more general result. Suppose that B is a family of open normal subgroups of K which forms a basis for the open neighborhoods of the identity. Then in the statement of Theorem 5.7, Clause (b) can be weakened to:

(b)′ Γ has Property (τ) with respect to the family F = {Γ ∩ N | N ∈ B}. Here the group Γ with finite generating set S is said to have Property (τ) with respect to the family F of finite index subgroups iff there exists an ϵ > 0 such that for each nontrivial irreducible unitary representation π : Γ → U(H) with M ⩽ ker π for some M ∈ F and each 0 ≠ v ∈ H, there exists an element γ ∈ S such that ||π(γ) · v − v|| ≥ ϵ||v||.

6. Dynamical properties of SL_{2}(ℤ[1/p_{1}, ⋯, 1/p_{t}])-actions

Throughout this section, let S = {p_{1}, ⋯, p_{t}} be a finite nonempty set of primes and let J be a (possibly infinite) nonempty set of primes such that S ∩J = ∅. Let L be a closed subgroup of K(J) = ∏_{p∈J} SL_{2}(ℤ_{p}) and let μ_{J} be the Haar probability measure on K(J)/L. We shall regard Γ_{S} = SL_{2}(ℤ[1/p_{1}, ⋯, 1/p_{t}]) as a subgroup of K(J) via the diagonal embedding.

In this section, we shall prove two theorems concerning important dynamical properties of the action of Γ_{S} on (K(J)/L, μ_{J}). The first result is an immediate consequence of Theorem 5.7. Theorem 6.1. The action of Γ_{S} on (K(J)/L, μ_{J}) is E_{0}-ergodic.

Proof. It is enough to check that the hypotheses of Theorem 5.7 are satisfied. Firstly, it is well-known that SL_{2}(ℤ_{p}) is a profinite group for every prime p. (For example, see [6, page 23].) It follows that K(J) = ∏_{p∈J} SL_{2}(ℤ_{p}) is also a profinite group. Secondly, we have already pointed out that the Strong Approximation Theorem [22, 7.12] implies that SL_{2}(ℤ) is a dense subgroup of K(J) and hence Γ_{S} is also a dense subgroup of K(J). Finally, arguing exactly as in Example 4.3.3.E of Lubotzky [19], we see that that Γ_{S} has Property (τ). (In order to apply Lubotzky’s argument, we need to make use of the well-known results that Γ_{S} is finitely generated [20, VIII.3.3] and has the congruence subgroup property [27].) □

PROPERTY (τ) AND COUNTABLE BOREL EQUIVALENCE RELATIONS 23

Remark 6.2. The action of SL_{2}(ℤ) on (K(J)/L, μ_{J}) is also E_{0}-ergodic. To see this, note that if B is the natural basis of open neighborhoods of the identity, then

F = {SL_{2}(ℤ) ∩ N | N ∈ B}

is a family of congruence subgroups of SL_{2}(ℤ). By Lubotzky [19, 4.3.3.D], SL_{2}(ℤ) has Property (τ) with respect to its family of congruence subgroups. Hence the result follows by Remark 5.9.

Before we can state the second result, we need to introduce the notion of a quotient of a Borel action. Definition 6.3. For each i = 1, 2, let Γ_{i} be a countable group and let X_{i} be a standard Borel Γ_{i}-space with an invariant ergodic probability measure μ_{i}. Then (X_{2}, Γ_{2}, μ_{2}) is a quotient of (X_{1}, Γ_{1}, μ_{1}) iff there exist

(i) a surjective homomorphism φ : Γ_{1} → Γ_{2} and (ii) a Borel function f : X_{1} → X_{2}

such that the following conditions are satisfied:

(a) f_{∗}μ_{1} = μ_{2}; and

(b) for each γ ∈ Γ_{1}, f(γ · x) = φ(γ) · f(x) for μ_{1}-a.e. x ∈ X_{1}.

The remainder of this section will be devoted to the proof of the following result. Theorem 6.4. Suppose that A is a finite nonempty subset of ℙ ∪ {∞} and that H_{A} = ∏_{p∈A} P SL_{2}(ℚ_{p}). Let M be a proper closed subgroup of H_{A} such that H_{A}/M has finite volume and let m_{A} be the Haar probability measure on H_{A}/M. Let Λ be a lattice in H_{A} and let Λ^{+} be a countable group such that Λ ⩽ Λ^{+} ⩽ H_{A}.

Then (H_{A}/M, Λ^{+}, m_{A}) is not a quotient of (K(J)/L, Γ_{S}, μ_{J}).

The proof of Theorem 6.4 is based upon the fact that (K(J)/L, Γ_{S}, μ_{J}) has no nontrivial strongly mixing quotients. (For example, see Adams [1, 7.2].) Recall that if G is a noncompact lcsc group and X is a standard Borel G-space with a G-invariant probability measure μ, then the action of G on X is said to be strongly mixing iff for any two Borel subsets A, B ⊆ X, we have that

μ(g(A) ∩ B) → μ(A)μ(B) as g → ∞ in G.

24 SIMON THOMAS

i.e. if ⟨g_{n} | n ≥ 0⟩ is a sequence of elements of G which contains no convergent subsequences, then

μ(g_{n}(A) ∩ B) → μ(A)μ(B) as n → ∞.

If ⟨g_{n} | n ≥ 0⟩ is such a sequence, then we write g_{n} → ∞ in G as n → ∞. Example 6.5. Let C be a finite nonempty subset of ℙ ∪ {∞} and let H_{C} = ∏_{p∈C} P SL_{2}(ℚ_{p}). Let Δ_{C} be an irreducible lattice in H_{C} and let m_{C} be the Haar probability measure on H_{C}/Δ_{C}. Then the action of H_{C} on (H_{C}/Δ_{C}, m_{C}) is strongly mixing. (For example, see Zimmer [34, Chapter 10].)

We are now ready to present the proof of Theorem 6.4. For each nonempty subset D ⊆ A, let H_{D} = ∏_{p∈D} P SL_{2}(ℚ_{p}) and let π_{D} : H_{A} → H_{A}/H_{A∖D} be the canonical surjection. By Margulis [20, II.6.2], since H_{A}/M has finite volume, there exists a nonempty subset B ⊆ A and a lattice Δ in H_{B} such that

M = Δ × ∏_{p∈A∖B} P SL_{2}(ℚ_{p}).

Let C ⊆ B be a minimal nonempty subset such that Δ_{C} = π_{C}(Δ) is a lattice in H_{A}/H_{A∖C}. Then, identifying H_{A}/H_{A∖C} with H_{C}, we have that Δ_{C} is an irreducible lattice in H_{C}. Let m_{C} be the Haar probability measure on H_{C}/Δ_{C}. Then (H_{C}/Δ_{C}, π_{C}(Λ^{+}), m_{C}) is a quotient of (H_{A}/M, Λ^{+}, m_{A}). Let \overline{π_{C}(Λ)} be the closure of π_{C}(Λ) in H_{C}. By Margulis [20, II.6.1], since Λ is a lattice in H_{A}, it follows that H_{C}/\overline{π_{C}(Λ)} has finite volume. In particular, \overline{π_{C}(Λ)} is not a compact subgroup of H_{C} and hence there exists a sequence ⟨δ_{n} | n ≥ 0⟩ of elements of π_{C}(Λ) such that δ_{n} → ∞ in H_{C} as n → ∞. Furthermore, since the action of H_{C} on (H_{C}/Δ_{C}, m_{C}) is strongly mixing, if ⟨δ_{n} | n ≥ 0⟩ is any such sequence and Z is any Borel subset of H_{C}/Δ_{C}, then

m_{C}(δ_{n}(Z) ∩ Z) → m_{C}(Z)^{2} as n → ∞.

Now suppose that (H_{A}/M, Λ^{+}, m_{A}) is a quotient of (K(J)/L, Γ_{S}, μ_{J}). Then clearly (H_{C}/Δ_{C}, π_{C}(Λ^{+}), m_{C}) is also a quotient of (K(J)/L, Γ_{S}, μ_{J}). Hence there exist

(i) a surjective homomorphism φ : Γ_{S} → π_{C}(Λ^{+}) and
(ii) a Borel function f : K(J)/L → H_{C}/Δ_{C}

such that the following conditions are satisfied:

PROPERTY (τ) AND COUNTABLE BOREL EQUIVALENCE RELATIONS 25

(a) f_{∗}μ_{J} = m_{C}; and
(b) for each γ ∈ Γ_{S}, f(γ · x) = φ(γ) · f(x) for μ_{J}-a.e. x ∈ K(J)/L.

Let ⟨δ_{n} | n ≥ 0⟩ be a sequence of elements of π_{C}(Λ) such that δ_{n} → ∞ in H_{C} as n → ∞. Let Z be a Borel subset of H_{C}/Δ_{C} such that m_{C}(Z) = 1/2 and let Y = f^{−1}(Z) ⊆ K(J)/L. Arguing as in the proof of Adams [1, 7.3], we see that there exists an open neighborhood U of 1_{K} in K(J) such that

|μ_{J}(k(Y) ∩ Y) − μ_{J}(Y)| < 1/8

for all k ∈ U. For each n ≥ 0, let γ_{n} ∈ Γ_{S} be an element such that φ(γ_{n}) = δ_{n}. After passing to a suitable subsequence if necessary, we can suppose that there exists k ∈ K(J) such that γ_{n} → k in K(J) as n → ∞. Furthermore, since Γ_{S} is dense in K(J), after replacing each γ_{n} by βγ_{n} and each δ_{n} by φ(β)δ_{n} for a suitably chosen element β ∈ Γ_{S}, we can suppose that k ∈ U. Note that μ_{J}(Y) = m_{C}(Z) and that μ_{J}(γ_{n}(Y) ∩ Y) = m_{C}(δ_{n}(Z) ∩ Z) for all n ≥ 0. In particular, if γ_{n} ∈ U, then

|m_{C}(δ_{n}(Z) ∩ Z) − 1/2| = |μ_{J}(γ_{n}(Y) ∩ Y) − μ_{J}(Y)| < 1/8.

But this contradicts the fact that

m_{C}(δ_{n}(Z) ∩ Z) → m_{C}(Z)^{2} = 1/4 as n → ∞.

This completes the proof of Theorem 6.4.

7. A measure classification theorem

In this section, we shall present a measure classification theorem for lattices in products of real and p-adic algebraic groups. First it is necessary to define the notion of an algebraic probability measure. Definition 7.1. Let H be a lcsc group and let L be a closed subgroup of H. Then a probability measure μ on H/L is said to be algebraic iff there exists a closed subgroup C of H such that

(a) μ is C-invariant; and

(b) μ is supported on a single C-orbit; i.e. there exists x ∈ H/L such that μ(C · x) = 1.

26 SIMON THOMAS

Remark 7.2. As pointed out in Witte [32], in order to show that the probability measure μ is algebraic, it is enough to find a (not necessarily closed) subgroup D of H such that μ is D-invariant and supported on a single D-orbit. To see this, note that

Stab_{H}(μ) = {h ∈ H | μ is h-invariant }

is a closed subgroup of H and clearly μ is Stab_{H}(μ)-invariant. Furthermore, since μ is supported on a single D-orbit, say D · x, and D ⩽ Stab_{H}(μ), it follows that μ is supported on the Stab_{H}(μ)-orbit Stab_{H}(μ) · x.

The following result, which is an immediate consequence of results of Margulis-Tomanov [21] and Witte Morris, will play a crucial role in the proof of Theorem 3.5. Theorem 7.3. Let H = ∏_{p∈S∪{∞}} P SL_{2}(ℚ_{p}), where S is a finite set of primes, and let L be a closed subgroup of H. If Γ is a lattice in H, then every Γ-invariant ergodic probability measure on H/L is algebraic.

Proof. Let Δ : H → H × H be the diagonal embedding. By Margulis-Tomanov [21, 3.1], every Δ(H)-invariant ergodic probability measure on (H/Γ) × (H/L) is algebraic. Hence the result follows from the following theorem, due to Dave Witte Morris, which will be proved in Appendix A. □

Theorem 7.4 (Witte Morris). Let H be a lcsc group and let L be a closed subgroup of H. Suppose that:

(a) Γ is a lattice in H;

(b) Δ : H → H × H is the diagonal embedding; and

(c) every Δ(H)-invariant ergodic probability measure on (H/Γ) × (H/L) is

algebraic. Then every Γ-invariant ergodic probability measure on H/L is algebraic.

In fact, the proof of Theorem 7.3 shows that the following more general result holds. Theorem 7.5. Let T be a finite subset of ℙ ∪ {∞}; and for each p ∈ T, let G_{p}(ℚ_{p}) be an algebraic ℚ_{p}-group which is generated by its unipotent algebraic ℚ_{p}-subgroups. Let H = ∏_{p∈T} G_{p}(ℚ_{p}) and let L be a closed subgroup of H. If Γ is a lattice in H, then every Γ-invariant ergodic probability measure on H/L is algebraic.

PROPERTY (τ) AND COUNTABLE BOREL EQUIVALENCE RELATIONS 27

□

8. The proof of Theorem 3.5

In this section, we shall present the proof of Theorem 3.5. As in Thomas [30], our strategy is based on that of Adams [1]. Throughout we shall assume that the reader is familiar with the basic machinery of Zimmer’s superrigidity theory [34], including the notions of Borel cocycles and induced actions. Clear accounts of this material can be found in Zimmer [34], Adams-Kechris [2] and Adams [1]. Notation 8.1. Throughout this section, for each finite nonempty set of primes S = {p_{1}, ⋯, p_{t}}, we shall write ℤ_{S} = ℤ[1/p_{1}, ⋯, 1/p_{t}], Γ_{S} = SL_{2}(ℤ_{S}) and Λ_{S} = P SL_{2}(ℤ_{S}). If S = ∅, then ℤ_{S} = ℤ, Γ_{S} = SL_{2}(ℤ) and Λ_{S} = P SL_{2}(ℤ).

Most of our effort will be devoted to proving the following result, which will also play a key role in Hjorth-Thomas [15]. As we shall see later, Theorem 3.5 is a straightforward consequence of Theorem 8.2. Theorem 8.2. Suppose that S is a finite nonempty set of primes and that T is a (possibly empty) finite set of primes . Let X be a standard Borel Γ_{S}-space with invariant probability measure μ_{1} and let Y be a standard Borel Λ_{T} -space. Let E_{1}, E_{2} be the orbit equivalence relations induced by the actions of Γ_{S}, Λ_{T} on X, Y respectively and suppose that f : X → Y is a Borel homomorphism from E_{1} to E_{2}. Suppose that the following conditions are satisfied:

(I) E_{1} is E_{0}-ergodic.

(II) Let G = ∏_{p∈S^{+}} SL_{2}(ℚ_{p}), where S^{+} = S ∪ {∞}, and let X̂ = X × G/Γ_{S} be the induced G-space. Then X̂ is an irreducible G-space; i.e. SL_{2}(ℚ_{p}) acts ergodically on X̂ for each p ∈ S^{+}.

(III) Suppose that A is a finite nonempty subset of ℙ ∪ {∞} and that H_{A} = ∏_{p∈A} P SL_{2}(ℚ_{p}). Let M be a proper closed subgroup of H_{A} such that H_{A}/M has finite volume and let m_{A} be the Haar probability measure on H_{A}/M. Let Λ be a lattice in H_{A} and let Λ^{+} be a countable group such that Λ ⩽ Λ^{+} ⩽ H_{A}. Then (H_{A}/M, Λ^{+}, m_{A}) is not a quotient of (X, Γ_{S}, μ_{1}).

(IV) Λ_{T} acts freely on Y.

(V) There does not exist a μ_{1}-measure 1 subset of X which is mapped by f into a single E_{2}-class.

28 SIMON THOMAS

Then S ⊆ T and there exist (i) a subgroup Γ_{S}^{0} of finite index in Γ_{S}, (ii) a Γ_{S}^{0}-invariant Borel subset X_{0} ⊆ X with μ_{1}(X_{0}) > 0, (iii) a homomorphism φ̄ : Γ_{S}^{0} → Λ_{T}, and (iv) a Borel map f̄_{1} : X → Y

such that the following conditions hold:

(a) φ̄(Γ_{S}^{0}) is a subgroup of finite index in P SL_{2}(ℤ_{S}).

(b) f̄_{1}(γ · x) = φ̄(γ) · f̄_{1}(x) for all γ ∈ Γ_{S}^{0} and x ∈ X_{0}.
(c) f̄_{1}(x) ∈ Λ_{T} · f(x) for all x ∈ X.

Remark 8.3. It is perhaps worth saying a few words about Hypothesis 8.2(II). By Margulis [20, IX.1.7(iii)], if we identify Γ_{S} with its image under the diagonal embedding into G, then Γ_{S} is an irreducible lattice in G. Let ν be the Haar probability measure on G/Γ_{S} and let μ̂_{1} = μ_{1} × ν. Then μ̂_{1} is G-invariant and Hypothesis 8.2(II) states that (X̂, μ̂_{1}) is an irreducible G-space.

We are now ready to begin the proof of Theorem 8.2. Assuming the above hypotheses, let

α : Γ_{S} × X → Λ_{T}

be the cocycle defined by

α(γ, x) · f(x) = f(γ · x).

Lemma 8.4. α is not equivalent to a cocycle taking values in an amenable subgroup of Λ_{T}.

Proof. Suppose that α is equivalent to a cocycle β taking values in an amenable subgroup A of Λ_{T}. Then there exists a Borel function b : X → Λ_{T} and a Γ_{S}- invariant Borel subset X_{0} ⊆ X with μ_{1}(X_{0}) = 1 such that for all γ ∈ Γ_{S} and x ∈ X_{0},

β(γ, x) = b(γ · x)α(γ, x)b(x)^{−1}.

Let f′ : X → Y be the Borel homomorphism from E_{1} to E_{2} defined by f′(x) = b(x) · f(x). Then f′ also satisfies Hypothesis 8.2(V). Furthermore, for all γ ∈ Γ_{S} and x ∈ X_{0}, we have

β(γ, x) · f′(x) = f′(γ · x).

PROPERTY (τ) AND COUNTABLE BOREL EQUIVALENCE RELATIONS 29

Thus f′ ↾ X_{0} is a Borel homomorphism from E_{1} ↾ X_{0} to the orbit equivalence relation F induced by the action of the amenable group A on Y. By Lemma 4.5, F is 1-amenable. Since E_{1} ↾ X_{0} is E_{0}-ergodic, Lemma 4.6 implies that E_{1} ↾ X_{0} is also F -ergodic and hence there exists a Borel subset X_{1} ⊆ X_{0} with μ_{1}(X_{1}) = 1 such that f′ maps X_{1} into a single F -equivalence class. But this contradicts the fact that f′ satisfies Hypothesis 8.2(V). □

Let S^{+} = S ∪ {∞} and G = ∏_{p∈S^{+}} SL_{2}(ℚ_{p}). Also let T^{+} = T ∪ {∞} and H = ∏_{p∈T^{+}} P SL_{2}(ℚ_{p}). Recall that if we identify Γ_{S} with its image under the diagonal embedding into G, then Γ_{S} is an irreducible lattice in G. Similarly, if we identify Λ_{T} with its image under the diagonal embedding into H, then Λ_{T} is also an irreducible lattice in H.

Now consider the induced action of G on X̂ = X × G/Γ_{S} and let

α̂ : G × X̂ → Λ_{T}

be the corresponding cocycle induced from α. (We shall explicitly recall the definition of an induced action later in the proof.) Let ν be the Haar probability measure on G/Γ_{S} and let μ̂_{1} = μ_{1} × ν. Let i : Λ_{T} → H be the inclusion map and consider the cocycle

i ∘ α̂ : G × X̂ → H.

For each p ∈ T^{+}, let π_{p} : H → P SL_{2}(ℚ_{p}) be the projection map into the pth factor and let

α̂_{p} : G × X̂ → P SL_{2}(ℚ_{p})

be the cocycle defined by α̂_{p} = π_{p} ∘ i ∘ α̂. Lemma 8.5. For each p ∈ T^{+}, α̂_{p} is not equivalent to a cocycle taking values in a proper algebraic ℚ_{p}-subgroup of P SL_{2}(ℚ_{p}).

Proof. Suppose that α̂_{p} is equivalent to a cocycle taking values in the proper algebraic ℚ_{p}-subgroup A(ℚ_{p}) of P SL_{2}(ℚ_{p}). Let

H_{0} = A(ℚ_{p}) × ∏_{q∈T^{+}∖{p}} P SL_{2}(ℚ_{q}) ⩽ H.

30 SIMON THOMAS

Then i ∘ α̂ is equivalent to a cocycle taking values in the closed subgroup H_{0} of H. By Adams-Kechris [2, 2.3], there exists a cocycle

α_{1} : Γ_{S} × X → H

such that α_{1} ∼ i ∘ α and α_{1} takes values in H_{0}.

Claim 8.6. There exists an element h ∈ H and a cocycle

α_{2} : Γ_{S} × X → Λ_{T}

such that α_{2} ∼ α and α_{2} takes values in the subgroup Λ_{T} ∩ hH_{0}h^{−1}.

Assuming that Claim 8.6 holds, we can complete the proof of Lemma 8.5 as follows. Let Λ_{0} = Λ_{T} ∩ hH_{0}h^{−1}. Recall that Λ_{T} = P SL_{2}(ℤ_{T}) is diagonally embedded in H = ∏_{q∈T^{+}} P SL_{2}(ℚ_{q}). Hence, by projecting Λ_{0} into P SL_{2}(ℚ_{p}), we obtain that Λ_{0} is isomorphic to a subgroup of A(ℚ_{p}). Since A(ℚ_{p}) is a proper algebraic ℚ_{p}-subgroup of P SL_{2}(ℚ_{p}), it follows that A(ℚ_{p}) is solvable-by-finite. Hence Λ_{0} is also solvable-by-finite. But this means that Λ_{0} is an amenable subgroup of Λ_{T}, which contradicts Lemma 8.4.

Thus it only remains to prove Claim 8.6. (The following argument is closely based on the proof of Adams-Kechris [2, 2.4].) We shall begin by showing that the orbit equivalence relation E arising from the action of Λ_{T} on H/H_{0} is 1-amenable. To see this, first notice that E is Borel bireducible with the orbit equivalence relation F arising from the action of H_{0} on H/Λ_{T}. (For example, see the proof of Adams-Kechris [2, 2.4].) Clearly F is Borel bireducible with the orbit equivalence relation F′ arising from the action of A(ℚ_{p}) on the complete Borel section

Z = {aΛ_{T} | a ∈ A(ℚ_{p})} ⊆ H/Λ_{T} .

Let Z_{0} be a complete countable Borel section for F′. Since A(ℚ_{p}) is a lcsc amenable group, Theorem 4.7 implies that F′ ↾ Z_{0} is 1-amenable. By Jackson-Kechris-Louveau [16, 2.15], since E ∼_{B} F′ ↾ Z_{0}, it follows that E is also 1-amenable.

By Adams-Kechris [2, 2.1], since i∘α is equivalent to a cocycle taking values in the closed subgroup H_{0} of H, it follows that there exists a Borel function g : X → H/H_{0} such that for all γ ∈ Γ_{S},

α(γ, x) · g(x) = g(γ · x) for μ_{1}-a.e. x ∈ X.

PROPERTY (τ) AND COUNTABLE BOREL EQUIVALENCE RELATIONS 31

Thus there exists a Borel subset X_{0} ⊆ X with μ_{1}(X_{0}) = 1 such that g ↾ X_{0} is a Borel homomorphism from E_{1} ↾ X_{0} to the orbit equivalence relation E arising from the action of Λ_{T} on H/H_{0}. Since E_{1} is E_{0}-ergodic and E is 1-amenable, Lemma 4.6 implies that there is a Borel subset X_{1} ⊆ X_{0} with μ_{1}(X_{1}) = 1 such that g maps X_{1} into a single E-equivalence class; say, Λ_{T} · hH_{0}. Arguing as in the proof of Adams-Kechris [2, 2.2], it follows that α is equivalent to a cocycle taking values in the stabilizer of hH_{0} under the action of Λ_{T}; i.e.

{λ ∈ Λ_{T} | λhH_{0} = hH_{0}} = Λ_{T} ∩ hH_{0}h^{−1}.

This completes the proof of Claim 8.6 and hence also the proof of Lemma 8.5. □

Now that we know that (X̂, G, μ̂_{1}) satifies both Hypothesis 8.2(II) and Lemma 8.5, we are able to apply Zimmer’s measure theoretic superrigidity results. For each q ∈ S^{+}, let pr_{q} : G → SL_{2}(ℚ_{q}) be the projection map into the qth factor. By Zimmer [34, 10.1.6], for each p ∈ T^{+}, one of the following two cases must hold:

(a) p ∈ S^{+} and there exists a ℚ_{p}-rational surjection φ_{p} : SL_{2}(ℚ_{p}) → P SL_{2}(ℚ_{p}) such that α̂_{p} is equivalent to the cocycle

β̂_{p} : G × X̂ → P SL_{2}(ℚ_{p})

defined by

β̂_{p}(g, x̂) = (φ_{p} ∘ pr_{p})(g).

(b) α̂_{p} is equivalent to a cocycle β̂_{p} taking values in a compact subgroup K_{p} of

P SL_{2}(ℚ_{p}). Note that Case (a) must hold when p = ∞. To see this, recall the well-known result that compact real matrix groups are the ℝ-points of algebraic ℝ-groups. (For example, see Baily [3].) Thus, when p = ∞, Lemma 8.5 rules out Case (b).

Let A = {p ∈ T^{+} | Case (a) holds }, B = T^{+} ∖ A, H_{A} = ∏_{p∈A} P SL_{2}(ℚ_{p}) and H_{B} = ∏_{p∈B} P SL_{2}(ℚ_{p}). Let φ : G → H_{A} be the surjective homomorphism defined by

φ(g) = ∏_{p∈A} φ_{p}(pr_{p}(g)).

Then i ∘ α̂ is equivalent to a cocycle

β̂: G × X̂ → H = H_{A} × H_{B}

32 SIMON THOMAS

given by

β̂(g, x̂) = ⟨φ(g), β̂_{B}(g, x̂)⟩,

where β̂_{B} : G × X̂ → H_{B} is a cocycle taking values in the compact subgroup K = ∏_{p∈B} K_{p} of H_{B}. Arguing as in the proof of Adams [1, 5.4], it follows that i ∘ α is equivalent to a cocycle

β : Γ_{S} × X → H = H_{A} × H_{B}

given by

β(γ, x) = ⟨φ(γ), β_{B}(γ, x)⟩,

where β_{B} : Γ_{S} × X → H_{B} is a cocycle taking values in the compact subgroup K of H_{B}. By Platonov-Rapinchuk [22, 3.14], we can suppose that each K_{p} is a maximal compact subgroup of P SL_{2}(ℚ_{p}). Since p ≠ ∞, this implies that K_{p} has countable index in P SL_{2}(ℚ_{p}). (Once again, see Platonov-Rapinchuk [22, 3.14].) Hence K has countable index in H_{B}.

Let U ⊆ H be a Borel transversal for H/Λ_{T} chosen so that H_{B} ⊆ U and identify U with H/Λ_{T} by identifying each u ∈ U with uΛ_{T}. Then the action of H on H/Λ_{T} induces a corresponding Borel action of H on U, defined by

h · u = the unique element in U ∩ huΛ.

The associated cocycle ρ : H × U → Λ_{T} is defined by

ρ(h, u) = the unique λ ∈ Λ_{T} such that (h · u)λ = hu
= (h · u)^{−1}hu

Recall that the induced action of H on

Ŷ = Y × U = Y × (H/Λ_{T})

is defined by

h · (y, u) = (ρ(h, u) · y, h · u).

Let j : Y → Ŷ be the Λ_{T} -equivariant Borel map defined by j(y) = (y, 1) and let f̂: X → Ŷ be the Borel map defined by f̂ = j ∘ f. Then for all γ ∈ Γ_{S} and x ∈ X,

(i ∘ α)(γ, x) · f̂(x) = f̂(γ · x).

PROPERTY (τ) AND COUNTABLE BOREL EQUIVALENCE RELATIONS 33

Let b : X → H be a Borel map such that for all γ ∈ Γ_{S},

β(γ, x) = b(γ · x)(i ∘ α)(γ, x)b(x)^{−1} for μ_{1}-a.e. x ∈ X;

and define f̄: X → Ŷ by f̄(x) = b(x) · f̂(x). Then for all γ ∈ Γ,

β(γ, x) · f̄(x) = f̄(γ · x) for μ_{1}-a.e. x ∈ X.

Next we shall study the distribution of f̄(X) within Ŷ. Since K is a compact group, it follows that K\H/Λ_{T} is a standard Borel space. Furthermore, since the actions of K and H_{A} on H/Λ_{T} commute, it follows that H_{A} acts on K\H/Λ_{T}. Let η : Ŷ → K\H/Λ_{T} be the map defined by η(y, uΛ_{T}) = KuΛ_{T} and let ω = (η ∘ f̄)_{∗}μ_{1}. Then ω is a φ(Γ_{S})-invariant ergodic probability measure on K\H/Λ_{T}. Furthermore, by definition, for any Borel subset A ⊆ K\H/Λ_{T},

ω(A) = μ_{1}({x ∈ X | (η ∘ f̄)(x) ∈ A}).

Since K has countable index in H_{B}, it follows that H_{A} has only countably many orbits on K\H/Λ_{T}. Hence, since φ(Γ_{S}) acts ergodically on K\H/Λ_{T}, it follows that ω is supported on a single H_{A}-orbit Ω on K\H/Λ_{T}. (The following proof is based on an unpublished argument of Dave Witte Morris, which appeared in an early version of Adams [1].) Lemma 8.7. A = S^{+} and ω is supported on a finite subset Ω_{0} ⊂ Ω.

Proof. Let C = Stab_{H_{A}}(ω) = {h ∈ H_{A} | ω is h-invariant }. Then C is a (topologically) closed subgroup of H_{A} such that φ(Γ_{S}) ⩽ C. Let A_{0} = A ∖ {∞} and let Γ_{A_{0}} = SL_{2}(ℤ_{A_{0}}). Since Γ_{A_{0}} is an irreducible lattice in ∏_{p∈A} SL_{2}(ℚ_{p}), it follows that φ(Γ_{A_{0}}) is an irreducible lattice in H_{A}. Hence, applying Theorem 7.3, since ω is a φ(Γ_{A_{0}})-invariant ergodic probability measure on the homogeneous H_{A}-space Ω, it follows that ω is supported on a single C-orbit. Because C contains φ(Γ_{A_{0}}), it follows that H_{A}/C has finite volume. Hence, by Margulis [20, II.6.2], since φ(Γ_{A_{0}}) is an irreducible lattice, one of the following two possibilities holds:

• C = H_{A}; or

• C is a lattice in H_{A}.

First suppose that C = H_{A}. Then there exists a proper closed subgroup M of H_{A} such that the φ(Γ_{S})-space (Ω, ω) is isomorphic to (H_{A}/M, m), where m is the Haar probability measure on H_{A}/M. In particular, φ(Γ_{S})-space (H_{A}/M, m) is a

34 SIMON THOMAS

quotient of the Γ_{S}-space (X, μ_{1}). However, since φ(Γ_{S}) contains the lattice φ(Γ_{A_{0}}), this contradicts Hypothesis 8.2(III). Hence C must be a lattice in H_{A}. In particular, C is a countable group and so ω is supported on a countable subset Ω_{0} of Ω. Since ω is a C-invariant probability measure, this implies that Ω_{0} is actually a finite set.

Finally, since C, φ(Γ_{A_{0}}) are lattices in H_{A} and φ(Γ_{A_{0}}) ⩽ φ(Γ_{S}) ⩽ C, it follows that

[φ(Γ_{S}) : φ(Γ_{A_{0}})] ≤ [C : φ(Γ_{A_{0}})] < ∞.

Since Γ_{S} ∩ ker φ = Γ_{A_{0}} ∩ ker φ = Z(Γ_{S}), this implies that [Γ_{S} : Γ_{A_{0}}] < ∞. Hence A_{0} = S and so A = S^{+}. □

Clearly Ω_{0} ⊂ K\H/Λ_{T} must be φ(Γ_{S})-invariant and so there exists a subgroup Γ_{S}^{0} ⩽ Γ_{S} such that [Γ_{S} : Γ_{S}^{0}] < ∞ and φ(Γ_{S}^{0}) acts trivially on Ω_{0}. Fix some element KuΛ_{T} ∈ Ω_{0} and let u = ab, where a ∈ H_{A} and b ∈ H_{B}. After replacing f̄: X → Ŷ by the map f̄′(x) = a^{−1}f̄(x) and φ : G → H_{A} by the surjective homomorphism φ′(g) = a^{−1}φ(g)a if necessary, we can suppose that u ∈ H_{B}. Since φ(Γ_{S}^{0}) fixes KuΛ_{T}, we have that φ(γ)u ∈ KuΛ_{T} for each γ ∈ Γ_{S}^{0}; and hence, applying the projection π_{A} : H → H_{A}, we obtain that

φ(γ) ∈ π_{A}(Λ_{T}) ⩽ H_{A}.

Note that π_{A}(Λ_{T}) is the image of P SL_{2}(ℤ_{T}) under the diagonal embedding into H_{A} = ∏_{p∈A} P SL_{2}(ℚ_{p}). In particular, there exists a fixed homomorphism

φ̄ : Γ_{S}^{0} → P SL_{2}(ℤ_{T})

such that (φ_{p} ∘ pr_{p}) ↾ Γ_{S}^{0} = φ̄ for all p ∈ A. Lemma 8.8. φ̄(Γ_{S}^{0}) is commensurable with P SL_{2}(ℤ_{S}).

Proof. Since Γ_{S}^{0} is an irreducible lattice in G = ∏_{p∈S^{+}} SL_{2}(ℚ_{p}), it follows that φ(Γ_{S}^{0}) is an irreducible lattice in H_{A} = ∏_{p∈S^{+}} P SL_{2}(ℚ_{p}). In other words, if we identify φ̄(Γ_{S}^{0}) with its image under the diagonal embedding, then φ̄(Γ_{S}^{0}) is an irreducible lattice in ∏_{p∈S^{+}} P SL_{2}(ℚ_{p}). Applying Margulis [20, IX.4.11], since φ̄(Γ_{S}^{0}) ⩽ P SL_{2}(ℚ), it follows that

[φ̄(Γ_{S}^{0}) : φ̄(Γ_{S}^{0}) ∩ P SL_{2}(ℤ_{S})] < ∞.

PROPERTY (τ) AND COUNTABLE BOREL EQUIVALENCE RELATIONS 35

Since P SL_{2}(ℤ_{S}) is also a lattice in ∏_{p∈S^{+}} P SL_{2}(ℚ_{p}), it follows that φ̄(Γ_{S}^{0}) and P SL_{2}(ℤ_{S}) are commensurable. □

From now on, after replacing Γ_{S}^{0} by a subgroup of finite index if necessary, we shall suppose that φ̄(Γ_{S}^{0}) ⩽ P SL_{2}(ℤ_{S}).

Let X_{0} = {x ∈ X | (η ∘ f̄)(x) = KuΛ_{T} }. Clearly ω({KuΛ}) = 1/|Ω_{0}| and so μ_{1}(X_{0}) = 1/|Ω_{0}| > 0. After deleting a μ_{1}-null subset of X_{0} if necessary, we can suppose that

β(γ, x) · f̄(x) = f̄(γ · x)

for all γ ∈ Γ_{S}^{0} and x ∈ X_{0}. Let f̄_{1} : X → Y be the Borel map obtained by projecting f̄(x) ∈ Ŷ = Y × U = Y × (H/Λ_{T}) into Y. Recall that H_{B} is contained in the distinguished Borel transversal U of H/Λ_{T}. Consequently, for each x ∈ X_{0}, there exists k_{x} ∈ K such that f̄(x) = (f̄_{1}(x), k_{x}u). Fix some x ∈ X_{0}. For each γ ∈ Γ_{S}^{0}, let λ_{γ} ∈ Λ_{T} be the element defined by

λ_{γ} = ⟨φ̄(γ), ⋯, φ̄(γ)⟩

and let c_{γ} ∈ K and d_{γ} ∈ H_{B} be the elements defined by β(γ, x) = φ(γ)c_{γ} and d_{γ} = φ(γ)λ_{γ}^{−1}. Notice that

φ(γ)c_{γ}k_{x}u = c_{γ}k_{x}ud_{γ}λ_{γ}

and so

(f̄_{1}(γ · x), k_{γ·x}u) = (φ̄(γ) · f̄_{1}(x), c_{γ}k_{x}ud_{γ}).

Hence the Borel map f̄_{1} : X_{0} → Y has the property that

f̄_{1}(γ · x) = φ̄(γ) · f̄_{1}(x)

for all γ ∈ Γ_{S}^{0} and x ∈ X_{0}. Finally to see that f̄_{1}(x) ∈ Λ_{T} · f(x) for all x ∈ X, note that (f̄_{1}(x), 1) ∈ H · f̂(x) and that H · f̂(x) ∩ j(Y) = Λ_{T} · f̂(x). This completes the proof of Theorem 8.2. Corollary 8.9. Suppose that S is a finite nonempty set of primes. Let X be a standard Borel Γ_{S}-space with invariant probability measure μ_{1} and let E_{1} be the orbit equivalence relation induced by the action of Γ_{S} on X. Suppose that:

(a) (X, Γ_{S}, μ_{1}) satisfies Hypotheses 8.2(I), (II) and (III); and

36 SIMON THOMAS

(b) F is a treeable countable Borel equivalence relation on the standard Borel

space Z. Then E_{1} is F -ergodic.

Proof. Arguing as in Jackson-Kechris-Louveau [16, Section 3.5], it follows that there exists a standard Borel space Y and a free action of Λ_{∅} = P SL_{2}(ℤ) on Y such that F is Borel bireducible with the orbit equivalence relation E_{2} induced by the action of Λ_{∅} on Y. Suppose that f : X → Y is a Borel homomorphism. Then Hypotheses 8.2(I), (II), (III) and (IV) hold. Since S ⊈ ∅, it follows that Hypothesis 8.2(V) fails. □

In the remainder of this section, we shall present the proof of Theorem 3.5. Suppose that S_{1}, S_{2} are finite nonempty sets of primes and that J_{1}, J_{2} are (possibly infinite) nonempty sets of primes such that S_{1} ∩ J_{1} = S_{2} ∩ J_{2} = ∅. For i = 1, 2, suppose that

• L_{i} is a closed subgroup of K(J_{i}) satisfying Hypotheses 3.2 and 3.3;

• X_{i} ⊆ F_{S_{i}}(J_{i}, L_{i}) is a Λ_{S_{i}}-invariant Borel subset such that μ_{J_{i}}(X_{i}) = 1; and • E_{i} is the Λ_{S_{i}}-orbit equivalence relation on X_{i}.

In order to conform with the notation of Theorem 8.2, we shall write S = S_{1}, T = S_{2}, X = X_{1}, Y = X_{2} and μ_{i} = μ_{J_{i}} for i = 1, 2. Also, we shall regard X as a Γ_{S}-space rather than a Λ_{S}-space. Of course, E_{1} is also the Γ_{S}-orbit equivalence relation on X.

Suppose that f : X → Y is a Borel reduction from E_{1} to E_{2}. We shall begin by checking that the hypotheses of Theorem 8.2 are satisfied. It is clear that Hypotheses 8.2(IV) and (V) hold. Next, by Theorems 6.1 and 6.4 respectively, it follows that Hypotheses 8.2(I) and (III) also hold. Thus it only remains to check that Hypotheses 8.2(II) holds. Lemma 8.10. (X̂, μ̂_{1}) is an irreducible G-space.

Proof. Recall that Γ_{S} is an irreducible lattice in G and that the ergodic Γ_{S}-space (X, μ_{1}) has discrete spectrum; i.e. the associated unitary representation of Γ_{S} on L^{2}(X, μ_{1}) is a direct sum of mutually orthogonal finite-dimensional irreducible representations. (For the latter point, see the proof of Theorem 5.7.) Hence we can argue exactly as in the proof of Zimmer [33, 2.4]. □

PROPERTY (τ) AND COUNTABLE BOREL EQUIVALENCE RELATIONS 37

Thus S ⊆ T and there exist Γ_{S}^{0}, X_{0}, φ̄ and f̄_{1} satisfying the conclusion of Theorem 8.2. Since μ_{1}(X_{0}) > 0, there exists an ergodic component X_{1} for the action of Γ_{S}^{0} on X_{0} such that X_{1} ⊆ X_{0}. Furthermore, by the ergodicity of the action of Γ_{S}^{0} on X_{1}, we can suppose that there exists an ergodic component Y_{1} for the action of φ̄(Γ_{S}^{0}) on Y such that f̄_{1}(X_{1}) ⊆ Y_{1}. (Since SL_{2}(ℤ_{S}) is dense in K(J_{2}), it follows that the action of the subgroup P SL_{2}(ℤ_{S}) ⩽ Λ_{T} on Y is uniquely ergodic. Hence Y decomposes into finitely many ergodic components under the action of φ̄(Γ_{S}^{0}) and the action of φ̄(Γ_{S}^{0}) on each component is also uniquely ergodic.) Since Γ_{S}^{0} preserves the probability measure (μ_{1})_{X_{1}} on X_{1}, it follows that φ̄(Γ_{S}^{0}) preserves the probability measure (f̄_{1})_{∗}(μ_{1})_{X_{1}} on Y_{1}. Since the action of φ̄(Γ_{S}^{0}) on Y_{1} is uniquely ergodic, this implies that (f̄_{1})_{∗}(μ_{1})_{X_{1}} = (μ_{2})_{Y_{1}}. Hence, after deleting a μ_{2}-null subset of Y_{1} if necessary, we can suppose that f̄_{1}(X_{1}) = Y_{1}. At this point, it is clear that (X, Λ_{S}, μ_{1}) and (Y, Λ_{S}, μ_{2}) are virtually isomorphic. Thus it only remains to prove that S = T and that μ_{2}(Λ_{T} · f(X)) = 1. Lemma 8.11. S = T.

Proof. Suppose not. Then [Λ_{T} : P SL_{2}(ℤ_{S})] = ∞ and hence [Λ_{T} : φ̄(Γ_{S}^{0})] = ∞. Let Δ = {g ∈ Λ_{T} : g(Y_{1}) = Y_{1}}. By Lemma 2.2(c), after deleting a μ_{2}-null subset of Y_{1} if necessary, we can suppose that [Λ_{T} : Δ] < ∞ and hence [Δ : φ̄(Γ_{S}^{0})] = ∞. Let

Δ = ⨆_{n=1}^{∞} φ̄(Γ_{S}^{0})g_{n}

be the corresponding coset decomposition. Let x ∈ X_{1} and let y_{n} = g_{n} · f̄_{1}(x) for each n ≥ 1. Since Λ_{T} acts freely on Y, it follows that

φ̄(Γ_{S}^{0}) · y_{n} ≠ φ̄(Γ_{S}^{0}) · y_{m}

for all n ≠ m. For each n ≥ 1, choose z_{n} ∈ X_{1} such that f̄_{1}(z_{n}) = y_{n}. Clearly

Γ_{S}^{0} · z_{n} ≠ Γ_{S}^{0} · z_{m}

for all n ≠ m. Since [Γ_{S} : Γ_{S}^{0}] < ∞, it follows that there exist n ≠ m such that

Γ_{S} · z_{n} ≠ Γ_{S} · z_{m}.

But this is impossible, since

Λ_{T} · f̄_{1}(z_{n}) = Λ_{T} · y_{n} = Λ_{T} · y_{m} = Λ_{T} · f̄_{1}(z_{m}).

38 SIMON THOMAS

□

Finally note that μ_{2}(f̄_{1}(X_{1})) > 0 and that f̄_{1}(x) ∈ Λ_{T} · f(x) for all x ∈ X. Since Λ_{T} acts ergodically on (Y, μ_{2}), it follows that μ_{2}(Λ_{T} · f(X)) = 1.

This completes the proof of Theorem 3.5.

Appendix A. The proof of Theorem 7.4

In this appendix, we shall present the proof of Theorem 7.4. More precisely, we shall prove the following generalization, which is also due to Dave Witte Morris. (As Witte Morris pointed out, the following proof is almost identical with that of Witte [32, 5.8].) Theorem A.1 (Witte Morris). Let G be a lcsc group and let H, L be closed subgroups of G. Suppose that:

(a) Γ is a lattice in H;

(b) Δ : H → H × G is the diagonal embedding; and

(c) every ergodic Δ(H)-invariant probability measure on (H/Γ) × (G/L) is al-

gebraic. Then every ergodic Γ-invariant probability measure on G/L is algebraic.

Proof. Let μ be an ergodic Γ-invariant probability measure on G/L. Identifying G/L with the subset {Γ} × (G/L) of (H/Γ) × (G/L), we can regard μ as an ergodic Δ(Γ)-invariant probability measure on (H/Γ) × (G/L). Let σ be the Haar probability measure on H/Γ and let μ′ be the probability measure on (H/Γ) × (G/L) defined by

(Int)  μ′(A) = ∫_{H/Γ} Δ(h)_{∗}μ(A ∩ [{hΓ} × (G/L)]) dσ(hΓ).

Then it is easily checked that μ′ is Δ(H)-invariant and ergodic. Notice that that Equation (Int) provides a decomposition of μ′ into probability measures supported on the G-orbits on (H/Γ)×(G/L); namely, for each h ∈ H, the probability measure supported on {hΓ} × (G/L) is the translate Δ(h)_{∗}μ. (For a clear account of the basic theory of measure decompositions, see Rohlin [25, §3].)

Let C = Stab_{H×G}(μ′) be the subgroup consisting of the elements of H × G which preserve μ′. By assumption, μ′ is supported on a single C-orbit; say, C · s, where s = (hΓ, gL). Note that Δ(H) ⩽ C. Hence, after replacing s by a suitable

PROPERTY (τ) AND COUNTABLE BOREL EQUIVALENCE RELATIONS 39

Δ(H)-translate, we can suppose that s = (Γ, gL). Furthermore, after replacing μ′ and s by G-translates (and Γ by a G-conjugate), we can suppose that s = (Γ, L).

We can also use the subgroup C to construct a second decomposition of μ′ as follows. Let M = C ∩ G. Then M ⊴ C; and it is easily checked that C = Δ(H)M and that C ∩ (Γ × L) ⩽ Δ(Γ)M = C ∩ (Γ × G). Clearly the action of C on C · s is isomorphic to its action on C/C ∩ (Γ × L). In particular, there exists a C-invariant probability measure on C/C ∩ (Γ × L). Since C ∩ (Γ × L) ⩽ Δ(Γ)M ⩽ C, it follows that there exists a Δ(Γ)M-invariant probability measure ρ on Δ(Γ)M/C ∩ (Γ × L) and that the C-invariant probability measure on C/C ∩(Γ×L) decomposes into Ctranslates of ρ. (For example, see the proof of Raghunathan [23, 1.6].) Identifying C/C ∩ (Γ × L) with C · s, we obtain a corresponding decomposition of μ′ into probability measures supported on the G-orbits

{hΓ} × (G/L) ⊇ [{hΓ} × (G/L)] ∩ C · s = Δ(h)Δ(Γ)M · s;

namely, for each h ∈ H, the probability measure supported on {hΓ} × (G/L) is the translate Δ(h)_{∗}ρ. By the essential uniqueness of such decompositions [25, §3], it follows that Δ(h)_{∗}μ = Δ(h)_{∗}ρ for σ-a.e. hΓ ∈ H/Γ and hence μ = ρ. By construction, ρ is supported on Δ(Γ)M · s and thus μ is supported on the single orbit ΓM · s = Δ(Γ)M · s of the μ-preserving subgroup ΓM of G. □

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

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