SUPERRIGIDITY AND COUNTABLE BOREL EQUIVALENCE RELATIONS

SIMON THOMAS

Abstract. We formulate a Borel version of a corollary of Furman’s superrigidity theorem for orbit equivalence and present a number of applications to the theory of countable Borel equivalence relations. In particular, we prove that the orbit equivalence relations arising from the natural actions of SL_{3}(ℤ) on the projective planes over the various p-adic fields are pairwise incomparable with respect to Borel reducibility.

1. Introduction

Let Γ be a lattice in a higher rank simple Lie group. Suppose that X is a standard Borel Γ-space with an invariant ergodic probability measure μ and that Γ acts freely on X; i.e. γ · x ≠ x for all x ∈ X and 1 ≠ γ ∈ Γ. Then, roughly speaking, Zimmer’s superigidity theorem [24] says that both the group Γ and its action on X are “almost uniquely determined” by the orbit equivalence relation E_{Γ}^{X} and the measure μ. (Strictly speaking, Zimmer’s result is not quite this strong; but this is a reasonably accurate summary of the later work of Furman [11].) In their fundamental paper [3], Adams-Kechris used Zimmer’s machinery to prove a purely Borel version of Zimmer’s superrigidity theorem; and since then, superrigidity techniques have been used to solve a number of well-known problems concerning countable Borel equivalence relations. (For example, see [3], [21], [22].) In particular, Adams [2] has recently used superrigidity, together with Ratner’s measure classification theorem [19, Theorem 3], to construct the first known examples of pairs of countable Borel equivalence relations E ⊆ F such that E ≰_{B} F.

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

Key words and phrases. Borel equivalence relation, superrigidity.

Research partially supported by NSF Grants.

2 SIMON THOMAS

On examining Adams’ paper, it quickly became clear that his techniques were applicable to a wide range of natural problems concerning countable Borel equivalence relations. For example, in Thomas [23], we pointed out that the equivalence relations in Adams [2] also provided the first known examples of Borel equivalence relations E such that E <_{B} E ⊕ E, as well as the first known examples of countable Borel equivalence relations E such that E × I(ℕ) ⋢_{B} E. In this paper, we shall continue our study of Adams [2]. In Section 4, we shall formulate the general Borel superrigidity theorem which is implicitly contained in Adams’ arguments; and then in the later sections, we shall present some easy but striking applications. In Section 5, we shall show that if n ≥ 3, then there exist uncountably many orbit equivalence relations arising from standard SL_{n}(ℤ)-spaces which are pairwise incomparable with respect to Borel reducibility; and in Section 6, we shall show that if n ≥ 3, then the orbit equivalence relations arising from the natural actions of SL_{n}(ℤ) on the (n − 1)-dimensional projective spaces over the various p-adic fields ℚ_{p} are pairwise incomparable with respect to Borel reducibility. The essential difference between the Adams-Kechris Borel superrigidity theorem and that in this paper is easily explained: the Adams-Kechris result is suitable for distinguishing between orbit equivalence relations of the form E_{Γ}^{X} and E_{Λ}^{Y}, where Γ and Λ are lattices in non-isogenous higher rank semisimple Lie groups; while the result in this paper is suitable for distinguishing between the orbit equivalence relations arising from different actions of the same lattice Γ.

This paper is organised as follows. In Section 2, we shall recall some basic notions and results concerning Borel equivalence relations, ergodic theory and lattices in Lie groups. Then in Section 3, we shall give a brief account of the orbit equivalence superrigidity theorems of Zimmer [24] and Furman [11]. Amongst other aims, this account is intended to explain the necessity of the somewhat technical hypotheses of the Borel superrigidity theorem which we shall present in Section 4. The final two sections illustrate the use of this theorem by means of the applications which were mentioned in the previous paragraph. This paper has been written so as to be intelligible to readers who are completely unfamiliar with the notions and techniques of superrigidity theory. In particular, there is no mention of Borel cocycles or cocycle reduction theorems within the main body of this paper. (However, this

SUPERRIGIDITY AND COUNTABLE BOREL EQUIVALENCE RELATIONS 3

material is a prerequisite for an understanding of the appendix, in which we sketch the main points of the proof of the Borel superrigidity theorem.) Acknowledgements: I would like to thank Scot Adams and Alexander Kechris for very helpful discussions concerning the material in this paper. Thanks are also due to the referee for pointing out a number of obscurities and inaccuracies in an earlier version of this paper.

2. Preliminaries

In this section, we shall recall some basic notions from the theory of Borel equivalence relations, ergodic theory and the theory of Lie 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 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 corresponding G-orbit equivalence relation on X, which we shall denote by E_{G}^{X}, is a Borel equivalence relation. In fact, by Kechris [15], E_{G}^{X} is Borel bireducible with a countable Borel equivalence relation. Conversely, by Feldman-Moore [9], 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}.

Let G be a lcsc group. Then the standard Borel G-space X is said to be free iff G acts freely on X; i.e. if g · x ≠ x for all x ∈ X and 1 ≠ g ∈ G. It is an important open problem whether for every countable Borel equivalence relation E,

4 SIMON THOMAS

there exists a countable group Γ and a free standard Borel Γ-space X such that E ∼_{B} E_{Γ}^{X}. (For a discussion of this problem, see Jackson-Kechris-Louveau [14, Section 5.2].)

Suppose that E is a Borel equivalence relation on the standard Borel space X. Then a Borel subset A ⊆ X is said to be a complete section iff A intersects every E-class. If A intersects every E-class in a unique element, then A is said to be a Borel transversal. For example, if H is a finite group and X is a standard Borel H-space, then E_{H}^{X} clearly has a Borel transversal. Less trivially, suppose that G is a lcsc group and that H is a closed subgroup of G. Let E_{H}^{G} be the orbit equivalence relation arising from the right action of H on G. Then by the Effros-Mackey cross section theorem [20, Theorem 5.4.2], there exists a Borel transversal for E_{H}^{G}; i.e. a Borel subset T ⊆ G which meets every right coset gH in a unique element. If T is a Borel transversal for E, then the set X/E of E-classes can be naturally regarded as a standard Borel space, by identifying each E-class [x]_{E} with the unique element t ∈ T ∩[x]_{E}. In particular, if G is a lcsc group and H is a closed subgroup, then the set of right cosets G/H is a standard Borel space. Furthermore, G has a natural Borel action on G/H via left translations. In this paper, unless we explicitly say otherwise, whenever we refer to a space of cosets G/H as a standard Borel G-space, we shall always mean with respect to the natural action of G via left translations.

A detailed development of the general theory of countable Borel equivalence relations can be found in Jackson-Kechris-Louveau [14]. Here we shall only recall some of the highlights of the theory.

The least complex countable Borel equivalence relations are those which are smooth; i.e. those countable Borel equivalence relations E on a standard Borel space X for which there exists a Borel function f : X → Y into a standard Borel space Y such that xEy iff f(x) = f(y). The Lusin-Novikov uniformization theorem [16, 18.10] implies that a countable Borel equivalence relation E is smooth iff E has a Borel transversal.

Next in complexity come the hyperfinite relations; i.e. those countable Borel equivalence relations E such that there exists an increasing sequence

F_{0} ⊆ F_{1} ⊆ ⋯ ⊆ F_{n} ⊆ ⋯

SUPERRIGIDITY AND COUNTABLE BOREL EQUIVALENCE RELATIONS 5

of finite Borel equivalence relations such that E = ⋃_{n∈ℕ} F_{n}. (Here an equivalence relation F is said to be finite iff every F -equivalence class is finite.) 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 an example of a nonsmooth hyperfinite relation. By Dougherty-Jackson-Kechris [8], if E is a nonsmooth hyperfinite relation, then E ∼_{B} E_{0}. Furthermore, a countable Borel equivalence relation E on a standard Borel space X is hyperfinite iff there exists a Borel action of ℤ on X such that E = E_{ℤ}^{X}.

Let E_{∞} be the orbit equivalence relation arising from the natural action of the free group F_{2} on two generators on its power set P(F_{2}) = 2^{F_{2}}. By Dougherty-Jackson-Kechris [8], E_{∞} is a universal countable Borel equivalence relation in the sense that F ≤_{B} E_{∞} for every countable Borel equivalence relation F. (Clearly this universality property uniquely determines E_{∞} up to Borel bireducibility.) Furthermore, E_{0} <_{B} E_{∞}. For many years, it was an open problem whether, up to Borel bireducibility, there existed infinitely many countable Borel equivalence relations E such that E_{0} <_{B} E <_{B} E_{∞}. This problem was finally resolved by Adams-Kechris [3], who used Zimmer’s superrigidity theory [24] to show that there are actually 2^{ω} such relations E up to Borel bireducibility. Of these, the only ones which have been extensively studied are the treeable relations, which were originally introduced by Adams [1]. The countable Borel equivalence relation E on the standard Borel space X is said to be treeable iff there exists a Borel relation R ⊆ X × X such that:

(a) ⟨X; R⟩ is an acyclic graph; and

(b) the connected components of ⟨X; R⟩ are precisely the E-equivalence classes. For example, if F is a countable free group and X is a free standard Borel F -space, then E_{F}^{X} is treeable. In particular, this is the case for E_{T ∞}, which is defined to be the restriction of E_{∞} to the Borel subset

{S ∈ P(F_{2}) | gS ≠ S for all 1 ≠ g ∈ F_{2}}

of P(F_{2}) where F_{2} acts freely. By Jackson-Louveau-Kechris [14],

E_{0} <_{B} E_{T ∞} <_{B} E_{∞};

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and E_{T ∞} is a universal treeable relation in the sense that E ≤_{B} E_{T ∞} for every treeable relation E. It remains an open question whether there exists a (necessarily treeable) countable Borel equivalence relation E such that E_{0} <_{B} E <_{B} E_{T ∞}. In Sections 5 and 6, we shall present uncountably many conjectural candidates for such a relation E.

2.2. Ergodic theory. Let G be a lcsc group and let X be a standard Borel Gspace. 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 G-invariant iff μ(g(A)) = μ(A) for every g ∈ G and Borel subset A ⊆ X. If μ is G-invariant, 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. In this case, we shall also say that μ is an ergodic probability measure. The following characterization of ergodicity is well-known.

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. (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 [5, 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 G-invariant and that Y is also a standard Borel G-space. Then the Borel function f : X → Y is said to be almost G-equivariant iff for all g ∈ G, g · f(x) = f(g · x) for ν-a.e. x ∈ X. In this case, it is easily checked that f_{∗}ν is also G-invariant; and we say that (Y, f_{∗}ν) is a factor of (X, ν). It is also easily checked that if ν is ergodic, then f_{∗}ν is ergodic.

SUPERRIGIDITY AND COUNTABLE BOREL EQUIVALENCE RELATIONS 7

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.

If G is a lcsc group, then there exists a (nonzero) Borel measure μ on G which is invariant under the action of G on itself via left translations. Such a measure μ is called a Haar measure on G. If μ′ is a second Haar measure on G, then there exists a positive real constant c such that μ′(A) = cμ(A) for every Borel subset A ⊆ G. It is well-known that G is compact iff μ(G) < ∞. In this case, there exists a unique Haar probability measure on G. The group G is said to be unimodular iff each Haar measure μ is also invariant under the action of G on itself via right translations. It is well-known that if G is either discrete, compact or a connected simple Lie group, then G is unimodular.

Suppose that K is a compact second countable group and that L is a closed subgroup. Then both K and L are unimodular. Hence there exists a unique Kinvariant probability measure μ on the standard Borel K-space K/L. (For example, see [18, Theorem 3.17].) The measure μ is called the Haar probability measure on K/L. The Haar probability measure μ on K/L can be described explicitly as follows. Let ν be the Haar probability measure on K and let π : K → K/L be the canonical surjection. Then μ = π_{∗}ν.

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.

8 SIMON THOMAS

(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}.

(c) Suppose that Λ ⊴ Γ is a normal subgroup such that [Γ : Λ] < ∞ and let C be the set of ergodic components for the action of Λ on K/L. Then Γ acts as a transitive permutation group on C.

Proof. (a) Since Γ is a dense subgroup of K, any Γ-invariant probability measure on K/L is also K-invariant; and so the result follows from the fact that the Haar probability measure is the unique K-invariant probability measure on K/L.

(b) Let N ⩽ H be a closed normal subgroup of K such that [K : N] < ∞. Then K acts transitively on the finite set {N · x | x ∈ K/L} of N-orbits; and so 0 < μ(N · x) ≤ μ(H · x) for each x ∈ K/L. If x = tL, then the actions of H on H · x and H/(tLt^{−1} ∩ H) are isomorphic. Hence by part (a), Λ acts ergodically on H · x.

(c) Let H = \overline{Λ} be the closure of Λ in K. Then we can suppose that C is the set of H-orbits on K/L. Since Λ ⊴ Γ, it follows that Γ normalizes H and so Γ permutes the set C of H-orbits. Since Γ acts ergodically on (K/L, μ), we must have that μ(Γ · Z) = 1 for each Z ∈ C and so Γ acts transitively on C.

2.3. Lattices in connected simple Lie groups. Throughout this paper, let Ω be a fixed algebraically closed field of characteristic 0 which contains ℝ and all of the p-adic fields ℚ_{p}. By an algebraic group, we shall always mean a subgroup H of some general linear group GL_{n}(Ω) which is Zariski closed in GL_{n}(Ω). If H can be defined using polynomials with coefficients in the subfield k of Ω, then we say that H is an algebraic k-group. For any subring R ⊆ Ω, we define

GL_{n}(R) = {(r_{ij}) ∈ GL_{n}(Ω) | r_{ij} ∈ R and det(r_{ij})^{−1} ∈ R};

and if H ⩽ GL_{n}(Ω) is an algebraic group, then we define

H(R) = H ∩ GL_{n}(R).

In particular, if R = k is a subfield of Ω and H is an algebraic k-group, then H(k) is the group of all matrices in H with entries in k. If H is an algebraic k-group, then

SUPERRIGIDITY AND COUNTABLE BOREL EQUIVALENCE RELATIONS 9

H is said to be k-simple if every proper normal k-subgroup is trivial and almost k-simple if every proper normal k-subgroup is finite.

Suppose that H ⩽ GL_{n}(Ω) is an algebraic K-group, where K is either ℝ or a finite extension of the field ℚ_{p} of p-adic numbers for some prime p. In this case, H(K) ⩽ GL_{n}(K) is a lcsc group with respect to the Hausdorff topology; i.e. the topology obtained by restricting the natural topology on K^{n^{2}} to H(K). Any topological notions concerning the group H(K) will always refer to the Hausdorff topology. In particular, if H is an ℝ-group, then H(ℝ)^{0} denotes the connected component of the identity in the Hausdorff topology.

Suppose that G is a connected centerless simple Lie group. Then there exists a ℚ-simple ℚ-group H such that G and H(ℝ)^{0} are isomorphic as Lie groups. (For example, see [24, Proposition 3.1.6].) The ℝ-rank of G is defined to be the maximal dimension of an abelian ℝ-subgroup of H which can be diagonlized over ℝ. For example, ℝ-rank(P SL_{n}(ℝ)) = n − 1. If G is an arbitrary connected simple Lie group, then we define ℝ-rank(G) = ℝ-rank(G/Z(G)).

Suppose that G is a connected simple Lie group and that Γ is a discrete subgroup of G. Then both G and Γ are unimodular. Hence there exists a (nonzero) Ginvariant Borel measure μ on the standard Borel G-space G/Γ, which is uniquely determined up to a positive real constant. Such a measure μ is called a Haar measure on G/Γ. (Once again, see [18, Theorem 3.17].) In this case, it is also easy to give an explicit description of μ as follows.

Let ν be a Haar measure on G and let T be a Borel transversal for G/Γ such that 1 ∈ T. Then we can identify T with G/Γ by identifying each t ∈ T with the coset tΓ; and the action of G on G/Γ induces a corresponding Borel action of G on T, defined by

g · t = the unique element in T ∩ gtΓ.

It is easily checked that the measure μ induced on G/Γ by ν ↾ T via this identification is G-invariant. (Furthermore, while μ is dependent on ν, it is independent of the choice of the transversal T.)

The discrete group Γ is said to be a lattice in G iff the covolume μ(G/Γ) < ∞. By Borel-Harish-Chandra [24, Theorem 3.1.7], if G is an almost ℚ-simple ℚ-group,

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then G(ℤ) is a lattice in G(ℝ). In particular, the groups SL_{n}(ℤ) and P SL_{n}(ℤ) are lattices in the connected simple Lie groups SL_{n}(ℝ), P SL_{n}(ℝ) respectively.

3. Orbit equivalence superrigidity

In this section, we shall discuss the orbit equivalence superrigidity theorems of Zimmer [24] and Furman [11]. Roughly speaking, these theorems say that, under suitable hypotheses, if X is a standard Borel Γ-space with an invariant ergodic probability measure μ, then both the group Γ and its action on X are “almost uniquely determined” by the orbit equivalence relation E_{Γ}^{X} and the measure μ. (The superrigidity results will not be stated in their full generality. In order to keep the discussion as transparent as possible, we shall concentrate on the case of lattices in connected centerless simple Lie groups.)

Definition 3.1. For each 0 ≤ i ≤ 1, let Γ_{i} be a lcsc group and let X_{i} be a standard Borel Γ_{i}-space with an invariant ergodic probability measure μ_{i}. Then (X_{0}, Γ_{0}, μ_{0}) and (X_{1}, Γ_{1}, μ_{1}) are orbit equivalent iff there exist Borel subsets Y_{i} ⊆ X_{i} with μ_{i}(Y_{i}) = 1 and a Borel bijection f : Y_{0} → Y_{1} such that

(a) f_{∗}μ_{0} = μ_{1}; and

(b) for all x, y ∈ Y_{0}, (x, y) ∈ E_{Γ_{0}}^{X_{0}} iff (f(x), f(y)) ∈ E_{Γ_{1}}^{X_{1}}.

For example, if (X_{0}, Γ_{0}, μ_{0}) and (X_{1}, Γ_{1}, μ_{1}) are isomorphic in the following sense, then it is clear that they are orbit equivalent.

Definition 3.2. For each 0 ≤ i ≤ 1, let Γ_{i} be a lcsc group and let X_{i} be a standard Borel Γ_{i}-space with an invariant ergodic probability measure μ_{i}. Then (X_{0}, Γ_{0}, μ_{0}) and (X_{1}, Γ_{1}, μ_{1}) 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_{0} → Y_{1} and an Borel isomorphism φ : Γ_{0} → Γ_{1} such that

(a) f_{∗}μ_{0} = μ_{1}; and

(b) f(γ · x) = φ(γ) · f(x) for all γ ∈ Γ_{0} and x ∈ Y_{0}.

The strongest conceivable superrigidity theorem would say that the converse also holds; i.e. if (X_{0}, Γ_{0}, μ_{0}) and (X_{1}, Γ_{1}, μ_{1}) are orbit equivalent, then they are isomorphic. Of course, in order for anything like this to be true, it is necessary to impose strong hypotheses on the groups involved. So from now on, suppose

SUPERRIGIDITY AND COUNTABLE BOREL EQUIVALENCE RELATIONS 11

that G_{0} and G_{1} are connected centerless simple Lie groups of ℝ-rank at least 2 and that Γ_{0}, Γ_{1} are lattices in G_{0}, G_{1} respectively. Unfortunately, even under these strong hypotheses, there exist examples of orbit equivalent (X_{0}, Γ_{0}, μ_{0}) and (X_{1}, Γ_{1}, μ_{1}) such that Γ_{0} ≇ Γ_{1}. However, while the lattice Γ_{0} is not necessarily uniquely determined by the orbit equivalence relation E_{Γ_{0}}^{X_{0}} and the measure μ_{0}, it turns out that the ambient Lie group G_{0} is uniquely determined. More precisely, for each 0 ≤ i ≤ 1, there is a naturally associated induced G_{i}-space

X̂_{i} = X_{i} × (G_{i}/Γ_{i})

with an invariant ergodic probability measure μ̂_{i}, defined as follows. Fix a Borel transversal T_{i} for G_{i}/Γ_{i} such that 1 ∈ T_{i} and identify T_{i} with G_{i}/Γ_{i} by identifying each t ∈ T_{i} with the coset tΓ_{i} ∈ G_{i}/Γ_{i}. Then we can define a G_{i} action on X̂_{i} = X_{i} × T_{i} by

g · (x, t) = (γ · x, s)

where γ ∈ Γ_{i} and s ∈ T_{i} are chosen so that gt = sγ. Let ν_{i} be the Haar probability measure on G_{i}/Γ_{i}. Then it is easily checked that μ̂_{i} = μ_{i} × ν_{i} is a G_{i}-invariant ergodic probability measure on X̂_{i}. (For example, see Adams-Kechris [3, Section 2].) We are now ready to state (a special case of) Zimmer’s superrigidity theorem.

Theorem 3.3 (Zimmer [24]). Assume the above hypotheses and also that Γ_{1} acts freely on X_{1}. If (X_{0}, Γ_{0}, μ_{0}) and (X_{1}, Γ_{1}, μ_{1}) are orbit equivalent, then (X̂_{0}, G_{0}, μ̂_{0}) and (X̂_{1}, G_{1}, μ̂_{1}) are isomorphic. In particular, G_{0} ≅ G_{1}.

It was an important insight of Adams-Kechris [3] that Zimmer’s methods could also be applied in a purely Borel context. Essentially Adams-Kechris proved that, under the above hypotheses, if Γ_{1} acts freely on X_{1} and E_{Γ_{0}}^{X_{0}} ≤_{B} E_{Γ_{1}}^{X_{1}}, then there exist Lie subgroups N ⊴ H ⩽ G_{1} such that G_{0} ≅ H/N and hence dim G_{0} ≤ dim G_{1}. In Section 6, we shall make use of the following variant of their theorem. (Throughout this paper, \overline{ℚ} denotes the algebraic closure of ℚ. If we strengthen the hypotheses by assuming that H ⩽ G(ℚ), then Theorem 3.4 is an easy consequence of [22, Theorem 2.3]. However, an examination of its proof shows that [22, Theorem 2.3] also holds under the weaker hypothesis that H ⩽ G(\overline{ℚ}).)

12 SIMON THOMAS

Theorem 3.4. Let n ≥ 3 and let X be a standard Borel SL_{n}(ℤ)-space with an invariant ergodic probability measure μ. Suppose that H ⩽ G(\overline{ℚ}), where G is an algebraic ℚ-group such that dim G < n^{2} − 1, and that Y is a free standard Borel H-space. If f : X → Y is a Borel function such that for all x, y ∈ X,

SL_{n}(ℤ) · x = SL_{n}(ℤ) · y implies H · f(x) = H · f(y),

then there exists an SL_{n}(ℤ)-invariant Borel subset M ⊆ X with μ(M) = 1 such that f maps M into a single H-orbit.

Remark 3.5. Theorem 3.4 is no longer true if we drop the hypothesis that H acts freely on Y. For example, consider the orbit equivalence relation E_{∞} arising from the natural action of the free group F_{2} on two generators on its powerset P(F_{2}). Then E_{∞} is a universal countable Borel equivalence relation; but it is well-known that F_{2} can be realised as a subgroup of P GL_{2}(\overline{ℚ}).

While results such as Theorem 3.4 are very effective in distinguishing between equivalence relations arising from the actions of lattices in non-isogenous higher rank simple Lie groups, they cannot be used to study the different actions of a single lattice. For example, they provide absolutely no information concerning the relationship between the natural actions of SL_{3}(ℤ) on the projective planes over the various p-adic fields. In order to understand this kind of problem, we shall next discuss some consequences of Furman’s superrigidity theorem [11]. But first it is necessary to introduce the notions of weak orbit equivalence and virtual isomorphism.

Definition 3.6. For each 0 ≤ i ≤ 1, 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_{0}, Γ_{0}, μ_{0}) and (X_{1}, Γ_{1}, μ_{1}) are weakly orbit equivalent iff there exist Borel subsets Y_{i} ⊆ X_{i} with μ_{i}(Y_{i}) > 0 and a Borel bijection f : Y_{0} → Y_{1} such that

(a) f_{∗}(μ_{0})_{Y_{0}} = (μ_{1})_{Y_{1}}; and

(b) for all x, y ∈ Y_{0}, (x, y) ∈ E_{Γ_{0}}^{X_{0}} iff (f(x), f(y)) ∈ E_{Γ_{1}}^{X_{1}}.

Remark 3.7. Recall that by Dougherty-Jackson-Kechris [8, Proposition 2.6], if E, F are countable Borel equivalence relations on the standard Borel spaces X, Y respectively, then E ∼_{B} F iff there exist complete sections A ⊆ X, B ⊆ Y such that

SUPERRIGIDITY AND COUNTABLE BOREL EQUIVALENCE RELATIONS 13

E ↾ A and F ↾ B are Borel isomorphic; i.e. there exists a Borel bijection f : A → B such that xEy iff f(x)F f(y). Thus weak orbit equivalence is the measure analogue of Borel bireducibility. However, we should point out that neither notion implies the other. For example, E_{∞} and E_{T ∞} are orbit equivalent, but E_{∞} ≁_{B} E_{T ∞}. On the other hand, let E, F be the orbit equivalence relations arising from the natural actions of SL_{2}(ℤ) and SL_{3}(ℤ) on their powersets P(SL_{2}(ℤ)), P(SL_{2}(ℤ)) respectively. Then E and F are not weakly orbit equivalent, but E ∼_{B} E_{∞} ∼_{B} F. Both of these examples rely upon the fact that the Borel complexity of a countable Borel equivalence relation often concentrates on a null subset. In fact, I do not know of an example of a non-hyperfinite countable Borel equivalence relation E on a standard Borel space X with an E-invariant probability measure μ for which I can prove that (E ↾ A) ∼_{B} E for every Borel subset A ⊆ X such that μ(A) = 1. Of course, it would be particularly interesting to know whether this is the case when E = E_{T ∞}.

Our definition of virtually isomorphic is slightly more stringent than that of Furman [11]. However, both definitions coincide in the special case when Γ_{0} and Γ_{1} are lattices in connected centerless simple Lie groups of ℝ-rank at least 2. (It is perhaps worth mentioning that, contrary to the remark in Furman [11], virtual isomorphism does not imply weak orbit equivalence. For example, it is easily checked that the equivalence relations in Adams [2] are virtually isomorphic but not weakly orbit equivalent.)

Definition 3.8. For each 0 ≤ i ≤ 1, 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_{0}, Γ_{0}, μ_{0}) and (X_{1}, Γ_{1}, μ_{1}) are virtually isomorphic iff there exist subgroups Λ_{i} ⩽ Γ_{i} with [Γ_{i} : Λ_{i}] < ∞ and ergodic components Z_{i} for the actions of Λ_{i} on X_{i} such that (Z_{0}, Λ_{0}, (μ_{0})_{Z_{0}}) and (Z_{1}, Λ_{1}, (μ_{1})_{Z_{1}}) are isomorphic.

Even when both Γ_{0} and Γ_{1} are assumed to be lattices in centerless simple Lie groups of ℝ-rank at least 2, there exist examples of weakly orbit equivalent (X_{0}, Γ_{0}, μ_{0}) and (X_{1}, Γ_{1}, μ_{1}) such that Γ_{0} and Γ_{1} are not virtually isomorphic. (As expected, two groups Γ_{0}, Γ_{1} are said to be virtually isomorphic iff there exist subgroups Λ_{i} ⩽ Γ_{i} with [Γ_{i} : Λ_{i}] < ∞ such that Λ_{0} ≅ Λ_{1}.) In fact, there exist canonical examples.

14 SIMON THOMAS

Example 3.9. Let G be a connected centerless simple Lie group with ℝ-rank(G) ≥ 2. Let Γ, Δ be lattices in G and let μ, ν be the Haar probability measures on G/Γ, G/Δ respectively. Then it is well-known that (G/Δ, Γ, ν) and (G/Γ, Δ, μ) are weakly orbit equivalent. (For example, see Furman [11, Example 2.10].) However, Γ and Δ are not necessarily virtually isomorphic. For example, suppose that Γ is a uniform lattice and that Δ is a nonuniform lattice; i.e. that the quotient space G/Γ is compact and that the quotient space G/Δ is noncompact. (For the existence of such lattices, see [17, page 92].) Suppose that Γ_{0} ⩽ Γ and Δ_{0} ⩽ Δ are subgroups with [Γ : Γ_{0}], [Δ : Δ_{0}] < ∞ and that π : Γ_{0} → Δ_{0} is an isomorphism. By the Mostow-Margulis Rigidity Theorem [24, Theorem 5.1.1], π extends to a Lie group automorphism π̃ : G → G. But then π̃ induces a homeomorphism between the compact space G/Γ_{0} and the noncompact space G/Δ_{0}, which is a contradiction.

Theorem 3.10 (Furman [11]). Let G be a connected centerless simple Lie group with ℝ-rank(G) ≥ 2 and let Γ be a lattice in G. Let X be a standard Borel Γ-space with an invariant ergodic probability measure μ such that (X, μ) does not have any factors of the form (G/Δ, ν), where Δ is a lattice in G and ν is the Haar probability measure on G/Δ. Suppose that H is an arbitrary countable group and that Y is a free standard Borel H-space with an invariant ergodic probability measure m. If (X, Γ, μ) and (Y, H, m) are weakly orbit equivalent, then there exists a finite normal subgroup N ⊴ H such that (X, Γ, μ) and (Y/N, H/N, p_{∗}m) are virtually isomorphic, where p : Y → Y/N is the canonical surjection.

In the next section, we shall formulate a Borel version of the following corollary of Furman’s superrigidity theorem.

Corollary 3.11. Suppose that G is a connected centerless simple Lie group with ℝ-rank(G) ≥ 2 and that Γ_{0}, Γ_{1} are lattices in G. For each 0 ≤ i ≤ 1, let X_{i} be a standard Borel Γ_{i}-space with an invariant ergodic probability measure μ_{i}. Suppose that (X_{0}, μ_{0}) does not have any factors of the form (G/Δ, ν), where Δ is a lattice in G and ν is the Haar probability measure on G/Δ, and that Γ_{1} acts freely on X_{1}. If (X_{0}, Γ_{0}, μ_{0}) and (X_{1}, Γ_{1}, μ_{1}) are weakly orbit equivalent, then they are virtually isomorphic.

SUPERRIGIDITY AND COUNTABLE BOREL EQUIVALENCE RELATIONS 15

Proof. By the Margulis-Kazhdan theorem [24, Theorem 8.1.2], since G is centerless, it follows that Γ_{1} has no nontrivial finite normal subgroups. Hence the result follows from Theorem 3.10.

Remark 3.12. It would be quite misleading to refer to Corollary 3.11 as a weak form of Furman’s superrigidity theorem, since the main point of his theorem is that H can be an arbitrary countable group.

4. A Borel superrigidity theorem

In this section, we shall formulate the Borel superrigidity theorem which is implicitly contained in Adams [2]. As we shall soon see, this result is essentially a Borel version of Corollary 3.11. We shall begin with a short discussion of the hypotheses of the theorem. Suppose that G is a connected centerless simple Lie group with ℝ-rank(G) ≥ 2. Let Γ be a lattice in G and let X be a standard Borel Γ-space with an invariant ergodic probability measure μ. In the remainder of this paper, we shall be concerned with the Borel superrigidity properties of those (X, Γ, μ) which also satisfy one or both of the following hypotheses.

Hypothesis 4.1. (X, μ) does not have any factors of the form (G/Δ, ν), where Δ is a lattice in G and ν is the Haar probability measure on G/Δ.

Hypothesis 4.2. The action of Γ on X is uniquely ergodic.

Fortunately there are many natural examples which satisfy both of these hypotheses. (For further examples satisfying Hypothesis 4.1, see Furman [10, Corollary B].)

Lemma 4.3. Let G be a connected centerless simple Lie group with ℝ-rank(G) ≥ 2 and let Γ be a lattice in G. Suppose that Γ can also be realised as a dense subgroup of the compact second countable group K. Let L be a closed subgroup of K and let μ be the Haar probability measure on K/L. Then (K/L, Γ, μ) satisfies Hypotheses 4.1 and 4.2.

Proof. By Lemma 2.2(a), the action of Γ on K/L is uniquely ergodic. Thus it is enough to show that (K/L, Γ, μ) satisfies condition 4.1. But this is an immediate consequence of the following two well-known dynamical results.

16 SIMON THOMAS

(i) If Δ is a lattice in G and ν is the Haar probability measure on G/Δ, then the action of Γ on (G/Δ) is strongly mixing.

(ii) (K/L, μ) has no nontrivial strongly mixing factors. (For the definition of a strongly mixing action, see Adams [2, Section 6], where a detailed proof of (i) can also be found. To see that (ii) holds, argue as in the proof of Adams [2, Theorem 7.2].)

We are now ready to state our Borel superrigidity theorem. Unfortunately the statement is rather technical and may initially be a little difficult to understand. Most applications only require the much more intelligible Corollary 4.7; but there are important applications, such as Corollary 4.9, which require all the technical information of Theorem 4.4. To see that Theorem 4.4 holds, it is essentially enough to carefully examine the proof of Adams [2, Lemma 4.6]. In order to make this paper relatively self-contained, we shall sketch the main points of the proof of Theorem 4.4 in Appendix A. (However, we should point out that the basic machinery of Borel cocycles and cocycle reduction theorems is a prerequisite for an understanding of Appendix A.)

Theorem 4.4. Suppose that G is a connected centerless simple Lie group with ℝ-rank(G) ≥ 2 and that Γ_{0}, Γ_{1} are lattices in G. For each 0 ≤ i ≤ 1, let X_{i} be a standard Borel Γ_{i}-space with an invariant ergodic probability measure μ_{i} and let E_{i} = E_{Γ_{i}}^{X_{i}} be the corresponding orbit equivalence relation. Suppose that

• (X_{0}, Γ_{0}, μ_{0}) satisfies Hypothesis 4.1; and that

• Γ_{1} acts freely on X_{1} and (X_{1}, Γ_{1}, μ_{1}) satisfies Hypothesis 4.2. Suppose that f : X_{0} → X_{1} is a Borel function such that

• for all x, y ∈ X_{0}, if xE_{0}y, then f(x)E_{1}f(y); and

• there does not exist a μ_{0}-measure 1 Borel subset of X_{0} which is mapped into

a single E_{1}-class. Then there exist

(i) a Lie group automorphism ψ ∈ Aut(G), (ii) a subgroup Λ_{0} ⩽ Γ_{0} such that [Γ_{0} : Λ_{0}] < ∞ and ψ(Λ_{0}) ⩽ Γ_{1}, (iii) ergodic components Z_{0} and Z_{1} for the actions of Λ_{0}, ψ(Λ_{0}) on X_{0}, X_{1} re-

spectively, and

SUPERRIGIDITY AND COUNTABLE BOREL EQUIVALENCE RELATIONS 17

(iv) a Borel function f̃: Z_{0} → Z_{1} such that the following conditions are satisfied:

(a) f̃_{∗}(μ_{0})_{Z_{0}} = (μ_{1})_{Z_{1}};

(b) f̃(γ · x) = ψ(γ) · f̃(x) for all γ ∈ Λ_{0} and x ∈ Z_{0}.

(c) f̃(x) ∈ Γ_{1} · f(x) for all x ∈ Z_{0}.

Remark 4.5. Notice that ψ(Λ_{0}) is necessarily a subgroup of finite index in Γ_{1}. To see this, first note that Λ_{0} is a lattice in G. Since the Haar measure ν of the connected centerless simple Lie group G is invariant under the action of the group Aut(G) of Lie group automorphisms, it follows that ψ(Λ_{0}) is also a lattice in G. It is now clear that [Γ_{1} : ψ(Λ_{0})] < ∞; in fact, ν(G/ψ(Λ_{0})) = [Γ_{1} : ψ(Λ_{0})]ν(G/Γ_{1}).

Remark 4.6. In general, there is no reason to suppose that f̃ is injective. (For example, consider the proof of Theorem 5.4.) However, if f is a Borel reduction, then f̃ is an injective Borel map; and hence, in this case, after replacing Z_{1} by the Borel subset f̃(Z_{0}) if necessary, we can suppose that f̃ is a Borel bijection.

Corollary 4.7. Suppose that G is a connected centerless simple Lie group with ℝ-rank(G) ≥ 2 and that Γ_{0}, Γ_{1} are lattices in G. For each 0 ≤ i ≤ 1, let X_{i} be a standard Borel Γ_{i}-space with an invariant ergodic probability measure μ_{i} and let E_{i} = E_{Γ_{i}}^{X_{i}} be the corresponding orbit equivalence relation. Suppose that

• μ_{0} is nonatomic and (X_{0}, Γ_{0}, μ_{0}) satisfies Hypothesis 4.1; and that

• Γ_{1} acts freely on X_{1} and (X_{1}, Γ_{1}, μ_{1}) satisfies Hypothesis 4.2. If E_{0} ≤_{B} E_{1}, then (X_{0}, Γ_{0}, μ_{0}) and (X_{1}, Γ_{1}, μ_{1}) are virtually isomorphic.

Proof. Let f : X_{0} → X_{1} be a Borel reduction from E_{0} to E_{1}. Since μ_{0} is nonatomic, there does not exist a μ_{0}-measure 1 Borel subset of X_{0} which is mapped into a single E_{1}-class. Consequently, Corollary 4.7 is an immediate consequence of Theorem 4.4, together with Remarks 4.5 and 4.6.

We shall end this section with an application that makes full use of the technical information contained in the statement of Theorem 4.4.

Definition 4.8. 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.

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 Thomas [23], we pointed out that each of Adams’ examples in [2] satisfies nE <_{B} (n + 1)E for all n ≥ 1. The next corollary shows that such equivalence relations E are relatively common; and, in fact, Section 5.1 provides uncountably many such relations which are pairwise incomparable with respect to Borel reducibility.

Corollary 4.9. Suppose that G is a connected centerless simple Lie group with ℝ-rank(G) ≥ 2 and that Γ is a lattice in G. Suppose that X is a free standard Borel Γ-space with an invariant ergodic probability measure μ such that (X, Γ, μ) satisfies Hypotheses 4.1 and 4.2; and let E = E_{Γ}^{X} be the corresponding orbit equivalence relation. Then nE <_{B} (n + 1)E for all n ≥ 1.

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

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

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

X_{ℓ,m} = {x ∈ X | f_{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.

Claim 4.10. For each ℓ < n + 1, μ(Γ · f_{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)}.) Before proving Claim 4.10, we shall show how to complete the proof of Corollary 4.9. Let k, ℓ < n + 1 be distinct integers such that m(k) = m(ℓ) = m. Then we must have that

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

which is a contradiction.

SUPERRIGIDITY AND COUNTABLE BOREL EQUIVALENCE RELATIONS 19

Thus it only remains to prove Claim 4.10. Fix some integer ℓ < n + 1. By the ergodicity of μ, it is enough to show that μ(Γ · f_{ℓ}(X_{ℓ,m(ℓ)})) > 0. Applying Theorem 4.4 and Remark 4.6 to the Borel reduction f_{ℓ} : X_{ℓ,m(ℓ)} → X × {m(ℓ)}, we find that there exist

(i) a Lie group automorphism ψ ∈ Aut(G), (ii) a subgroup Λ ⩽ Γ such that [Γ : Λ] < ∞ and ψ(Λ) ⩽ Γ, (iii) ergodic components Z_{0}, Z_{1} for the actions of Λ, ψ(Λ) respectively on X, and (iv) a Borel bijection f̃: Z_{0} → Z_{1} such that f̃(x) ∈ Γ · f_{ℓ}(x) for all x ∈ Z_{0}. In particular, we have that

Z_{1} = f̃(Z_{0}) ⊆ Γ · f_{ℓ}(X_{ℓ,m(ℓ)})

and so μ(Γ · f_{ℓ}(X_{ℓ,m(ℓ)})) ≥ μ(Z_{1}) > 0.

5. The Gefter-Golodets Examples

In this section, we shall present the Borel version of a theorem of Gefter-Golodets [13, Theorem A.8], which says that if G is a connected almost ℚ-simple ℚ-group with ℝ-rank(G) ≥ 2 and Γ is a lattice in G(ℝ) which is commensurable with G(ℤ), then Γ has 2^{ω} many free ergodic actions on standard Borel spaces which are pairwise not weakly orbit equivalent. (Here two subgroups A, B of a group G are said to be commensurable iff A ∩ B has finite index in both A and B.) In order to keep our presentation as transparent as possible, we shall initially only consider the case when G = SL_{n} and Γ = SL_{n}(ℤ) for some odd integer n ≥ 3. (We have chosen this example because SL_{n} is a connected, (algebraically) simply connected, almost ℚsimple ℚ-group, which allows us to apply the Strong Approximation Theorem [18, Theorem 7.12]; and because SL_{n}(ℝ) is centerless when n is odd, which allows us to directly apply Corollary 4.7.) Fix some odd integer n ≥ 3 and for each nonempty subset J ⊆ ℙ of primes, let

K(J) = ∏_{p∈J} SL_{n}(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 as a subgroup of K(J) via the diagonal embedding. Furthermore, by the Strong Approximation Theorem, Γ is a dense subgroup of K(J). Let μ_{J} be the Haar probability measure on K(J) and let E_{J} be the orbit equivalence relation arising from the free action of Γ on K(J) via left translations.

Theorem 5.1. With the above hypotheses, if J_{0} ≠ J_{1} are distinct nonempty subsets of ℙ, then E_{J_{0}} and E_{J_{1}} are incomparable with respect to Borel reducibility.

Suppose that J_{0} ≠ J_{1} and that E_{J_{0}} ≤_{B} E_{J_{1}}. By Lemma 4.3, both (K(J_{0}), Γ, μ_{J_{0}}) and (K(J_{1}), Γ, μ_{J_{1}}) satisfy Hypotheses 4.1 and 4.2; and so by Corollary 4.7, they are virtually isomorphic. Hence there exist

• subgroups Λ_{i} ≤ Γ with [Γ : Λ_{i}] < ∞,

• ergodic components X_{i} for the action of Λ_{i} on K(J_{i}),

• a Borel bijection f : X_{0} → X_{1}, and

• an isomorphism φ : Λ_{0} → Λ_{1} such that the following conditions are satisfied:

(a) f_{∗}(μ_{J_{0}})_{X_{0}} = (μ_{J_{1}})_{X_{1}}; and

(b) f(γ · x) = φ(γ) · f(x) for all γ ∈ Λ_{0} and x ∈ X_{0}. (We should point out that the following argument is essentially a proper subset of the proof of Gefter-Golodets [13].) For each 0 ≤ i ≤ 1, let H_{i} = \overline{Λ}_{i} be the closure of Λ_{i} in K(J_{i}) and let μ_{i} = (μ_{J_{i}})_{X_{i}}. Then H_{i} is an open subgroup of K(J_{i}); and by Lemma 2.2, we can suppose that X_{i} ⊆ H_{i} and that μ_{i} is the Haar probability measure on H_{i}. Let Aut(X_{i}, Λ_{i}, μ_{i}) be the group of (equivalence classes of) Borel bijections β : X_{i} → X_{i} satisfying β(γ · x) = γ · β(x) for all γ ∈ Λ_{i} and x ∈ X_{i}, where we identify two such functions β and β′ if β(x) = β′(x) for μ_{i}-a.e. x ∈ X_{i}. Clearly each element t ∈ H_{i} induces a corresponding element ρ_{t} ∈ Aut(X_{i}, Λ_{i}, μ_{i}) such that

ρ_{t}(x) = xt for μ_{i}-a.e. x ∈ X_{i};

and the next lemma shows that

Aut(X_{i}, Λ_{i}, μ_{i}) = {ρ_{t} | t ∈ H_{i}}.

Lemma 5.2 (Gefter-Golodets [13]). Suppose that β : X_{i} → X_{i} is a Borel function such that β(γ · x) = γ · β(x) for all γ ∈ Λ_{i} and x ∈ X_{i}. Then there exists t ∈ H_{i} such that β(x) = xt for μ_{i}-a.e. x ∈ X_{i}.

SUPERRIGIDITY AND COUNTABLE BOREL EQUIVALENCE RELATIONS 21

Proof. For each x ∈ X_{i}, let h(x) ∈ H_{i} be the element such that β(x) = xh(x). Then h : X_{i} → H_{i} is a Λ_{i}-invariant Borel map; and since Λ_{i} acts ergodically on X_{i}, there exists a Borel subset M ⊆ X_{i} with μ_{i}(M) = 1 such that h ↾ M is a constant map.

Hence, by considering the Borel bijection β : X_{1} → X_{1} defined by β(f(x)) = f(xa), we can define a group isomorphism θ : H_{0} → H_{1} by

θ(a) = b iff f(xa) = f(x)b for μ_{0}-a.e. x ∈ X_{0}.

By [16, Theorem 17.25], θ is a Borel map; and hence by [16, Theorem 9.10], θ is continuous. But by Gefter-Golodets [13, Proposition A.5], since J_{0} ≠ J_{1}, the groups K(J_{0}) and K(J_{1}) do not contain open subgroups which are isomorphic as topological groups. This completes the proof of Theorem 5.1.

Remark 5.3. Theorem 5.1 is also true if n ≥ 4 is an even integer. To see this, let Z = {±I} be the center of SL_{n}(ℤ). Then for each ∅ ≠ J ⊆ ℙ, E_{J} is Borel bireducible with the orbit equivalence relation arising from the corresponding action of P SL_{n}(ℤ) on K(J)/Z; and Corollary 4.7 is applicable to (K(J)/Z, P SL_{n}(ℤ), \overline{μ}_{J}), where \overline{μ}_{J} is the corresponding P SL_{n}(ℤ)-invariant ergodic measure on K(J_{0})/Z. In a similar fashion, the various reductions in the proof of Gefter-Golodets [13, Theorem A.8] adapt easily to the Borel context; and we see that if G is a connected almost ℚ-simple ℚ-group with ℝ-rank(G) ≥ 2 and Γ is a lattice in G(ℝ) which is commensurable with G(ℤ), then Γ has 2^{ω} many free ergodic actions on standard Borel spaces which are pairwise incomparable under Borel reducibility.

We shall end this section with two applications of Theorem 5.1 to the general theory of countable Borel equivalence relations. Firstly, if E and F are Borel equivalence relations on the standard Borel spaces X, Y respectively, then the relation E ≤_{B} F is sometimes interpreted to mean that the quotient space X/E has “Borel cardinality” less than or equal to that of Y/F; and, not surprisingly, there turns out to be a number of similarities between the theories of “Borel cardinality” and cardinality without the axiom of choice. From this point of view, it is natural to say that the “Borel cardinality” of X/E dominates that of Y/F iff there exists a surjective Borel function f : X → Y such that xEy implies f(x)F f(y); and then to ask whether this implies that F ≤_{B} E. Of course, it is hard to imagine that anyone would expect a positive answer to this question, even in the case when both E and F are countable Borel equivalence relations.

Theorem 5.4. There exists a pair of countable Borel equivalence relations E, F on standard Borel spaces X, Y respectively such that

(a) there exists a surjective Borel function f : X → Y such that xEy implies f(x)F f(y); and

(b) F ≰_{B} E.

Proof. Let J_{0} and J_{1} be nonempty subsets of ℙ such that J_{1} ⊊ J_{0}. Let E = E_{J_{0}}, F = E_{J_{1}} and let f : K(J_{0}) → K(J_{1}) be the canonical projection map.

Secondly, a well-known open problem of Hjorth asks whether whenever E ⊆ F are countable Borel equivalence relations and E is universal, then F is necessarily also universal. Once again, a positive answer to this question seems very unlikely; but in Thomas [22], we pointed out that it was not even known whether there existed a pair E ⊆ F of countable Borel equivalence relations such that E ≰_{B} F. Soon afterwards, Adams [2] remedied this situation by constructing a pair E ⊆ F of countable Borel equivalence relations such that E and F were incomparable with respect to Borel reducibility. The next result shows that there also exist examples E ⊆ F such that F is strictly less complex than E.

Theorem 5.5. There exists a pair of countable Borel equivalence relations E ⊆ F on a standard Borel space X such that F <_{B} E.

Proof. Consider the action of Γ = SL_{n}(ℤ) on G = SL_{n}(ℤ_{p}), where p ∈ ℙ and n ≥ 3 is an odd integer. Let Λ = ker φ and H = ker ψ be the congruence subgroups of Γ, G respectively arising from the canonical surjections

φ : Γ → SL_{n}(ℤ/pℤ) ≅ SL_{n}(𝔽_{p})

and

ψ : G → SL_{n}(ℤ_{p}/pℤ_{p}) ≅ SL_{n}(𝔽_{p}).

(Here 𝔽_{p} denotes the field with p elements.) Then H is the closure of Λ in G; and there exist elements t_{i} ∈ Γ for 1 ≤ i ≤ d = |SL_{n}(𝔽_{p})| such that {t_{i} | 1 ≤ i ≤ d} is

SUPERRIGIDITY AND COUNTABLE BOREL EQUIVALENCE RELATIONS 23

simultaneously a set of coset representatives for Λ in Γ and for H in G. We shall show that F = E_{Γ}^{G} and E = E_{Λ}^{G} satisfy our requirements.

For each 1 ≤ i ≤ d, let F_{i} be the orbit equivalence relation arising from the action of Λ on the ergodic component Ht_{i} ⊆ G. (In the proof of the following claim, we shall make repeated use of the fact that gH = Hg for all g ∈ G.)

Claim 5.6. F ∼_{B} F_{i} for each 1 ≤ i ≤ d.

Proof of Claim 5.6. First note that the inclusion map Ht_{i} → G is a Borel reduction from F_{i} to F. To see this, suppose that x, y ∈ Ht_{i} and that xF y. Then there exists γ ∈ Γ such that γx = y. But since

γHt_{i} ∩ Ht_{i} = Hγt_{i} ∩ Ht_{i} ≠ ∅,

it follows that γ ∈ H ∩ Γ = Λ and so xF_{i}y.

Next for each 1 ≤ j ≤ d, let f_{j} : Ht_{j} → Ht_{i} be the Borel map defined by f_{j}(x) = t_{i}t_{j}^{−1}x. Then a similar argument shows that f = f_{1} ∪ ⋯ ∪ f_{d} is a Borel reduction from F to F_{i}.

It is now easy to complete the proof of Theorem 5.5. Clearly E ∼_{B} F_{1} ⊕ ⋯ ⊕ F_{d} and so E ∼_{B} dF. By Corollary 4.9, F <_{B} dF and so F <_{B} E.

It would be very interesting to know whether the results in this section also hold for n = 2. Recall that by Jackson-Kechris-Louveau [14, Proposition 3.4], whenever X is a free standard Borel SL_{2}(ℤ)-space, then the corresponding orbit equivalence relation is treeable. In particular, for each nonempty subset J ⊆ ℙ of primes, let E_{J}^{T} be the orbit equivalence relation arising from the free action of SL_{2}(ℤ) on ∏_{p∈J} SL_{2}(ℤ_{p}) via left translations. Then E_{J}^{T} is a treeable relation. Also, by Jackson-Kechris-Louveau [14, Proposition 1.7], since SL_{2}(ℤ) preserves the Haar probability measure on ∏_{p∈J} SL_{2}(ℤ_{p}), it follows that E_{J}^{T} is not hyperfinite. Thus E_{0} <_{B} E_{J}^{T} ≤_{B} E_{T ∞}.

Conjecture 5.7. If J_{0} ≠ J_{1} are distinct nonempty subsets of ℙ, then E_{J_{0}}^{T} and E_{J_{1}}^{T} are incomparable with respect to Borel reducibility.

Also note that for each nonempty subset J ⊆ ℙ and each n ≥ 1, the relation nE_{J}^{T} is treeable and hence

E_{J}^{T} ≤_{B} nE_{J}^{T} ≤_{B} (n + 1)E_{J}^{T} ≤_{B} E_{T ∞}.

24 SIMON THOMAS

Conjecture 5.8. If ∅ ≠ J ⊆ ℙ, then nE_{J}^{T} <_{B} (n + 1)E_{J}^{T} for all n ≥ 1.

6. Groups acting on p-adic projective spaces

If n ≥ 2, then P G(n−1, ℚ_{p}) denotes the (n−1)-dimensional projective space over the p-adic field ℚ_{p}; i.e. P G(n−1, ℚ_{p}) is the space of 1-dimensional vector subspaces of the n-dimensional vector space ℚ_{p}^{n}. In this section, we shall prove that if n ≥ 3, then the orbit equivalence relations arising from the natural actions of SL_{n}(ℤ) on the various p-adic projective spaces P G(n − 1, ℚ_{p}) are pairwise incomparable with respect to Borel reducibility.

For the rest of this section, fix some integer n ≥ 3. Let Γ = SL_{n}(ℤ) and for each p ∈ ℙ, let K_{p} = SL_{n}(ℤ_{p}). Then K_{p} is a compact group and Γ is a dense subgroup of K_{p}. The next result will enable us to identify P G(n − 1, ℚ_{p}) with the space of cosets K_{p}/L_{p}, where L_{p} is a suitably chosen closed subgroup of K_{p}. Throughout we shall identify each linear transformation g ∈ GL_{n}(ℚ_{p}) with the corresponding matrix with respect to the standard basis e_{1}, …, e_{n} of ℚ_{p}^{n}. In particular, GL_{n}(ℤ_{p}) will be identified with the setwise stabilizer of the ℤ_{p}-submodule M = ⊕_{i=1}^{n}ℤ_{p}e_{i}.

Lemma 6.1. Suppose that 1 ≤ k ≤ n and that ⟨v_{1}, …, v_{k}⟩ and ⟨w_{1}, …, w_{k}⟩ are k-dimensional subspaces of ℚ_{p}^{n}. Then there exists an element g ∈ SL_{n}(ℤ_{p}) such that g · ⟨v_{1}, …, v_{k}⟩ = ⟨w_{1}, …, w_{k}⟩.

Proof. Clearly it is enough to prove the result for the case when v_{i} = e_{i} for each 1 ≤ i ≤ k. After multiplying each w_{i} by a suitable power of p if necessary, we can suppose that N = ⊕_{i=1}^{k}ℤ_{p}w_{i} is a ℤ_{p}-submodule of M = ⊕_{i=1}^{n}ℤ_{p}e_{i}. By the invariant factor theorem [6, Theorem 16.8], there exists a basis b_{1}, …, b_{n} of the ℤ_{p}-module M and integers 0 ≤ a_{1} ≤ ⋯ ≤ a_{k} such that p^{a_{1}}b_{1}, …, p^{a_{k}}b_{k} is a basis of the ℤ_{p}-module N. Clearly there exist an element g ∈ GL_{n}(ℤ_{p}) such that g(e_{i}) = b_{i} for each 1 ≤ i ≤ n. Hence g ∈ GL_{n}(ℤ_{p}) satisfies g · ⟨v_{1}, …, v_{k}⟩ = ⟨w_{1}, …, w_{k}⟩; and, after multiplying g by a suitable diagonal matrix if necessary, we can suppose that g ∈ SL_{n}(ℤ_{p}).

Fix some point x_{p} ∈ P G(n−1, ℚ_{p}) and let L_{p} be the stabilizer of x_{p} in K_{p}. Then L_{p} is a closed subgroup of K_{p} and we can identify the K_{p}-spaces P G(n − 1, ℚ_{p}) and K_{p}/L_{p} via the Borel bijection

y ↦ {g ∈ K_{p} | g · x_{p} = y}.

Slightly abusing notation, we shall denote both the Haar probability measure on K_{p}/L_{p} and the corresponding probability measure on P G(n − 1, ℚ_{p}) by μ_{p}. Let

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

In other words, P G^{∗}(n−1, ℚ_{p}) is the Borel subset of P G(n−1, ℚ_{p}) where P SL_{n}(ℤ) acts freely. For each subspace W of the vector space ℚ_{p}^{n}, let

[W] = {x ∈ P G(n − 1, ℚ_{p}) | x ⩽ W }

be the corresponding subspace of P G(n − 1, ℚ_{p}). Then it is easily checked that μ_{p}([W]) = 0 for every proper subspace W of ℚ_{p}^{n}.

Lemma 6.2. μ_{p}(P G^{∗}(n − 1, ℚ_{p})) = 1.

Proof. If x ∈ P G(n − 1, ℚ_{p}) ∖ P G^{∗}(n − 1, ℚ_{p}), then there exists γ ∈ Γ ∖ Z(Γ) and an eigenspace E of γ such that x ∈ [E]. Since E is a proper subspace of ℚ_{p}^{n}, it follows that μ_{p}([E]) = 0. Hence P G(n − 1, ℚ_{p}) ∖ P G^{∗}(n − 1, ℚ_{p}) is the union of countably many μ_{p}-null sets.

Lemma 6.3. If p ≠ q ∈ ℙ, then (P G(n − 1, ℚ_{p}), Γ, μ_{p}) and (P G(n − 1, ℚ_{q}), Γ, μ_{q}) are not virtually isomorphic.

Of course, Lemma 6.3 also implies that (P G^{∗}(n − 1, ℚ_{p}), P SL_{n}(ℤ), μ_{p}) and (P G^{∗}(n − 1, ℚ_{q}), P SL_{n}(ℤ), μ_{q}) are not virtually isomorphic.

Let Λ ⩽ Γ be a subgroup with [Γ : Λ] < ∞ and let X be an ergodic component for the action of Λ on P G(n − 1, ℚ_{p}). For each subgroup Δ ⩽ Λ with [Λ : Δ] < ∞, let e(Δ) be the number of ergodic components for the action of Δ on X. Then Lemma 6.3 is an immediate consequence of the following two claims.

Claim 6.4. If Δ ⊴ Λ is a normal subgroup with [Λ : Δ] < ∞, then

e(Δ) = bp^{r}

for some r ≥ 0 and some divisor b of |SL_{n}(𝔽_{p})|.

Claim 6.5. For each integer N ≥ 0, there exists a normal subgroup Δ ⊴ Λ with [Λ : Δ] < ∞ such that e(Δ) ≥ N.

If m ≥ 1 and t ≥ 0, then Γ(m) = ker φ and K_{p}(p^{t}) = ker ψ_{t} denote the congruence subgroups of Γ, K_{p} respectively arising from the canonical surjections

φ : Γ = SL_{n}(ℤ) → SL_{n}(ℤ/mℤ)

and

ψ_{t} : K_{p} = SL_{n}(ℤ_{p}) → SL_{n}(ℤ_{p}/p^{t}ℤ_{p}).

Proof of Claim 6.4. Let H = \overline{Λ} be the closure of Λ in K_{p}. Then we can suppose that X is an orbit of H on P G(n − 1, ℚ_{p}). By Bass-Lazard-Serre [4], there exists an integer m ≥ 1 such that Γ(m) ⩽ Δ. Let m = ap^{t}, where (a, p) = 1. Then \overline{Γ(m)} = K_{p}(p^{t}) and the ergodic components for the action of Γ(m) on X are precisely the K_{p}(p^{t})-orbits; say, C = {Y_{i} | 1 ≤ i ≤ d}. Since K_{p}(p^{t}) ⊴ H, it follows that H acts as a transitive permutation group on C. Hence d is a divisor of [H : K_{p}(p^{t})], which is itself a divisor of [K_{p} : K_{p}(p^{t})]. Clearly we can suppose that m was chosen so that t ≥ 1; and this implies that [K_{p} : K_{p}(p^{t})] = |SL_{n}(𝔽_{p})|p^{ℓ} for some integer ℓ ≥ 0. (For example, see [7, Section 5.1].) Now let D = {Z_{j} | 1 ≤ j ≤ e(Δ)} be the collection of ergodic components for the action of Δ on X. Then each Z_{j} decomposes into the union of a certain number of elements of C. Furthermore, since Δ ⊴ Λ, it follows that Λ acts as a transitive permutation group on D. Hence each Z_{j} contains the same number of elements of C and so e(Δ) is a divisor of d. This completes the proof of Claim 6.4.

Proof of Claim 6.5. Arguing as in the proof of Claim 6.4, we see that it is enough to show that for each N ≥ 0, there exists t ≥ 1 such that K_{p}(p^{t}) has at least N orbits on P G(n − 1, ℚ_{p}). Let e_{1}, …, e_{n} be the standard basis of ℚ_{p}^{n} and let M = ⊕_{i=1}^{n}ℤ_{p}e_{i}. Fix some t ≥ 1. Let x, y ∈ P G(n − 1, ℚ_{p}) and suppose that g ∈ SL_{n}(ℤ_{p}) satisfies g · x = y. Then, under the induced action of SL_{n}(ℤ_{p}) on M/p^{t}M, we have that

(x ∩ M) + p^{t}M ↦^{g} (y ∩ M) + p^{t}M.

For each 0 ≤ r ≤ p^{t}, let v_{r} = e_{1} + re_{2}. Since K_{p}(p^{t}) acts trivially on M/p^{t}M, it follows that K_{p}(p^{t}) · ⟨v_{r}⟩ ≠ K_{p}(p^{t}) · ⟨v_{s}⟩ for all 0 ≤ r < s ≤ p^{t}. This completes the proof of Claim 6.5.

Definition 6.6. For each p ∈ ℙ, let E_{p} be the orbit equivalence relation arising from the action of Γ on P G(n − 1, ℚ_{p}).

Theorem 6.7. If p ≠ q ∈ ℙ, then E_{p} and E_{q} are incomparable with respect to Borel reducibility.

Suppose not and let f : P G(n−1, ℚ_{p}) → P G(n−1, ℚ_{q}) be a Borel function such that xE_{p}y iff f(x)E_{q}f(y). First suppose that there exists a Γ-invariant Borel subset X ⊆ P G^{∗}(n − 1, ℚ_{p}) with μ_{p}(X) = 1 such that f(x) ∈ P G^{∗}(n − 1, ℚ_{q}) for all x ∈ X. Applying Corollary 4.7, we obtain that (P G^{∗}(n − 1, ℚ_{p}), P SL_{n}(ℤ), μ_{p}) and (P G^{∗}(n − 1, ℚ_{q}), P SL_{n}(ℤ), μ_{q}) are virtually isomorphic, which contradicts Lemma 6.3. Hence we can suppose that there exists a Γ-invariant Borel subset X ⊆ P G^{∗}(n − 1, ℚ_{p}) with μ_{p}(X) = 1 such that f(x) ∉ P G^{∗}(n − 1, ℚ_{q}) for all x ∈ X.

Definition 6.8. The subspace E ⩽ ℚ_{q}^{n} is said to be a \overline{ℚ}-subspace iff there exists a (possibly empty) collection of vectors b_{1}, …, b_{t} ∈ \overline{ℚ}^{n} such that E = ⟨b_{1}, …, b_{t}⟩.

Clearly if E, F ⩽ ℚ_{q}^{n} are \overline{ℚ}-subspaces, then E ∩ F is also a \overline{ℚ}-subspace. In particular, for each point z ∈ P G(n − 1, ℚ_{q}), there exists a unique minimal \overline{ℚ}- subspace V such that z ∈ [V].

Lemma 6.9. For each x ∈ X, there exists a proper \overline{ℚ}-subspace E of ℚ_{q}^{n} such that f(x) ∈ [E].

Proof. For each x ∈ X, there exists an element γ ∈ Γ ∖ Z(Γ) and an eigenspace E of γ such that f(x) ∈ [E]. Clearly E is a proper \overline{ℚ}-subspace of ℚ_{q}^{n}.

Recall that the Effros Borel space on ℚ_{q}^{n} is the standard Borel space consisting of the set

F (ℚ_{q}^{n}) = {Z ⊆ ℚ_{q}^{n} | Z is a closed subset of ℚ_{q}^{n}}

equipped with the σ-algebra generated by the sets of the form

{Z ∈ F (ℚ_{q}^{n}) | Z ∩ U ≠ ∅},

where U varies over the open subsets of ℚ_{q}^{n}. For each x ∈ X, let V_{x} ∈ F (ℚ_{q}^{n}) be the unique minimal \overline{ℚ}-subspace such that f(x) ∈ [V_{x}]; and let m : X → F (ℚ_{q}^{n})

28 SIMON THOMAS

be the Borel map defined by m(x) = V_{x}. Since there are only countably many possibilities for V_{x}, there exists a Borel subset Y_{0} ⊆ X with μ_{p}(Y_{0}) > 0 and a fixed \overline{ℚ}-subspace V such that V_{x} = V for all x ∈ Y_{0}. Let X_{0} = Γ·Y_{0}. Since μ_{p} is ergodic, it follows that μ_{p}(X_{0}) = 1. After slightly adjusting f if necessary, we can suppose that V_{x} = V for all x ∈ X_{0}. (More precisely, let c : X_{0} → X_{0} be a Borel function such that c(x) ∈ Γ · x ∩ Y_{0} for all x ∈ X_{0}. Then we can replace f with f′ = f ∘ c.)

Now suppose that x, y ∈ X_{0} and that y ∈ Γ · x. Then there exists γ ∈ Γ such that γ · f(x) = f(y). We claim that

γ ∈ Γ_{{V }} = {φ ∈ Γ | φ(V) = V }.

To see this, note that γ(V) is also a \overline{ℚ}-subspace and that f(y) ∈ [V ∩γ(V)]. Hence, by the minimality of V, we must have that γ(V) = V.

Next let x ∈ X_{0} and suppose that φ ∈ Γ_{{V }} satisfies φ · f(x) = f(x). Then f(x) is contained in the eigenspace E of φ corresponding to some eigenvalue λ ∈ \overline{ℚ}. By the minimality of V, we must have that V ⩽ E and so φ(v) = λv for all v ∈ V. Let H ⩽ P GL(V) be the group of projective linear transformations induced on [V] by Γ_{\{V \}}. Then H is clearly isomorphic to a subgroup of P GL_{d}(\overline{ℚ}), where d = dim V < n; and we have just shown that

• H acts freely on the standard Borel space H · f(X_{0}); and

• if x, y ∈ X_{0} and y ∈ Γ · x, then there exists an element h ∈ H such that

h · f(x) = f(y). By Theorem 3.4, there exists a Γ-invariant Borel subset M ⊆ X_{0} with μ_{p}(M) = 1 such that f maps M into a single H-orbit. But this means that f maps M into a single Γ-orbit, which is a contradiction. This completes the proof of Theorem 6.7.

Once again, it would be very interesting to know whether Theorem 6.7 also holds when n = 2.

Conjecture 6.10. The orbit equivalence relations arising from the natural actions of SL_{2}(ℤ) on the projective lines over the various p-adic fields are pairwise incomparable with respect to Borel reducibility.

Appendix A. The proof of Theorem 4.4

In this appendix, we shall sketch the proof of Theorem 4.4. It should be stressed that most of the following argument is taken directly from Adams [2, Section 8].

SUPERRIGIDITY AND COUNTABLE BOREL EQUIVALENCE RELATIONS 29

Throughout we shall assume that the reader is familiar with the basic machinery of Zimmer’s superrigidity theory [24], including the notions of Borel cocycles and induced actions. Clear accounts of this material can be found in Zimmer [24], Adams-Kechris [3] and Adams [2].

Let G be a connected centerless simple Lie group with ℝ-rank(G) ≥ 2 and let Γ_{0}, Γ_{1} be lattices in G. For each 0 ≤ i ≤ 1, let X_{i} be a standard Borel Γ_{i}-space with an invariant ergodic probability measure μ_{i}; and let E_{i} = E_{Γ_{i}}^{X_{i}} be the corresponding orbit equivalence relation. Suppose that

• (X_{0}, Γ_{0}, μ_{0}) satisfies Hypothesis 4.1; and that

• Γ_{1} acts freely on X_{1} and (X_{1}, Γ_{1}, μ_{1}) satisfies Hypothesis 4.2. Suppose that f : X_{0} → X_{1} is a Borel function such that

• for all x, y ∈ X_{0}, if xE_{0}y, then f(x)E_{1}f(y); and

• there does not exist a μ_{0}-measure 1 Borel subset of X_{0} which is mapped into

a single E_{1}-class. Let α : Γ_{0} × X_{0} → Γ_{1} be the corresponding cocycle defined by

α(γ, x) · f(x) = f(γ · x)

for all γ ∈ Γ_{0} and x ∈ X_{0}. Since there does not exist a μ_{0}-measure 1 Borel subset of X_{0} which is mapped into a single E_{1}-class, a standard argument shows that α is not equivalent to a cocycle taking values in a finite subgroup of Γ_{1}. (For example, see Adams [2, Claim 8.2].) Let i : Γ_{1} → G be the inclusion map. By Adams [2, Corollary 5.4], since α is not equivalent to a cocycle taking values in a finite subgroup of Γ_{1}, there exists a Lie group automorphism φ ∈ Aut(G) such that i ∘ α : Γ_{0} × X_{0} → G is equivalent to the cocycle α_{φ} : Γ_{0} × X_{0} → G defined by α_{φ}(γ, x) = φ(γ). Let h : X_{0} → G be a Borel map such that for all γ ∈ Γ_{0},

φ(γ) = h(γ · x)α(γ, x)h(x)^{−1} for μ_{0}-a.e. x ∈ X_{0}.

Thus h is a “Borel adjustment” which transforms our initial function f into a function f_{1} which corresponds to the “simpler” cocycle α_{φ}. Unfortunately, this simplification comes at a price: the function f_{1} takes values in the induced G-space X̂_{1} = X_{1} × (G/Γ_{1}).

In more detail, let T ⊂ G be a Borel transversal for G/Γ_{1} such that 1 ∈ T and identify T with G/Γ_{1} by identifying each t ∈ T with tΓ_{1}. Let

X̂_{1} = X_{1} × T = X_{1} × (G/Γ_{1})

be the G-space induced from the Γ_{1}-space X_{1}. Let j : X_{1} → X̂_{1} be the Γ_{1}- equivariant map defined by j(x) = (x, Γ_{1}) and let f̂ = j ∘ f : X_{0} → X̂_{1}. Then

α(γ, x) · f̂(x) = f̂(γ · x)

for all γ ∈ Γ_{0} and x ∈ X_{0}; and for all x, y ∈ X_{0},

Γ_{0} · x = Γ_{0} · y implies G · f̂(x) = G · f̂(y).

Now define f_{1} : X_{0} → X̂_{1} by f_{1}(x) = h(x) · f̂(x). Then for all γ ∈ Γ_{0},

φ(γ) · f_{1}(x) = f_{1}(γ · x) for μ_{0}-a.e. x ∈ X_{0}.

Also for later use, note that G · f_{1}(x) = G · f̂(x) for all x ∈ X_{0}.

We next need to study the distribution of f_{1}(X_{0}) within X̂_{1}. Ideally, we would like to have that f_{1}(X_{0}) is contained in one of the “twisted” copies

X_{(t)} = {(x, tΓ_{1}) | x ∈ X_{1}}

of X_{1} in X̂_{1} = ⨆_{t∈T} X_{(t)}. Fortunately, this is almost true. As we shall soon see, Ratner’s measure classification theorem [19, Theorem 3] implies that either there exists finitely many elements t_{1}, …, t_{ℓ} ∈ T such that

f_{1}(x) ∈ X_{(t_{1})} ⊔ ⋯ ⊔ X_{(t_{ℓ})} for μ_{0}-a.e. x ∈ X_{0};

or f_{1}(X_{0}) is “uniformly distributed” throughout X̂_{1} = ⨆_{t∈T} X_{(t)}. Furthermore, Hypothesis 4.1 turns out to be exactly what is needed to eliminate the second possibility. (It is probably worth mentioning that this dichotomy also plays a fundamental role in the proof of Furman’s superrigidity theorem.)

More precisely, let π : X̂_{1} → G/Γ_{1} be the map defined by π(x, tΓ_{1}) = tΓ_{1}; and regard X̂_{1} and G/Γ_{1} as Γ_{0}-spaces via the actions defined by

(x, tΓ_{1}) ↦^{γ} φ(γ) · (x, tΓ_{1}) and tΓ_{1} ↦^{γ} φ(γ)tΓ_{1}

SUPERRIGIDITY AND COUNTABLE BOREL EQUIVALENCE RELATIONS 31

respectively. Then the Borel map π ∘ f_{1} : X_{0} → G/Γ_{1} is almost Γ_{0}-equivariant; and so it follows that ω = (π ∘ f_{1})_{∗}μ_{0} is a Γ_{0}-invariant ergodic probability measure on G/Γ_{1}. Furthermore, by definition, for any Borel subset A ⊆ G/Γ_{1},

ω(A) = μ_{0}({x ∈ X_{0} | (π ∘ f_{1})(x) ∈ A}).

Thus understanding the distribution of f_{1}(X_{0}) within X̂_{1} = ⨆_{t∈T} X_{(t)} is the same as understanding the factor (G/Γ_{1}, ω) of (X_{0}, μ_{0}). To accomplish this, we shall make use of the following result of Furman [10, Lemma 4.6], which is a straightforward consequence of Ratner’s measure classification theorem [19, Theorem 3].

Theorem A.1 (Furman [10]). Let G be a noncompact centerless simple Lie group and let Λ, Γ be lattices in G. Then any Λ-invariant ergodic probability measure on G/Γ is either G-invariant (i.e. Haar) or else is supported on a finite set.

We cannot directly apply Theorem A.1 to the factor (G/Γ_{1}, ω). In the statement of Theorem A.1, G/Γ is regarded as a Λ-space under the natural action via left translations; while the action of Γ_{0} on G/Γ_{1} is slightly “twisted”. However, this difficulty is easily dealt with. Let Γ′_{1} = φ^{−1}(Γ_{1}) and regard G/Γ′_{1} as a Γ_{0}-space via its natural action by left translations. Then φ^{−1} induces a Γ_{0}-equivariant map G/Γ_{1} → G/Γ′_{1}, which we shall also denote by φ^{−1}, via

φ^{−1}(tΓ_{1}) = φ^{−1}(t)Γ′_{1};

and ω′ = φ_{∗}^{−1}ω is a Γ_{0}-invariant ergodic probability measure on G/Γ′_{1}. Obviously, (G/Γ′_{1}, ω′) is also a factor of (X_{0}, μ_{0}); and since (X_{0}, Γ_{0}, μ_{0}) satisfies Hypothesis 4.1, it follows that ω′ is not the Haar measure on G/Γ′_{1}. Hence, by Theorem A.1, the support of ω′ must be a finite subset S′ ⊂ G/Γ′_{1}; and so the support of ω is the finite subset S = φ(S′) ⊂ G/Γ_{1}. Clearly the set S must be Γ_{0}-invariant and so there exists a subgroup Λ_{0} ⩽ Γ_{0} such that [Γ_{0} : Λ_{0}] < ∞ and Λ_{0} acts trivially on S. Fix some element tΓ_{1} ∈ S, where t ∈ T. Since Λ_{0} fixes tΓ_{1}, we have that φ(γ)tΓ_{1} = tΓ_{1} for all γ ∈ Λ_{0} and so

φ(Λ_{0}) ⩽ tΓ_{1}t^{−1}.

Furthermore, since ω(tΓ_{1}) = 1/|S|, it follows that

Y_{0} = {x ∈ X_{0} | (π ∘ f_{1})(x) = tΓ_{1}}

32 SIMON THOMAS

satisfies μ_{0}(Y_{0}) = 1/|S|. Let ψ ∈ Aut(G) be the Lie group automorphism defined by ψ(g) = t^{−1}φ(g)t and let f_{2} : X_{0} → X̂_{1} be the Borel function defined by f_{2}(x) = t^{−1} · f_{1}(x). Then the following conditions are satisfied.

(a) ψ(Λ_{0}) ⩽ Γ_{1}.

(b) Y_{0} = {x ∈ X_{0} | f_{2}(x) ∈ j(X_{1})}.

(c) For all γ ∈ Γ_{0}, ψ(γ) · f_{2}(x) = f_{2}(γ · x) for μ_{0}-a.e. x ∈ X_{0}.

(d) G · f_{2}(x) = G · f̂(x) for all x ∈ X_{0}. (As we pointed out in Remark 4.5, ψ(Λ_{0}) is necessarily a subgroup of finite index in Γ_{1}.) Let p : j(X_{1}) → X_{1} be the Γ_{1}-equivariant map defined by p(x, Γ_{1}) = x and let f_{3} : Y_{0} → X_{1} be the Borel function defined by f_{3}(x) = (p ∘ f_{2})(x). Then there exists an ergodic component Z_{0} for the action of Λ_{0} on X_{0} such that Z_{0} ⊆ Y_{0} and

ψ(γ) · f_{3}(x) = f_{3}(γ · x)

for all γ ∈ Λ_{0} and x ∈ Z_{0}. Furthermore, by the ergodicity of the action of Λ_{0} on Z_{0}, we can suppose that there exists an ergodic component Z_{1} for the action of ψ(Λ_{0}) on X_{1} such that f_{3}(Z_{0}) ⊆ Z_{1}. Let f̃ = f_{3} ↾ Z_{0}. Since Λ_{0} preserves the probability measure (μ_{0})_{Z_{0}} on Z_{0}, it follows that ψ(Λ_{0}) preserves the probability measure f̃_{∗}(μ_{0})_{Z_{0}} on Z_{1}; and since the action of ψ(Λ_{0}) on Z_{1} is uniquely ergodic, this implies that f̃_{∗}(μ_{0})_{Z_{0}} = (μ_{1})_{Z_{1}}.

It only remains to show that f̃(x) ∈ Γ_{1} · f(x) for all x ∈ Z_{0}. To see this, recall that f_{2}(x) ∈ G · f̂(x) for all x ∈ X_{0} and that G · f̂(x) ∩ j(X_{1}) = Γ_{1} · f̂(x). If x ∈ Z_{0}, then f_{2}(x) ∈ j(X_{1}) and so f_{2}(x) ∈ Γ_{1} · f̂(x), which implies that f̃(x) ∈ Γ_{1} · f(x). This completes the proof of Theorem 4.4.

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

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