Rotation Equivalence and Cocycle Superrigidity for Compact Actions

ROTATION EQUIVALENCE AND COCYCLE SUPERRIGIDITY FOR COMPACT ACTIONS

FILIPPO CALDERONI

Abstract. We analyze Euclidean spheres in higher dimensions and the corresponding orbit equivalence relations induced by the group of rational rotations from the viewpoint of descriptive set theory. We show that such equivalence relations are not treeable in dimension greater than 2. Then we show that the rotation equivalence in dimension n ≥ 5 is not Borel reducible to the one in dimension 3, and is not Borel reducible to the restriction of the rotation equivalence to its free part in any lower dimension. We also apply our techiniques to give a short new proof of the existence of uncountably many pairwise incomparable equivalence relations up to Borel reducibility. Our methods combine a cocycle superrigidity result from the works of Furman and Ioana with the superrigidity theorem for S-arithmetic groups of Margulis.

1. Introduction

An equivalence relation E on the standard Borel space X is said to be Borel if E ⊆ X ×X is a Borel subset of X ×X. A Borel equivalence relation E is said to be countable if every E-equivalence class is countable. If E and F are countable Borel equivalence relations on the standard Borel spaces X and Y , respectively, then a Borel map f : X → Y is said to be a homomorphism from E to F if for all x, y ∈ X,

x E y =⇒ f(x) F f(y).

A weak Borel reduction from E to F is a countable-to-one Borel homomorphism. Whenever f satisﬁes the stronger property that for all x, y ∈ X,

x E y ⇐⇒ f(x) F f(y),

2010 Mathematics Subject Classiﬁcation. Primary: 03E15, 37A20, 54H05. We would like to thank Alekos Kechris for pointing out Theorem 5.3 and its relevance to this project. Also, we would like to thank Dave Morris and Alex Furman for some insightful discussions about Margulis superrigidity theorem. 1 2 F. CALDERONI then we say that E is Borel reducible to F , that f is a Borel reduction, and we write E ≤B F . Most of the Borel equivalence relations that we will consider in this paper arise from group actions as follows. Let Γ be a countable discrete group. Then a standard Borel Γ-space is a standard Borel space X equipped with a Borel action Γ × X →

X, (g, x) 7→ g · x of Γ on X. We denote by R(Γ y X) the corresponding orbit equivalence relation on X, whose classes are the Γ-orbits. That is,

2 R(Γ y X) := {(x, y) ∈ X | ∃g ∈ Γ(g · x = y)}.

Whenever Γ is a countable group, it is clear that R(Γ y X) is a countable Borel equivalence relation. A classical result of Feldman and Moore [9] establishes that all countable Bore equivalence relations arise in this manner. Precisely, if E is an arbitrary countable Borel equivalence relation on some standard Borel space

X, then there exist a countable group Γ and a Borel action of Γ y X such that E = R(Γ y X). The structure of the class of countable Borel equivalence relations up to Borel reducibility has been studied intesively over the last few decades. By a classical dichotomy by Silver [29] the collection of all (countable) Borel equivalence relations with uncountably many classes has a minimum element, the identity relation on real numbers, denoted by =R. A Borel equivalence relation is called concretely classiﬁable (or smooth) if it is Borel reducible to =R. An example of countable Borel equivalence relation that is not concretly clas- siﬁable is the relation of eventual equality E0 on 2N, the Cantor space of binary sequences equipped with the product topology. In fact, Harrington, Kechris and Louveau [14] showed that if E is a (countable) Borel equivalence relation and E is not smooth, then E0 ≤B E. Therefore, E0 is an immediate successor of =R up to Borel reducibility. At the other extreme we have the phenomenon of universality. A countable

Borel equivlence relation F is said to be universal if and only if E ≤B F for every countable Borel equivalence relation E. The universal countable Borel equivalence relation E∞ is clearly unique up to Borel bi-reducibility, and has many diﬀerent natural realization throughout mathematics. E.g., see [5, 8, 11, 12, 35, 36]. 3

While =R, E0, and E∞ are easily seen to be linearly ordered by ≤B, the structure of the interval [E0,E∞] is far from settled. A technical obstacle to separate

E0 from other nonsmooth Borel equivalence relations was noted first by Hjorth and Kechris [18], who showed that every countable Borel equivalence relation is Borel reducible to E0 when restricted on a comeager subset. Therefore, showing that a countable Borel equivalence relation lies strictly between E0 and E∞ is “generically” hard, and descriptive theoretical methods alone are not enough. The subject fluo- rished after the groundbreaking work of Adams and Kechris [2] who showed that the structure of countable Borel equivalence relations under Borel reducibility is extremely rich. In fact, there are uncountable many pairwise incomparable countable Borel equivalence relations up to Borel reducibility. Since then more results of incomparability have been discovered in the work of Thomas [32, 33, 34], Hjorth and Kechris [17, 19], and Coskey [6, 7]. In this paper, we use the theory of countable equivalence relation to analyze the equivalence relation induced by countable groups of rotations on the spheres in higher dimensions. This new approach is poised to shed light on the long-term project of better understanding the structure of the interval [E0,E∞]. For n > 1, let Sn−1 be the (hyper)sphere in the n-dimensional Euclidean space. As usual, we define Sn−1 = {x ∈ Rn : ||x|| = 1}. Since the linear transormations n n−1 of SOn(R) preserves the dot product of R , we let SOn(R) act on S for all n > 1. More specifically, we consider the action of the countable subgroup of n−1 rational rotations SOn(Q) on S and the induced orbit equivalence relation. To n−1 simplify our notation, we let Rn := R(SOn(Q) y S ). Since SO2(Q) is abelian,

R2 is easily seen to be reducible to E0. (Every countable Borel equivalence relation induced by the action of an abelian group is necessarily hyperﬁnite by a result of

Gao and Jackson [13].) Then, it is natural to investigate the complexity of Rn in higher dimension. Our results in Section 4 show that for no n > 2, the orbit equivalence relation Rn is Borel reducible to E0. In fact, we prove the following stronger result.

Theorem 1.1. Rn is not treeable for all n ≥ 3.

To see why Theorem 1.1 implies that no Rn is hyperﬁnite for n > 2, we recall that the class of treeable countable Borel equivalence relations is downword closed 4 F. CALDERONI under Borel reducibility. That is, if E is Borel reducible to F and F is treeable, then E is treeable too. Moreover, it is well-known that E0 is treeable. In light of the results of Adams and Kechris [2] and Thomas [31] it seems natural to conjecture that the complexity of Rn increases with the dimension. While we still do not know whether Rn is Borel reducible to Rn+1 for n > 2, we make the following conjecture:

Conjecture 1.2. If n > m, then Rn is not Borel reducible to Rm.

Notice that conjecture 1.2 implies that no Rn is universal. Also, if m = 2,

Conjecture 1.2 follows easily from the fact that R2 is hyperﬁnite and Theorem 1.1.

When X is a standard Borel Γ-space denote by FrΓ X the free part of the action

Γ y X. That is, FrΓ X := {x ∈ X : ∀g 6= 1Γ(g · x 6= x)}. Whenever Γ is clear from the context we let Fr X = FrΓ X. Since Fr X is a Γ-invariant Borel set, we denote ∗ ∗ n−1 by Rn the restriction of Rn to the free part. That is, Rn := R(SOn(Q) y Fr S ). In Section 5 we prove the following weaker form of Conjecture 1.2.

∗ Theorem 1.3. If n ≥ 5 and m < n, then Rn is not Borel reducible to Rm.

The following is a straightforward consequence our result.

∗ Corollary 1.4. For all n ≥ 2, the countable Borel equivalence relation Rn is not universal.

For dimension three we have a better understanding of the nonfree part of the m−1 action SOm(Q) y S . So, using Theorem 1.3, we can verify Conjecture 1.2 for all n ≥ 5 and m = 3.

Theorem 1.5. If n ≥ 5, then Rn is not Borel reducible to R3.

Our proofs combine techniques from ergodic theory such as the analysis of cocycles associated to Borel homomorphisms and free actions, Kazhdan property (T), and Margulis’ superrigidity results [25]. The main novelty of our argument is using a cocycle rigidity result for compact actions which follows directly from the work of Furman [10] and Ioana [20]. We believe that this could be useful for further applications in the theory of countable Borel equivalence relations. 5

In the last section we discuss some further application of our methods. We present a short new proof of the existence of uncountably many pairwise incomparable countable Borel equivalence relations.

2. Preliminaries

2.1. Ergodic theory and descriptive set theory. Suppose that Γ is a countable group and that X is a standard Borel Γ-space. If µ is a Γ-invariant probability measure on X, then the action of Γ on (X, µ) is said to be ergodic if for every Γ-invariant Borel subset A ⊆ X, either µ(A) = 0 or µ(A) = 1.

Theorem 2.1. Let µ be a Γ-invariant probability measure on the standard Borel Γ-space X, then the following statements are equivalent:

(i) The action of Γ on (X, µ) is ergodic. (ii) If Y is a standard Borel space and f : X → Y is a Γ-invariant Borel function, then there is a Γ-invariant Borel subset M ⊆ X with µ(M) = 1 such that

f M is a constant function.

When Γ is a countable discrete group and Λ < Γ is a subgroup, the following are equivalent.

(1) [Γ : Λ] < ∞. (2) There exists a constant c > 0 such that for any measure-preserving ergodic action of Γ on a standard probability space (X, µ) there is a measurable Λ-

invariant subset Z ⊆ X with µ(Z) ≥ c and Λ acts ergodically on (Z, µZ ),

where µZ is the probability measure deﬁned by µZ (A) = µ(A)/µ(Z).

When Z is as in clause (2) we say that Z is an ergodic component for the action of Λ on X. Suppose that Γ is a countable group and that X is a standard Borel Γ-space with an invariant probability measure µ. Until further notice let E = R(Γ y X) and F be a Borel equivalence relation on a standard Borel space Y . Then (E, µ) is said to be F -ergodic if for every Borel homomorphism f : X → Y from E to F , there exists a Γ-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 will usually say that E is F -ergodic. 6 F. CALDERONI

Notice that ergodicity is a strong obstruction to Borel reducibility. When the action of Γ on (X, µ) is ergodic, then E is not Borel reducible to =R. Moreover, provided that E is F -ergodic, then E is not weakly Borel reducible to F . In particular, E is not Borel reducible to F . Recall that an equivalence relation relation F is said to be finite if every F - equivalence class is finite. A countable Borel equivalence relation E on the standard Borel space X is said to be hyperfinite if there exists an increasing sequence

F0 ⊆ F1 ⊆ · · · ⊆ Fn ⊆ · · ·

S of ﬁnite Borel equivalence relations on X such that E = Fn. n∈N

Proposition 2.2. Let Γ be a countable nonamenable group and let X be a standard

Borel Γ-space and µ be a Γ-invariant probability measure on X. Then R(Γ y X) is not hyperﬁnite.

A long-standing open problem in descriptive set theory asks whether any orbit equivalence relation generated by a Borel action of a countable amenable group is hyperﬁnite. Below we will mention the following important result by Gao and Jackson.

Theorem 2.3 (Gao–Jackson [13]). If Γ is a countable abelian group and Γ acts on a standard Borel X in a Borel fashion, then R(Γ y X) is hyperﬁnite.

A graphing of an equivalence relation is a graph whose connected components coincide with the equivalence classes. We say that a countable Borel equivalence relation E on a standard Borel space is treeable if there is a Borel acyclic graphing of E. If Fn is the free group on n generators for some n ∈ N ∪ {∞}, and Fn acts freely on the standard Borel space X, then R(Fn y X) is treeable. The class of treeable countable Borel equivalence relations admits a universal

2 element, which si deﬁned as the orbit equivalence relation R(F2 y Fr 2F ) induced

2 by the shift action of F2 on the free part of 2F := x | x: F2 → {0, 1} . Clearly,

2 R(F2 y Fr 2F ) is not hyperfinite for Proposition 2.2. The following are some well-known properties of hyperfinite and treeable equivalence relations. (E.g., see [21, Proposition 1.3–3.3].) An immediate consequence is that every hyperfinite equivalence relation is treeable. 7

Proposition 2.4. Let E and F be countable Borel equivalence relations.

(a) If E ⊆ F and F is hyperﬁnite (respectively treeable), then E is hyperﬁnite (respectively treeable).

(b) If E ≤B F and F is hyperﬁnite (respectively treeable), then E is hyperﬁnite (respectively treeable).

2.2. Algebraic groups and lattices. Let Ω to be a fixed algebraically closed field of characteristic 0 containing R and all p-adic fields Qp, with p prime.

An algebraic group G is a subgroup of the general linear group GLn(Ω) which is

Zariski closed in GLn(Ω), i.e., G consists of all matrices M in GLn(Ω) which satisfy a set of equations f1(M) = 0, . . . , fk(M) = 0, where each fi is a polynomial in

Ω[x1, . . . , xn]. When the equations defining G have coefficients in a subfield k ⊆ Ω, we call G a k-group. If R is a subring of Ω, we let

−1 GLn(R) = {(aij) ∈ GLn(Ω) : aij ∈ R and (det(aij)) ∈ R}, and deﬁne for any algebraic group G ⊆ GLn(Ω),

G(R) = G ∩ GLn(R).

In particular, if G is a k-group and R = k, then G(k) is the group of all matrices in

G with coeﬃcients in k. Typical examples include the special linear group SLn(k) and the special orthogonal group SOn(k). Moreover, when p is a prime number we 1 1 1 denote by Z p the ring generated by p . Then, SOn(Z p ) denotes the groups of special orthogonal matrices in SOn(R) with rational entries, whose denominators are powers of p. If G is an algebraic k-group, then G 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 G ≤ GLn(Ω) is an algebraic k-group, where k is either R or the

ﬁeld Qp of p-adic numbers for some prime p. In this case, G(k) ≤ GLn(k) is a lcsc group with respect to the Hausdorﬀ topology; i.e., the topology obtained by

2 restricting the natural topology on the product space kn to G(k). Unless otherwise speciﬁed any topological notion about G(k) will refer to the Hausdorﬀ topology. 8 F. CALDERONI

Let G be a connected algebraic group. The (solvable) radical Rad(G) is the maximal normal connected solvable subgroup of G. The connected algebraic group G is called semisimple if and only if Rad(G) is trivial. Algebraic groups considered in this paper are generally semisimple. When G is a semisimple algebraic group defined over k, then the k-rank of G, in symbols rankk(G), is defined as the maximal dimension of an abelian k-subgroup of G which is k-split, i.e., which can be diagonalized over k. Recall that if G is a locally compact second countable group and Γ ≤ G is a subgroup, we say that Γ is a lattice in G if and only if Γ is discrete and there is an invariant Borel probability measure for the canonical action of G on G/Γ defined by (g, hΓ) 7→ ghΓ. Useful examples of lattices arise in the context of arithmetic groups. (See also [39, Chapter 10].) We briefly describe a few examples that will appear later in this paper.

Let S = {p1, . . . , pt} be a ﬁnite nonempty set of primes. For every p ∈ S, we denote by Qp the ﬁeld of p-adic numbers. When G is an algebraic Q-group, we can 1 1 identify ΓS := G [ ,..., ] with its image under the diagonal embedding into Z p1 pn

G(R) × G(Qp1 ) × · · · × G(Qpt ).

The group ΓS is a typical example of S-arithmetic group. We shall deﬁne S- arithmetic groups and discuss some relevant properties in Section 5.2. Here we recall the following result.

Theorem 2.5 (A. Borel [4],). If G is a connected semisimple Q-group, then ΓS is a lattice in G(R) × G(Qp1 ) × · · · × G(Qpt ).

1 In particular, we shall use the fact that SOn Z p is a lattice in SOn(R) ×

SOn(Qp) in Section 4.

3. A descriptive view of the n-spheres

In this section we introduce the measure theoretic machinery that is fundamental to the main results of this paper. As a warm up, we shall discuss the proof of the following proposition. 9

2 Proposition 3.1. The orbit equivalence relation R3 = R(SO3(Q) y S ) is not hyperﬁnite.

Clearly, in order to apply Proposition 2.2 we need to ﬁnd an invariant probability measure on the sphere.

3.1. Invariant ergodic measures on the n-spheres. If G is a lcsc group, then there exists a (nonzero) Borel measure mG on G which is invariant under the action of G on itself via left translations. Such a measure is called a Haar measure on G. 0 If mG is a second Haar measure on G, then there exists a positive real constant c 0 such that mG(A) = cmG(A) for every Borel subset A ⊆ G. It is well-known that

G is compact if and only if mG(G) < ∞. In this case, there exists a unique Haar probability measure on G. The group G is said to be unimodular if and only if each Haar measure on G 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 < K is a closed subgroup. Then both K and L are unimodular. Hence there exists a unique K- invariant probability measure on the standard Borel K-space K/L. (For example, see [28, Theorem 3.17].) The measure is called the Haar probability measure on K/L. The Haar probability measure on K/L can be described explicitly as the pushforward of the Haar probability measure mK on K through the canonical surjection π : K → K/L.

Lemma 3.2 (See [32, 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 action of Γ on K/L is uniquely ergodic; i.e., µ is the unique Γ-invariant probability measure on K/L.

Note that if µ is the unique Γ-invariant probability measure and therefore the set of invariant measures consists of only one point, then µ is necessarily ergodic.

Lemma 3.2 can be used to deﬁne the invariant measure on Sn. It is well-known n−1 that the compact group SOn(R) acts transitively on S . Then the stabilizer of n−1 each point of S in SOn(R) is easily seen to be isomorphic to SOn−1(R). (In fact, n−1 since SOn(R) acts transitively, the stabilizers of all points of S are conjugate 10 F. CALDERONI

n−1 n−1 subgroups of SOn(R).) Hence, given any ﬁxed x0 ∈ S , we can identify S with the coset space SOn(R)/ SOn−1(R) via the map x 7→ {g ∈ SOn(R): g · x0 = x}. Slightly abusing notation, we shall denote both the Haar probability measure on n−1 SOn(R)/ SOn−1(R) and the corresponding probability measure on S by µn and call µn the spherical measure. Since SOn(Q) is dense in SOn(R) for all n ∈ N, n it follows that µn is the unique SOn(Q)-invariant measure on S by Lemma 3.2 . n−1 Thus, the action SOn(Q) y (S , µn) is ergodic.

Notice that it is possible to deﬁne µn in terms of Lebesgue measure. In fact, if n λn is the Lebegue measure in R and B1(0) is the unitary n-dimensional ball, then

1 µn(A) = λn({tx : x ∈ A, 0 ≤ t ≤ 1}). λn(B1(0))

For more details about this approach we refer the interested reader to [26, The- orem 3.4–3.7]. Now we have all ingredients to prove that R3 is not hyperﬁnite. (Proposition 3.1.)

2 Proof of Proposition 3.1. Let µ = µ3 be the spherical measure on S , and consider 2 the orbit relation R3 induced by the action of SO3(Q) on S . Because of the previous 2 discussion SO3(Q) acts on (S , µ) ergodically. It is well-known that SO3(Q) contains a free subgroup Γ of rank 2. E.g., as pointed out by Tao [30] we can deﬁne Γ = hσ, τi the group generated by the matrices     3 4 0 5 0 0 1   1       σ = −4 3 0 and τ = 0 3 4 . 5   5   0 0 5 0 −4 3

Now, by Proposition 2.4 it suffices to show that the orbit equivalence relation induced by the action of Γ is not hyperfinite. Let Z be the set of points in S2 that are fixed by some non-identity transformation in Γ, that is Z = {x ∈ S2 : ∃γ ∈ Γ r {1} γ · x = x}. Each non-identity trasformation in Γ is a rotation with exactly two fixed points on S2, namely, the ones along its rotation axis. Consequently, Z and its saturation Γ · Z are countable sets, thus µ(Γ · Z) = 0. Let M := S2 \ Γ · Z. It is clear that M is Γ-invariant. Since Γ is not amenable, the induced equivalence relation R(Γ y M) is not hyperfinite by Proposition 2.2. It follows that R3 is not hyperfinite by Proposition 2.4 (b) as desired. 11

Notice that SO2(Q) is the group of rotations of the plane about the origin with rational entries. Hence SO2(Q) isomorphic to a subgroup of the abelian group R/Z. 1 It readily follows that the orbit equivalence relation induced by SO2(Q) y S is hyperﬁnite because of Theorem 2.3.

The fact that R2 is not concretely classiﬁable is a consequence of the following classical result in descriptive set theory.

Proposition 3.3 (E.g., see [16, Corollary 3.5]). Let G be a countable group acting by homeomorphisms on a Polish space X. If there is a dense orbit and every orbit is meager, then the induced orbit equivalence relation R(G y X) is not smooth.

Proposition 3.4. R2 is not concretely classiﬁable. Thus, R2 ∼B E0.

1 4 −3 Proof. Let A = 5 3 4 . It is easily checked that A ∈ SO2(Q), and A is the rotation 1 3 4 2πir of S by the angle θ with sin θ = 5 and cos θ = 5 . That is, A · x = xe for all x ∈ S1 and for r ∈ [0, 1) with θ = 2πr. By Niven’s theorem [27, Corollary 3.12], it follows that r is an irrational number. So, let Γ = hAi be the subgroup of SO2(Q) generated by A. Since r is irrational, we know that for any point x ∈ S1 the orbit Γ · x is dense in S1. It follows that R(Γ y S1) is not smooth by Proposition 3.3.

Consequently, R2 is not smooth by Proposition 2.4 (a).

The results of next section generalize and strengthen Proposition 3.1. In fact, we shall prove that for all n ≥ 3 the orbit equivalence relation of Rn is not treeable.

4. Proof of Theorem 1.1

We recall the definition of property (T) introduced by Kazhdan [22]. Let G be a lcsc group and let π : G → U(H) be a unitary representation of G on the separable Hilbert space H. We say that π almost admits invariant vectors if for every > 0 and every compact subset K ⊆ G, there exists a unit vector v ∈ H such that kπ(g) · v − vk < for all g ∈ K. We say that G has Kazhdan property (T) if for every unitary representation π of G, if π almost admits invariant vectors, then π has a non-zero invariant vector. It is well-known that if Γ is a countable group and Γ has property (T), then Γ is finitely generated. The first ingredient in the proof of Theorem 1.1 is the following antitreeability result that was isolated by Hjorth and Kechris in [18, Theorem 10.5]. 12 F. CALDERONI

Theorem 4.1 (essentially Adams-Spatzier [1]). Let Γ be a countable group with Kazhdan’s property (T), X a standard Borel Γ-space and µ a Γ-invariant, nonatomic, ergodic measure on X. Then R(Γ y X) is not treeable. (In fact, if F is any treeable countable Borel equivalence relation, then R(Γ y X) is F -ergodic.)

Remark 4.2. Hjorth [15] pointed out that the hypothesis that the action of Γ y (X, µ) be ergodic is not necessary. The weaker assumption that R(Γ y X) is not smooth suﬃces.

Also, we shall use the next result, which is contained also in [3, Theorem 6.4.4].

Theorem 4.3 (Margulis [24]). Let p be a prime number with p ≡ 1 (mod 4). 1 For n > 4, the group SOn Z p is a dense subgroup of SOn(R) with Kazhdan’s property (T).

Lemma 4.4. For n ≥ 5, Rn is not treeable.

1 Proof. Fix n ≥ 5 and let Γ = SOn Z 5 . Since the class of treeable equivalence relations is closed under containement it suﬃces to show that the orbit equivalence relation R(Γ y X) is not treeable. We emphasize that Γ is a dense subgroup of n−1 SOn(Q) by Theorem 4.3, therefore the action of Γ y (S , µn) is ergodic because of Lemma 3.2. Then, since Γ has property (T), it follows R(Γ y X) is not treeable by Theorem 4.1.

The proof of Lemma 4.4 does not generalize to all n ≥ 3. In fact, Zimmer [38] proved that SOn(R) contains no inﬁnite countable Kazhdan subgroup for n = 3, 4. However we can use a diﬀerent anti-treeability result of Kechris [23] to settle the remaining cases. We recall that a countable group Γ is said to be anti-treeable if and only if for every Borel action of Γ on a standard Borel space X, which is free and admits an invariant probability Borel measure, the induced equivalence relation R(Γ y X) is not treeable.

Theorem 4.5 (Kechris [23, Theorem 9]). Let Γ be a countable group which is a lattice in a product group G = G1 × G2 of two locally compact second countable groups, each of which contains an inﬁnite amenable discrete group. Then, if Γ is not amenable, Γ is anti-treeable. 13

Before proving the analogue of Proposition 4.4 in dimension n = 3, 4, we point out the following:

n−1 Proposition 4.6. For all n ≥ 2 the free part of the action of SOn(Q) on S is n−1 conull. That is, µn Fr(S ) = 1.

n Proof. Consider the action of SOn(Q) on the n-dimensional euclidean space R . n Let µ = µn be the spherical measure and denote by λ the Lebesgue measure in R . For any nonidentity A ∈ Γ, the linear subspace of ﬁxed vectors {x ∈ Rn | Ax = x} is an eigenspace with eigenvalue 1, thus has dimension less than n. It follows that, n λ({x ∈ R | Ax = x}) = 0. If A ∈ Γ is not the identity, denote by FixA = {x ∈ Sn−1 : Ax = x}. Clearly

1 n−1 µ(FixA) = λ({tx | x ∈ S , Ax = x and t ∈ [0, 1]}) = 0. λ(B1(0)) Therefore,

n−1 n−1 [ µ(Fr(S )) = µ S \ FixA = 1. A∈Γ\{1}

Proposition 4.7. For n = 3, 4 the orbit equivalence relation Rn is not treeable.

1 Proof. As we explained in Section 2 we can consider SOn Z 5 as a lattice in 1 n−1 SOn(R) × SOn(Qp). Obviously the action SOn Z 5 y S is free on the free n−1 1 part Fr S , and the measure µn is SOn Z 5 -invariant. 1 Next, notice that SOn Z 5 satisﬁes the assumptions of Theorem 4.5. For 1 n = 3, the matrices σ, τ as in the proof of Proposition 3.1 belongs to SO3 Z 5 , 1 which is not amenable as it contains a copy of F2. For n = 4, we see that SO4 Z 5 is not amenable the same way. Arguing exactly as in Tao [30] one can prove that the matrices     3 4 0 0 5 0 0 0         1 −4 3 0 0 1 0 3 4 0 σ =   and τ =   . 5   5    0 0 5 0 0 −4 3 0     0 0 0 5 0 0 0 5

1 generates a nonabelian free subgroup of SO4 Z 5 . In both cases, Theorem 4.5, ∗ ∗ yields that the orbit equivalence relation R is not treeable. Since Rn is obviously

Borel reducible to Rn, it follows that Rn is not treeable by Proposition 2.4 (b). 14 F. CALDERONI

5. Irreducibility results

Before proving the main results of this paper we brieﬂy introduce the notion of cocycle.

5.1. Cocycles. Let Γ be a countable group and let X be a standard Borel Γ-space with an invariant probability measure µ.

Deﬁnition 5.1. Suppose that ∆ is a countable group. Then a Borel function α:Γ × X → ∆ is called a cocycle if for all g, h ∈ Γ

α(hg, x) = α(h, g · x)α(g, x) µ-a.e.(x).

Moreover, the cocycle is called strict if this equation holds for all g, h ∈ Γ and x ∈ X.

Suppose that X0 ⊆ X is a Γ-invariant set and α:Γ × X0 → ∆ is a strict cocycle. Then we can obviously extend α to a strict cocycle α¯ :Γ × X → ∆ by setting

α¯(g, x) = α(g, x) for all x ∈ X0, and α¯(g, x) = 1∆ for all x∈ / X0. Further, since Γ is assumed to be countable, for every cocycle α:Γ × X → ∆, there is a strict cocycle α0 such that for all g ∈ Γ

α(g, x) = α0(g, x) µ-a.e.(x).

In fact, if we let Ag,h := {x | α(hg, x) = α(h, g · x)α(g, x)}, for all g, h ∈ Γ, and T S A := g,h∈Γ Ag,h, then X0 = g∈Γ g · A is a Γ-invariant Borel set with µ(X0) = 1 0 and α Γ×X0 is strict. Then we can deﬁne α by extending α Γ×X0 as described above. Therefore, we shall assume that every cocycle is strict. Cocycles typically arise in the following manner. Let Γ and ∆ be countable discrete groups. Let X, Y be a standard Borel Γ-space and a ∆-space, respectively.

Also, suppose that the action ∆ y Y is free. If f : X → Y is a Borel homomorphism from R(Γ y X) to R(∆ y Y ), then we can deﬁne a Borel cocycle α:Γ × X → ∆ by letting α(g, x) be the unique element of ∆ such that

α(g, x) · f(x) = f(g · x).

In this case we say that α is the cocycle associated to f. Further, notice that whenever α(g, x) = α(g) only depends on g, then α: G → H is a group homomorphism. Suppose now that B : X → ∆ is a Borel function and that f 0 : X → Y is deﬁned 15 by f 0(x) = B(x) · f(x). Then f 0 is a Borel homomorphism from R(Γ y X) to R(∆ y Y ) as well, and the corresponding cocycle β :Γ × X → ∆ satisﬁes

β(g, x) = B(g · x)α(g, x)B(x)−1 for all g ∈ Γ and x ∈ X. The above equation states that α and β are equivalent as cocycles (or cohomologous). This notion is made precise by the following deﬁnition.

Deﬁnition 5.2. Suppose that ∆ is a countable group. Then the cocycles α, β :Γ× X → ∆ are cohomologous (or equivalent) iﬀ there exist a Borel function B : X → H and a Γ-invariant Borel subset X0 ⊆ X with µ(X0) = 1 such that

β(g, x) = B(g · x)α(g, x)B(x)−1 for all g ∈ Γ and x ∈ X0.

Analyzing cocycles that arise from Borel homomorphisms could be useful to prove an irreducibility result, namely, to prove that a given countable Borel equivalence relation is not Borel reducible to another one. With the same notation as above, suppose that f : X → Y is a Borel homomorphism from R(Γ y X) to R(∆ y Y ) and α is the cocycle associated to f. If α is cohomologous to β and there are further restrictions on β, we might be able to use ergodicity to ﬁnd serious obstructions for a Borel reduction from R(Γ y X) to R(∆ y Y ).

Now let G be a connected compact group with Haar measure mG. Suppose that Γ is a countable group together with a dense embedding Γ ,→ G so that the left-translation action Γ y (G, mG) is ergodic. Let α:Γ × G → Λ be a cocycle.

If Γ has Kazhdan property (T), then there exist a neighborhood V of 1G and a constant C ∈ (31/32, 1) such that mG({x ∈ G | α(g, xt) = α(g, x)}) ≥ C for all g ∈ Γ and every t in V . (See Ioana [20, p. 2742].) This “local uniformity” condition has many useful consequences. In particular, we are interested in the following:

Theorem 5.3. Let Γ ≤ G be a dense subgroup of a connected, compact group G. Suppose that Γ has Kazhdan property (T). Consider the left translation action

Γ y (G, mG), where mG is the Haar measure of G. Assume that π1(G), the fundamental group of G, is ﬁnite. Let Λ be a countable group and α:Γ × G → Λ be a cocycle. If any group homomorphism π1(G) → Λ is trivial, then α is cohomologous to a homomorphism δ :Γ → Λ. 16 F. CALDERONI

Theorem 5.3 is a special case of a result of Ioana [20, Theorem 3.2], that was abstracted from an argument of Furman [10, Theorem 5.21]. Also, in the proof of Theorem 1.3 we shall use the following well-known result. (E.g., see [3, Corollary 1.3.5].)

Proposition 5.4. Let G be a topological group with Kazhdan Property (T), and let H be a locally compact amenable group. If ρ: G → H is a continuous homomorphism then ρ(G) is relatively compact.

5.2. S-arithmetic groups. Before introducing S-arithmetic groups and discussing their properties we recall some basic facts and terminology about valuations. A global field is a finite extension of the field of rational numbers. Throughout this section let k be a global field. A valuation on k is a function v : k → R+ ∪ {0} such that

• v(x) = 0 if and only if x = 0; • v(xy) = v(x)v(y) for all x, y ∈ k; • v(x + y) ≤ v(x) + v(y) for all x, y ∈ k.

To avoid trivial valuation we also assume that v(x) 6= 1 for some x ∈ k, x 6= 0.

Notice that any valuation v induces a metric dv on k by setting dv(x, y) = v(x − y) for all x, y ∈ k. We denote by kv the completion of k relative to this metric. Two valuations v and w on the same field are said to be equivalent precisely when the corresponding metric dv and dw induce the same topologies. A valuation v is said to be non-archimedean if and only if it satisfies the ultramet- ric inequality. Otherwise, we say that v is archimedean. A valuation is archimedean if and only if kv is archimedean. (A local field k is said to be archimedean if and only if it is isomorphic to R or C.) Also, we notice that k is not archimedean if and only if it is totally disconnected. With our assumption we have char(k) = 0, so the set of archimedean valuation is non-empty. The set {x ∈ k | v(x) ≤ 1} is called the ring of integers k. This is the unique maximal compact subring k. When v is non-archimedean the ring of integers of kv is usually denoted by Ov. More generally, suppose that S is a set of valuation of k. An element x ∈ k is said to be S-integral if and only if v(x) ≤ 1 for each non-archimedean valuation in S. The set of S-integral elements is a subring of k and is denoted by k(S). 17

In this paper we mainly deal with the case k = Q. To discuss valuations uniformly, it we let Q∞ = R.

For p = ∞, let vp(x) = |x| be the standard absolute value. If p is a prime −` number, then vp is the standard p-adic valuation. That is, we set vp(x) = p , ` a where x = p b , and a, b are coprime with p. (We let vp(0) = ∞). It is well-known that every valuation on Q is equivalent to vp for some p ∈ {p | p is prime} ∪ {∞}.

Clearly, vp is archimedean if and only if p = ∞. An element of x ∈ Q is S-integral m if and only if x = n where m ∈ Z and n ∈ N and the prime factors of n are in S. 1 For instance, if S = {p, ∞}, then Q(S) = Z p . If G is a connected algebraic Q-group, then we have a natural embedding G(Q(S)) Q embeds into p∈S G(Qp) as a discrete subgroup. In fact, G(Q(S)) is a lattice in Q p∈S G(Qp). More generally, we consider the following deﬁnition.

Deﬁnition 5.5. Let S be a ﬁnite subset of V and let G be a Q-group. A subgroup of G is said to be S-arithmetic if and only if it is commensurable with G(k(S)).

Suppose that G, H are k-groups. A k-homomorphism f : G → H is said to be a k-isogeny if and only if it is surjective and has ﬁnite kernel. We recall that for every connected semisimple G there exist algebraic groups Ge, and an isogeny π : Ge → G so that for every isogeny ρ: H → G, there is an isogeny πρ : Ge → H such that the composition ρ ◦ πρ = π. The group Ge is called the (algebraic) universal covering of G. If G is a k-group, so is Ge, and π is deﬁned over k. Next proposition is a special case of Margulis [25, Theorem 3.2.9, Chapter 1].

Proposition 5.6. Let S ⊆ V be a ﬁnite set of valutations of the ﬁeld k. Suppose G is a connected simple k-group. If Λ is an S-arithmetic subgroup of G, then π−1(Λ) is an S-arithmetic subgroup of Ge.

The following is known as Margulis’ normal subgroup theorem for S-arithmetic groups. We let X rankS G = rankkp G. p∈S∪{∞}

Theorem 5.7 (Margulis, [25, Theorem A, Chapter VIII]). Suppose that G is a connected almost k-simple k-group. Let Γ be an S-arithmetic subgroup of G and let

N be a normal subgroup of Γ. Suppose rankS G ≥ 2 and that either G is connected 18 F. CALDERONI or S is finite. Then either N lies in the centre of G or the quotient Γ/N is finite (in which case N has finite index in Γ).

We state a particular case of the famous Margulis’ superrigidity theorem for S-arithmetic groups ([25, Theorem 5.14, Chapter VII]).

Theorem 5.8 (Margulis). Suppose that for each p ∈ S, Gp is a connected, semisimple Qp-group such that Gp(Qp) has no anisotropic factors. Suppose also that Q rankS Gp ≥ 2 and Γ is an irreducible lattice in G = p∈S∪{∞} Gp(Qp). Let k be Qq for some prime number q and H be a connected almost k-simple k-group. Suppose that ρ:Γ → H(k) is a homomorphism with ρ(Γ) Zariski dense in H.

(a) If q∈ / S, then ρ(Γ) is relatively compact in H(k). (b) If q ∈ S and ρ(Γ) is not relatively compact in H(k), then there is a uniquely determined continuous homomorphism ρˆ: G → H(k) and a uniquely determined homomorphism ν :Γ → Z(H) such that ρ(g) = ν(g) · ρˆ(g), for all g ∈ Γ.

5.3. Proof of Theorem 1.3. Recall that we identify Sn−1 with the coset space

SOn(R)/ SOn−1(R). If x ∈ SOn(R), we denote by [x] the coset x SOn−1(R) for readability. Let n ≥ 5 and m < n. Suppose that f : Sn−1 → Sm−1 is a Borel map ∗ n−1 such that [x] Rn [y] ⇐⇒ f([x]) Rm f([y]) for all [x], [y] ∈ S . 1 Put Γ = SOn Z 5 , µ = µn, and ∆ = SOm(Q). Clearly, f is also a countable- n−1 ∗ ∗ to-one homomorphism from R(Γ y S ) into Rm. Since Rm is free, let ω :Γ × Sn−1 → ∆ be the Borel cocycle associated to f.

Let G := SOn(R) and denote by mG the Haar measure on G. Since the action n−1 of Γ on S = SOn(R)/ SOn−1(R) is a quotient of the left-translation action of

Γ on SOn(R), the cocycle ω induces a Borel cocycle α:Γ × G → SOm(Q) for the left-translation action Γ y (G, mG), which is deﬁned by setting α(g, x) = ω(g, [x]). As usual let Ge be the universal covering group of G, and let p: Ge → G the 2-1 covering map. Also let Γe = p−1(Γ) be the inverse image of Γ. We can lift α to a cocycle αe: Γe × Ge → ∆ by setting αe(g, x) = α(p(g), p(x)). Now it is easily checked that the hypothesis of Theorem 5.3 are satisﬁed. Since π1(Ge) = {e} is trivial, the only group homomorphism π1(Ge) → ∆ is obviously the trivial one. Moreover, since Ge is connected, it follows that Γe < Ge is dense. Therefore, by Theorem 5.3 there 19 are a group homomrphism ϕ: Γe → ∆ and a Borel map B : Ge → ∆ such that

ϕ(g) = B(g · x)α(g, x)B(x)−1 for all g ∈ Γe and almost every x ∈ Ge. Recall that Ge = Spin(n) whose center Z(Ge) is ﬁnite. (E.g., see [37, page 40]) Let K = ker ϕ. Because of Proposition 5.6, Γe is S-arithmetic in Ge for S = {∞, 5}.

Moreover, isogenous groups have the same S-rank, so rankS Ge ≥ 2. (This observa- tion follows from [25, Corollary 1.4.6 (a), Chapter 1] and the fact that rankQp SOn = n b 2 c when p ≡ 1 mod 4.) By Margulis’ normal subgroup theorem for S-arithmetic groups (see Theorem 5.7) it follows that either:

(1) [Γ:e K] < ∞, or else (2) ϕ: Γe → ∆ is a virtual embedding; i.e., K is ﬁnite.

Now we analyze the two cases separately. In case [Γ:e K] < ∞, then there is a

K-invariant measurable Z ⊆ Ge with me G(Z) > 0 such that K acts ergodically on (Z, µZ ), where µZ (A) = me G(A)/me G(Z). Then consider the cocycle

αe (K × Z): K × Z → ∆.

Next, for all x ∈ X we can deﬁne the “adjusted” Borel homomorphism by f 0(x) = 0 0 B(x) · f(x) so that ϕ is the cocycle associated to f Z and f Z is a weak Borel ∗ 0 0 0 reduction from R(K y Z) to Rm. It follows that f (k · x) = ϕ(k) · f (x) = f (x) for all k ∈ K and x ∈ Z. This shows that f 0 : Z → Y is a K-invariant map. Since 0 K acts ergodically on (Z, µZ ), it follows that f is constant on a measure 1 set, which is a contradiction. On the other hand, the following lemma shows that there is no virtual embedding

ρ: Γe → SOm(Q). So, we can exclude case (2).

Lemma 5.9. For every group homomorphism ρ: Γe → SOm(Q) is a homomorphism, then ρ(Γ)e is ﬁnite.

Proof. Since Γe is finitely generated, there are only finitely many prime numbers p1, . . . , pn that appear in the denominators in the entries of the matrices in ϕ(Γ)e . So, it suffices to show that the power of each prime appearing in the denominators of the matrix entries in ϕ(γ) ∈ ϕ(Γ)e is uniformly bounded over γ ∈ Γe. This will 20 F. CALDERONI yield some N ∈ N so that

ϕ(Γ)e ⊆ (aij) ∈ SOm(Q) | aij has denominator less than N .

It will follow that ϕ(Γ)e is discrete in SOn(R), therefore ϕ(Γ)e is ﬁnite because SOn(R) is compact.

For every q ∈ {p1, . . . , pn}, let SOm(Q) → SOm(Qq) be the canonical embedding so that we can view ρ as a map ρq : Γe → SOm(Qq).

If q 6= 5, then let ρq : Γe → SOm(Qq). Clearly, ρq(Γ)e need not be Zariski closed in SOm(Qq) so we cannot apply Theorem 5.8 directly. However, the Zariski closure of ρq(Γ)e is an algebraic group over Qp, so let Mq be the Zariski closure of ρq(Γ)e .

Here notice that Mq need not be semisimple, so we consider the semisimple group deﬁned as Lp = Mq/R(Mq) where R(Mq) is the (solvable) radical of Mq, and is deﬁned to be the maximal normal connected solvable subgroup of Mq. Clearly Lq is a semisimple Qq-group and let σ : Mq → Lq be the quotient map.

Now the map σ ◦ ρq is a group homomorphism with Zariski dense image. There- fore, if we let Kq = σ ◦ ρq(Γ)e we have that Kq is relatively compact in Lq by −1 Theorem 5.8. It follows that σ (Kq) is an extension of a compact group by a solvable group, so it is amenable. Therefore, since Γe has property (T), we have that ρq(Γ)e is relatively compact because of Proposition 5.4. Otherwise, if q = 5, the statement follows by Theorem 5.8–((b)), the fact that

Z(SOm) is ﬁnite, and the fact that every continuous homomorphism Ge → SOm is trivial because m < n.

This concludes the proof of Theorem 1.3. Now, we can leverage Theorem 1.3 to show that Rn 6≤B R3, for n ≥ 5.

n−1 2 Proof of Theorem 1.5. Suppose that n ≥ 5 and f : S → S such that x Rn n−1 y ⇐⇒ f(x) R3 f(y). Let µ = µn be the spherical measure on S . Clearly A := −1 2 n−1 f (Fr S ) is a SOn(Q)-invariant Borel subset of S . Since the action of SOn(Q) n−1 on (S , µn) is ergodic, we obtain that A is either null or conull. If µn(A) = 1, n−1 2 then f(S ) is almost contained in the free part of the action SO3(Q) y S . So we can argue as in the proof of Theorem 1.3 to obtain a contradiction. Therefore n−1 we can suppose that µn(A) = 0, without losing generality. Let X0 = S r A. 2 2 It follows that µn(X0) = 1 and f(X0) is contained in S r Fr(S ), which is the 21

2 non-free part of the action SO3(Q) y Fr(S ). Since every rotation in dimension 3 has a ﬁxed axis, each rotation ﬁxes exacly two points on the sphere, thus f(X0) is a countable sets. This is a contradiction because f is countable-to-one and µn is not atomic.

5.4. Incomparable CBERs via left-translation. Adams and Kechris [2] found uncountably many pairwise incomparable equivalence relations up to Borel reducibility. In particular, they analyzed the equivalence relations induced by the 1 shift action of the groups ΓS := SO7 Z S on the free part of the space of the space 2ΓS equipped with the product topology, for diﬀerent sets of prime numbers S. Our proof of Theorem 1.3 implicitely gives a proof of another result of incomparability, that avoids the machinery of Zimmer’s superrigidity theory.

Denote by P be the set of prime numbers p such that p ≡ 1 (mod 4). If S ⊆ P, 1 then we let Γn,S = SOn(Z[ S ]). Whenever S = {p}, we write Γn,p = Γn,{p}.

Fix any n ≥ 5 and let K = SOn(R). Let mK be the Haar measure of K and consider the action Γn,S y (K, mK ) by left-translations. For each S ⊆ P, we simplify our notation and let ΓS = Γn,S and XS := FrΓS K. We focus on the orbit equivalence relation TS := R(ΓS y XS) induced by left-translation.

Theorem 5.10. Suppose S, T ⊆ P. If S * T , then TS is not Borel reducible to TT .

Sketch of the proof. Let p ∈ S r T . Any Borel reduction from TS to TT can be regarded as a weak Borel reduction from R(Γn,p y XS ) to TT . Therefore, since TT is free, let α:Γp × XS → ΓT be the associated cocycle. Then, α lifts to a cocycle −1 α˜ : Γep × XS → ΓT where Γep = π (Γp) and π : Ge → G is the 2–1 cover map. Then the proof continues exactly as the one of Theorem 1.3. Notice that by Margulis superrigidity there is no virtual embedding ρ: Γep → ΓT .

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Department of Mathematics, Statistics, and Computer Science, University of Illi- nois at Chicago, Chicago IL 60607, USA Email address: [email protected]