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Rotation Equivalence and Cocycle Superrigidity for Compact Actions

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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 corre- sponding 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 equiv- alence 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 incom- parable 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 2 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 2 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) := f(x; y) 2 X j 9g 2 Γ(g · x = y)g: 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 E1 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 E1 are easily seen to be linearly ordered by ≤B, the struc- ture of the interval [E0;E1] 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 E1 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 count- able 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;E1]. For n > 1, let Sn−1 be the (hyper)sphere in the n-dimensional Euclidean space. As usual, we define Sn−1 = fx 2 Rn : jjxjj = 1g. 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 := fx 2 X : 8g 6= 1Γ(g · x 6= x)g. 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 cocy- cles 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 incompa- rable 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.
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