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Borel Superrigidity for Actions of Low Rank Lattices BOREL SUPERRIGIDITY FOR ACTIONS OF LOW RANK LATTICES BY SCOTT SCHNEIDER A dissertation submitted to the Graduate School—New Brunswick Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Program in Mathematics Written under the direction of Simon Thomas and approved by New Brunswick, New Jersey October, 2009 ABSTRACT OF THE DISSERTATION Borel Superrigidity for Actions of Low Rank Lattices by Scott Schneider Dissertation Director: Simon Thomas A major recent theme in Descriptive Set Theory has been the study of countable Borel equivalence relations on standard Borel spaces, including their structure under the partial ordering of Borel reducibility. We shall contribute to this study by proving Borel incomparability results for the orbit equivalence relations arising from Bernoulli, profinite, and linear actions of certain subgroups of P SL2(R). We employ the techniques and general strategy pioneered by Adams and Kechris in [3], and develop purely Borel versions of cocycle superrigidity results arising in the dynamical theory of semisimple groups. Specifically, using Zimmer’s cocycle superrigidity theorems [58], we will prove Borel superrigidity results for suitably chosen actions of groups of the form P SL2(O), where O is the ring of integers inside a multi-quadratic number field. In particular, for suitable primes p 6= q, we prove that the orbit equivalence relations arising from the √ natural actions of P SL2(Z[ q]) on the p-adic projective lines are incomparable with respect to Borel reducibility as p, q vary. Furthermore, we also obtain Borel non- reducibility results for orbit equivalence relations arising from Bernoulli actions of the groups P SL2(O). In particular, we show that if Ep denotes the orbit equivalence re- √ lation arising from a nontrivial Bernoulli action of P SL2(Z[ p ]), then Ep and Eq are incomparable with respect to Borel reducibility whenever p 6= q. ii Acknowledgements This thesis would not have been possible without the dedicated assistance of my advisor, Simon Thomas, to whom I express my deepest gratitude. I am also extremely grateful for the countless mathematical interactions I have had with Samuel Coskey, with whom, and often from whom, I have learned the subject. Additionally I would like to thank Gregory Cherlin, Paul Ellis, Alex Furman, Richard Lyons, and Chuck Weibel for many helpful discussions. Finally I would like to thank my family for all the support they have given me over the years, and I would like to extend a special word of thanks to Tamar Swerdel for assistance in all matters non-mathematical during the time in which this thesis was written. iii Table of Contents Abstract ........................................ ii Acknowledgements ................................. iii 1. Introduction to countable Borel equivalence relations ......... 1 1.1. Background from Descriptive Set Theory . 1 1.2. Classification problems and equivalence relations . 3 1.3. Borel reducibility . 4 1.4. Borel equivalence relations and the ≤B partial order . 7 1.5. Countable Borel equivalence relations . 8 1.6. Superrigidity: a first glance . 10 1.7. The basic structure of countable Borel equivalence relations under Borel reducibility . 11 1.8. Outline of thesis . 14 2. Precise statements of main results ..................... 15 2.1. Profinite and linear actions of SL2(O)................... 15 2.2. A Borel superrigidity theorem . 19 2.3. Bernoulli actions of P SL2(O)........................ 22 2.4. A more abstract Borel superrigidity theorem . 24 2.5. Notation . 25 3. Some preliminaries from the theory of algebraic groups ........ 27 3.1. Polish groups and Haar measure . 27 3.2. Semisimple Lie groups and lattice subgroups . 29 3.3. Amenability . 33 iv 3.4. Property (T ) and property (τ) ....................... 34 3.5. Totally real number fields . 36 3.6. A measure classification theorem . 39 4. Some preliminaries from measurable dynamics ............. 42 4.1. Ergodicity . 42 4.2. Strong ergodicity . 47 4.3. Isomorphism, factor, and quotient . 50 4.4. Bernoulli actions . 52 4.5. Entropy . 54 5. Superrigidity ................................... 57 5.1. Orbit equivalence and Borel reducibility . 57 5.2. Borel cocycles . 59 5.3. Induced spaces, actions, and cocycles . 63 5.4. Zimmer superrigidity . 66 5.5. E0-ergodicity, 1-amenability, and the cocycle hypothesis . 68 6. Proof of Theorem 2.4.1 ............................ 74 6.1. An application of Zimmer cocycle superrigidity . 75 6.2. Adjusting a permutation group homomorphism . 78 6.3. Coming down on the left . 82 7. Proofs of Theorems 2.2.1 and 2.3.1 ..................... 85 7.1. Some computations involving subgroups of P SL2(R) . 86 7.2. Proof of Theorem 2.3.1 . 89 7.2.1. E0-ergodicity of orbit equivalence relations arising from Bernoulli actions . 89 7.2.2. Irreducibility of spaces induced from Bernoulli actions . 90 7.2.3. Entropy and factors of Bernoulli shifts . 92 7.2.4. Studying the image of f˜ in Yˆ ................... 92 v 7.2.5. Untwisting (f,˜ ϕ)........................... 93 7.3. Proof of Theorem 2.2.1 . 94 7.3.1. Property (τ) and E0-ergodicity . 95 7.3.2. Controlling the factors of ΛS y K(J)/L . 96 7.3.3. Studying the image of f˜ in Yˆ .................... 99 7.3.4. Obtaining the virtual isomorphism . 100 7.4. Suggestions for future research . 102 References ....................................... 104 Vita ........................................... 108 vi 1 Chapter 1 Introduction to countable Borel equivalence relations This thesis is a contribution to the study of Borel equivalence relations on standard Borel spaces. The organizing framework for this theory comes from Descriptive Set Theory, which classically has consisted of the study of definable sets and functions in complete, separable metric spaces. Recently a new direction of research has emerged in the field, aimed at understanding the definable equivalence relations on these spaces together with their resulting quotients. As we will see, such quotients arise naturally as spaces of invariants for classification problems in many different areas of mathematics. When realized in this way as definable equivalence relations, such classification prob- lems admit a natural notion of “relative complexity” that induces a partial pre-ordering on the collection of all such definable equivalence relations. While applications to the complexity theory of classification problems constitute an important, and indeed moti- vating, branch of the subject, much recent attention has been focused on understanding the structure of the partial ordering of the complexity classes itself. This thesis con- tributes to the study of this structure, and to the development of techniques for its continued study. In this introductory chapter we will develop preliminary notions and provide a context for our results by highlighting some elements of the basic theory of countable Borel equivalence relations. We begin with some elementary definitions and facts from Descriptive Set Theory. 1.1 Background from Descriptive Set Theory Formally, a Polish space is a topological space that admits a complete, separable metric, and a standard Borel space is a measurable space (X, B) that admits a Polish topology 2 whose σ-algebra of Borel sets is B. If (X, B) and (Y, C) are standard Borel spaces, then a function f : X → Y is called Borel if f −1(C) ∈ B for every C ∈ C.A Borel isomorphism is a bijective function f : X → Y such that both f and f −1 are Borel. By a classical theorem due to Kuratowski, any two uncountable standard Borel spaces are Borel isomorphic. Examples of standard Borel spaces include the unit interval [0, 1], the analytic spaces n R, R , and C, the p-adic numbers Qp, and also the descriptive set-theoretic spaces such as Baire space NN and Cantor space 2N = P(N). Any Borel set A ⊆ X in a standard Borel space (X, B) can be regarded as a standard Borel space in its own right with the induced σ-algebra B A = {B ∩ A | B ∈ B}, although in general it may be necessary to take a different underlying Polish topology. For instance, the open unit interval (0, 1) ⊆ R with its subspace σ-algebra is standard Borel, even though it is not complete in the subspace topology. Informally, we think of Borel sets and functions as those that are “explicitly” con- structed or defined, and we think of the unique (up to isomorphism) uncountable stan- dard Borel space as the abstract setting in which “explicit” or “concretely definable” mathematics takes place. Working in the Borel setting will often resemble working without the full power of the Axiom of Choice. If (X, B) and (Y, C) are standard Borel spaces, then X ×Y is a standard Borel space in the product σ-algebra B ×C, which coincides with the Borel σ-algebra of the product topology τ × σ for any Polish topologies τ and σ that generate B and C, respectively. In particular, an equivalence relation E ⊆ X × X on the standard Borel space X is called Borel if E is Borel in the product space X × X. Furthermore, if X and Y are standard Borel spaces then a function f : X → Y is Borel if and only if its graph is Borel as a subset of X × Y (for instance, see [49, 4.5.2]). For general background from Descriptive Set Theory see [30] or [49]. 3 1.2 Classification problems and equivalence relations Abstractly, we might say that a “classification problem” in mathematics consists of: (1) a collection of mathematical objects to be classified, and (2) some notion of equivalence up to which the classification is to be made. For example, we might wish to classify some collection of groups up to group isomor- phism, or graphs up to graph isomorphism, or topological spaces up to homeomorphism. The study of definable equivalence relations on standard Borel spaces is motivated by the fact that for many such abstract classification problems, the collection of objects to be classified can be given the structure of a standard Borel space, on which the given notion of equivalence is a definable equivalence relation.
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