DESCRIPTIVE SET THEORY AND THE ERGODIC THEORY OF COUNTABLE GROUPS Thesis by Robin Daniel Tucker-Drob In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy California Institute of Technology Pasadena, California 2013 (Defended April 17, 2013) ii c 2013 Robin Daniel Tucker-Drob All Rights Reserved iii Acknowledgements I would like to thank my advisor Alexander Kechris for his invaluable guidance and support, for his generous feedback, and for many (many!) discussions. In addition I would like to thank Miklos Abert,´ Lewis Bowen, Clinton Conley, Darren Creutz, Ilijas Farah, Adrian Ioana, David Kerr, Andrew Marks, Benjamin Miller, Jesse Peterson, Ernest Schimmerling, Miodrag Sokic, Simon Thomas, Asger Tornquist,¨ Todor Tsankov, Anush Tserunyan, and Benjy Weiss for many valuable conversations over the past few years. iv Abstract The primary focus of this thesis is on the interplay of descriptive set theory and the ergodic theory of group actions. This incorporates the study of turbulence and Borel re- ducibility on the one hand, and the theory of orbit equivalence and weak equivalence on the other. Chapter 2 is joint work with Clinton Conley and Alexander Kechris; we study measurable graph combinatorial invariants of group actions and employ the ultraproduct construction as a way of constructing various measure preserving actions with desirable properties. Chapter 3 is joint work with Lewis Bowen; we study the property MD of resid- ually finite groups, and we prove a conjecture of Kechris by showing that under general hypotheses property MD is inherited by a group from one of its co-amenable subgroups. Chapter 4 is a study of weak equivalence. One of the main results answers a question of Abert´ and Elek by showing that within any free weak equivalence class the isomorphism relation does not admit classification by countable structures. The proof relies on affirm- ing a conjecture of Ioana by showing that the product of a free action with a Bernoulli shift is weakly equivalent to the original action. Chapter 5 studies the relationship between mixing and freeness properties of measure preserving actions. Chapter 6 studies how ap- proximation properties of ergodic actions and unitary representations are reflected group theoretically and also operator algebraically via a group’s reduced C∗-algebra. Chapter 7 is an appendix which includes various results on mixing via filters and on Gaussian actions. v Contents Acknowledgements iii Abstract iv Contents v Chapter 1. Introduction 2 1. Borel reducibility and classification 3 2. Approximation and classification in the ergodic theory of countable groups 4 3. Invariants of weak equivalence and measurable combinatorics 6 4. Co-induction and weak containment 8 5. Automatic freeness 8 6. Expressions of non-amenability in ergodic theory and representation theory 9 Chapter 2. Ultraproducts of measure preserving actions and graph combinatorics 14 1. Introduction 14 2. Preliminaries 20 3. Ultraproducts of standard measure spaces 21 4. Ultraproducts of measure preserving actions 25 5. Characterizing factors of ultraproducts 29 6. Graph combinatorics of group actions 37 7. Brooks’ Theorem for group actions 41 vi 8. Matchings 51 9. Independence numbers 56 10. Sofic actions 58 11. Concluding remarks 65 Chapter 3. On a co-induction question of Kechris 69 1. Introduction 69 2. The space of actions and proof of Theorem 1.4 73 3. The Rohlin Lemma 74 4. Proof of Theorem 1.1 76 5. Consequences of Theorem 1.1 77 6. Gaussian actions 80 7. Induced representations and the proof of Theorem 1.3 83 Chapter 4. Weak equivalence and non-classifiability of measure preserving actions 87 1. Introduction 88 2. Preliminaries and notation 95 2.1. Measure algebras and standard probability spaces 95 2.2. Measure preserving actions 96 2.3. The space of measure preserving actions 96 3. Proofs of Theorems 1.1 and 1.2 97 3.1. Weak containment and shift-invariant factors 97 3.2. Convexity in the space of actions 101 3.3. Ergodic decomposition and weak containment 103 4. Consequences of Theorem 1.2 and applications to MD and EMD 108 4.1. Free, non-ergodic weak equivalence classes 108 4.2. The properties MD and EMD 110 5. Weak equivalence and invariant random subgroups 112 5.1. Invariant random subgroups 112 vii 5.2. The compact space of weak equivalence classes 113 5.3. Random Bernoulli shifts 120 5.4. A sufficient condition for weak containment 126 5.5. Independent joinings over an IRS and the proof of Theorem 1.5 128 6. Non-classifiability 134 6.1. Non-classifiability by countable structures of ∼=, ∼=w, and ∼=U on free weak equivalence classes 134 6.2. Extending Theorem 1.7 136 7. Types and amenability 138 7.1. The space COS(Γ) 139 7.2. Proof of Theorem 1.8 140 8. Ultraproducts of measure preserving actions 142 9. Stable weak containment 146 Chapter 5. Mixing actions of countable groups are almost free 148 1. Introduction 148 2. Definitions and notation 151 3. Proof of Theorem 1.4 152 4. An example 154 5. A question 154 Chapter 6. Shift-minimal groups, fixed Price 1, and the unique trace property 155 1. Introduction 155 2. Preliminaries 162 2.1. Group theory 162 2.2. Ergodic theory 163 2.3. Invariant random subgroups 164 3. Shift-minimality 165 3.1. Seven characterizations of shift-minimality 165 viii 3.2. NA-ergodicity 167 3.3. Amenable invariant random subgroups 171 4. Permanence properties 172 4.1. Invariant random subgroups with trivial intersection 172 4.2. Finite index subgroups 174 4.3. Direct sums 177 4.4. Other permanence properties 178 5. Examples of shift-minimal groups 181 5.1. Free groups 181 5.2. Property (BP) 182 5.3. Linear groups 187 ∗ 5.4. Unique tracial state on Cr (Γ) 188 6. Cost 191 6.1. Notation and background 191 6.2. Cost and weak containment in infinitely generated groups 192 6.3. The cost of a generic action 203 6.4. Cost and invariant random subgroups 205 6.5. Fixed price 1 and shift-minimality 211 7. Questions 212 7.1. General implications 212 7.2. Cost and pseudocost 215 7.3. Other questions 216 8. Appendix: Invariant random subgroups as subequivalence relations 217 8.1. Invariant random partitions 218 8.2. Normalized subequivalence relations 221 9. Appendix: The amenable radical of a countable group 226 9.1. Basic properties of ARΓ 226 9.2. Groups with trivial amenable radical 228 ix Chapter 7. Appendix: Mixing via filters and Gaussian actions 232 1. Milding mixing = IP∗-mixing for groups 232 2. F-mixing 239 3. Permanence properties of F-mixing 247 4. Gaussian actions 253 1 2 Chapter 1 Introduction The questions addressed in this thesis lie at the interface of several fields including de- scriptive set theory, ergodic theory, representation theory, probability theory, and measur- able group theory. A unified approach to studying these questions is facilitated by a global perspective which was initiated and greatly developed in [Kec10]. From this perspective, problems in ergodic theory may be seen as topological-dynamical and descriptive problems concerning continuous actions of the Polish group A = A(X; µ) of automorphisms of a standard (usually non-atomic) probability space (X; µ). Likewise, representation theory may be studied via continuous actions of the Polish group U = U(H) of unitary operators on a separable (usually infinite-dimensional) Hilbert space H. More concretely, if Γ is a countable group then the set A(Γ; X; µ) of all measure pre- serving actions of Γ on (X; µ) naturally forms a Polish space on which A acts continuously by conjugation. What is significant here is that the natural ergodic theoretic notion of iso- morphism (”conjugacy”) of measure preserving actions of Γ is exactly the orbit equivalence relation generated by this action of the Polish group A; analogous remarks hold for unitary representations of Γ and the Polish group U. Descriptive set theorists have developed a general theory of Borel reducibility, which studies the set theoretic complexity of equiva- lence relations such as those arising from Polish group actions. Applications of this theory to actions of A and U have led to deep and surprising insights into the nature of conjugacy 3 in ergodic theory and of unitary equivalence in representation theory. We begin with a brief introduction to the basic notions of this framework. 1. Borel reducibility and classification If E and F are equivalence relations on standard Borel spaces X and Y , respectively, then E is called Borel reducible to F , denoted E ≤B F , if there is a Borel map : X ! Y satisfying xEy , (x)F (y) for all x; y 2 X. Such a map is called a Borel reduction from E to F . The substance of this notion lies in the requirement that this map be definable in some sense, and there are theoretical reasons for choosing Borel definability. The resulting richness of the ordering ≤B and its continuing success in comparing naturally occurring equivalence relations in mathematics may be taken as further justifications for this choice. A Borel reduction from E to F may be seen as providing an explicitly definable classification of elements of X up to E-equivalence using the F -classes as invariants. An equivalence relation is said to be classifiable by countable structures if it is Borel re- ducible to the isomorphism relation on some standard Borel space of countable structures, for example, countable graphs, groups, or partial orders. More precisely, E admits classifi- cation by countable structures if there exists a countable language L and a Borel reduction from E to isomorphism on the standard Borel space XL of all L-structures with universe N.