The Title of the Dissertation

UNIVERSITY OF CALIFORNIA, SAN DIEGO

Some Structural Results for Measured Equivalence Relations and Their Associated von Neumann Algebras

A dissertation submitted in partial satisfaction of the requirements for the degree Doctor of Philosophy

Mathematics

Daniel J. Hoﬀ

Committee in charge:

Professor Adrian Ioana, Chair Professor Benjamin Grinstein Professor Bill Helton Professor John McGreevy Professor Alireza Salehi Golseﬁdy Professor Hans Wenzl

Chair

University of California, San Diego

2016

iii DEDICATION

For Burcu,

Keeper of my spirit,

Home to my joy

iv TABLE OF CONTENTS

Signature Page...... iii

Dedication...... iv

Table of Contents...... v

Acknowledgements...... viii

Vita...... x

Abstract of the Dissertation...... xi

Chapter 1 Introduction...... 1 1.1 Background...... 1 1.1.1 Organization...... 2 1.2 Preliminaries...... 2 1.2.1 Von Neumann algebras...... 2 1.2.2 Tracial von Neumann algebras...... 3 1.2.3 Amenable von Neumann algebras...... 4 1.2.4 Probability measure preserving equivalence relations...... 4 1.2.5 Von Neumann algebras of equivalence relations..5 1.2.6 Amenable equivalence relations...... 6 1.2.7 Orbit equivalence relations...... 6 1.2.8 Ampliﬁcations...... 7

Chapter 2 Von Neumann Algebras of Equivalence Relations with Non- trivial One-Cohomology...... 8 2.1 Introduction...... 8 2.1.1 Background and statement of results...... 8 2.1.2 Organization and strategy...... 12 2.2 Preliminaries...... 12 2.2.1 Representations of equivalence relations...... 12 2.2.2 Orbit equivalence relations...... 17 2.2.3 Relative mixingness and weak containment of bimodules...... 18 2.2.4 Relative amenability...... 20 2.2.5 Ampliﬁcations...... 20 2.2.6 Popa’s intertwining by bimodules...... 21 2.3 Deducing Primeness from an s-Malleable Deformation...... 21

v 2.4 Gaussian Extension of R and s-Malleable Deformation of L(R)...... 25 2.4.1 Gaussian extension of R ...... 25 2.4.2 s-Malleable deformation of L(R)...... 27 2.5 Primeness of L(R)...... 28 2.5.1 L(R)-L(R) bimodules arising from representations of R ...... 29 2.5.2 Proof of Theorem 2A...... 32 2.5.3 Remark...... 33 2.6 Unique Prime Factorization...... 33 2.6.1 An obstruction to unique factorization...... 33 2.6.2 Unique prime factorization via s-malleable deformation...... 38 2.6.3 Unique prime factorization for equivalence relations...... 42 2.7 Application to Measure Equivalent Groups...... 45

Chapter 3 Von Neumann’s Problem and Extensions of Non-amenable Equiv- alence Relations...... 48 3.1 Introduction and statement of main results...... 48 3.1.1 Background...... 48 3.1.2 Von Neumann’s problem for Bernoulli extensions 49 3.1.3 Uncountably many non-isomorphic extensions.. 51 3.1.4 Actions of locally compact groups...... 53 3.1.5 Outline of the proof of Theorem 3A...... 54 3.1.6 Organization...... 55 3.2 Preliminaries...... 55 3.2.1 Amenable equivalence relations...... 55 3.2.2 Extensions and expansions of equivalence relations 56 3.2.3 Graphed equivalence relations and isoperimetric constants...... 59 3.2.4 Cost of equivalence relations...... 60 3.3 Bernoulli extensions of equivalence relations...... 61 3.3.1 Bernoulli extensions and ergodicity...... 61 3.3.2 Isomorphisms of Bernoulli extensions...... 62 3.3.3 Bernoulli extensions restricted to subequivalence relations...... 63 3.3.4 Compressions of Bernoulli extensions...... 64 3.4 Bernoulli percolation on graphed equivalence relations...... 68 3.4.1 Bernoulli percolation on graphs...... 69 3.4.2 Inﬁnitely many inﬁnite clusters...... 70 3.4.3 Proof of Theorem 3.4.3...... 72

vi 3.5 Ergodicity of the cluster equivalence relation...... 74 3.6 Proofs of Theorem 3A and Corollary 3B...... 78 3.6.1 A generalization of Theorem 3A...... 78 3.6.2 Proof of Theorem 3.6.1...... 78 3.6.3 Proof of Corollary 3B...... 83 3.7 Uncountably many ergodic extensions of nonamenable R ...... 84 3.7.1 Co-induced equivalence relation...... 84 3.7.2 A separability argument...... 88 3.7.3 Proof of Theorem 3C...... 90 3.8 Actions of locally compact groups...... 91 3.8.1 Proof of Corollary 3D...... 94 3.8.2 Deducing [GM15, Theorem B] from Theorem 3A. 95

References...... 97

vii ACKNOWLEDGEMENTS

There are many people without whom this dissertation would not exist. First and foremost I would like to thank Adrian Ioana for proposing the topics of study herein and for the countless hours he spent sharing both his knowledge and his wisdom as my advisor. I cannot overstate how much I appreciated our many stimulating discussions when I had progress to report, and perhaps most importantly, his encouragement, patience, and unfailing kindness when I struggled. I am also grateful for the time and efforts of the other committee members, and for all of the faculty whose stimulating courses I had the opportunity to enjoy. I would also like to thank RémiBoutonnet for organizing many engaging learning seminars and for patiently answering my many questions regardless of their quality. I am so grateful to have been able to show him the full extent of my ignorance, and for his selfless efforts to whittle away at it. I would like to thank Daniel Drimbe for his always-encouraging belief in me. I am also very indebted to my fellow graduate students, especially to Jay, Rob, and Tait, for celebrating in the good times, commiserating in the bad, and providing all kinds of rejuvenating diversions in between. I owe much to all those I studied with, especially the algebros, Grimm, Rob, and Robert. A special thanks is due to Rob, without whose efforts our class would not have enjoyed such synergy. Away from math, thank you to all members of the UN of Windansea, especially Darlene, David, Giulio, Michael, Moses, and Paul, who provided so many of my best memories in San Diego. A great thanks is due to my perpetually ahead brother and sister who gave me the stubborn desire to understand that which I do not. I thank Eric Lauer- Hunt for sparking my interest in math, and my many excellent math teachers for building it, especially Mrs. Eldredge, who sent me to college with a love of math. Once there I owe so much to Peter Olver, who patiently guided my transition to research, and provided the passion for discovery that has driven me ever since. It is only the very fortunate whose lives allow for the pursuit of these studies. For this opportunity I thank my parents, who have given me so much, and on whose unconditional love and support I rely in all things.

viii Chapter 2 is, in part, a reprint of the material as it appears in [Ho15] Daniel J. Hoff, Von Neumann algebras of equivalence relations with nontrivial one-cohomology, J. Funct. Anal. 270 (2016), no. 4, 1501–1536. MR 3447718. of which the dissertation author was the primary investigator and author. Chapter 3 is, in part, material submitted for publication as it appears in [BHI15] Lewis Bowen, Daniel J. Hoff, and Adrian Ioana, Von Neumann’s problem and extensions of non-amenable equivalence relations, preprint arXiv:1509.01723, 2015. of which the dissertation author was one of the primary investigators and authors. This material is based upon work supported by the National Science Foun- dation Graduate Research Fellowship Program under Grant No. DGE-1144086. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

ix VITA

2011 B.S. in Mathematics University of Minnesota, Twin Cities

2011-2013 Graduate Teaching Assistant University of California, San Diego

2013 M.A. in Mathematics University of California, San Diego

2013-2016 NSF Graduate Research Fellow University of California, San Diego

2014 C.Phil. in Mathematics University of California, San Diego

2016 Ph.D. in Mathematics University of California, San Diego

x ABSTRACT OF THE DISSERTATION

Some Structural Results for Measured Equivalence Relations and Their Associated von Neumann Algebras

Daniel J. Hoﬀ Doctor of Philosophy in Mathematics

University of California San Diego, 2016

Professor Adrian Ioana, Chair

Using Popa’s deformation/rigidity theory, we investigate prime decomposi- tions of von Neumann algebras of the form L(R) for countable probability measure preserving (pmp) equivalence relations R. We show that L(R) is prime whenever R is non-amenable, ergodic, and admits an unbounded 1-cocycle into a mixing orthogonal representation weakly contained in the regular representation. This is accomplished by constructing the Gaussian extension R˜ of R and subsequently an s-malleable deformation of the inclusion L(R) ⊂ L(R˜). We go on to note a general obstruction to unique prime factorization, and avoiding it, we prove a unique prime factorization result for products of the form L(R1) ⊗ L(R2) ⊗ · · · ⊗ L(Rk). As a corollary, we get a unique factorization result in the equivalence relation setting for products of the form R1 × R2 × · · · × Rk. We then study extensions of pmp equivalence relations R following the joint work [BHI15] with Lewis Bowen and Adrian Ioana. By extending the techniques of Gaboriau and Lyons [GL07], we prove that if R is non-amenable and ergodic, it has an extension R˜ which contains the orbits of a free ergodic pmp action of the free group F2. This allows us to prove that any such R admits uncountably many ergodic extensions which are pairwise not (stably) von Neumann equivalent. We further deduce that any non-amenable unimodular locally compact second countable group admits uncountably many free ergodic pmp actions which are pairwise not von Neumann equivalent (hence, pairwise not orbit equivalent).

xi Chapter 1

Introduction

1.1 Background

The most fundamental questions in the study of von Neumann algebras ask to what extent countable groups Γ and probability measure preserving (pmp) equivalence relations R can be recovered from the associated von Neumann algebras L(Γ) and L(R). The greatly celebrated work of Alain Connes [Co75b] shows that in many cases we find a striking lack of rigidity: all amenable Γ with infinite conjugacy classes (icc) and all amenable ergodic R give rise to the same von Neumann algebra, known as the hyperfinite II1 factor R. For the next three decades, analysis of the non-amenable case proved largely elusive, until the breakthrough work of Sorin Popa in [Po01a, Po01b, Po03, Po04], whose powerful deformation/rigidity theory has since given rise to many beautiful rigidity results and solutions to problems that before appeared intractable. As will become evident, the results of this dissertation rest heavily on the framework of Popa’s breakthrough theory and the powerful techniques that have arisen from it. Looking more broadly beyond the field, the study of von Neumann algebras has had remarkable and often surprising impacts in diverse areas of mathematics. Perhaps the most famous example of this is Vaughan Jones’ work in von Neumann algebras, which led to his discovery of the powerful knot invariant now known as the Jones Polynomial, and for which he was awarded the Fields Medal in 1990. In another direction, many recent impressive results in the the-

1 2 ory of groups and measured equivalence relations which do not a priori involve operator algebras were obtained using operator algebraic techniques, for example [Po04, Po05, Po06a, Io08, PS09, Io13, Io14, BIS-G15]. This theme is an important one in this dissertation as well—Corollary 2E, Theorem 3C, and Corollary 3D below are further examples of exactly this phenomenon.

1.1.1 Organization

We begin by establishing the necessary preliminaries in Section 1.2. Chap- ter2 explores the consequences of nontrivial one-cohomology in pmp equivalence relations and their von Neumann algebras, following [Ho15]. Chapter3 then gives the results of joint work with Lewis Bowen and Adrian Ioana on extensions of pmp equivalence relations [BHI15], beginning by studying the analogue of von Neumann’s problem in this setting.

1.2 Preliminaries

1.2.1 Von Neumann algebras

Let H be a separable Hilbert space1 and let B(H) denote the ∗-algebra of bounded operators on H, where for T ∈ B(H), T ∗ is the adjoint of T , deﬁned by the equation hT ξ, ηi = hξ, T ∗ηi for all ξ, η ∈ H.A von Neumann algebra M is a unital2 ∗-subalgebra of B(H) that is closed in the weak operator topology (wot): i.e., if {xn} ⊂ M and x ∈ B(H) with hxnξ, ηi → hxξ, ηi as n → ∞ for all ξ, η ∈ H, then we must have x ∈ M. Basic examples are M = B(H) itself, or given a standard probability space (X, µ), the abelian algebra M = L∞(X) operating on H = L2(X) by pointwise multiplication. One can check that given any subset D ⊂ B(H) with D∗ = D,

1We need not always take H to be separable, but to avoid technicalities we will do so throughout this dissertation. 2i.e. M contains the identity operator on H which we denote by 1 ∈ B(H). 3 then the commutant D0 ⊂ B(H) deﬁned by

D0 = {T ∈ B(H): T x = xT for all x ∈ D} is a von Neumann algebra. In fact, von Neumann’s Bicommutant Theorem tells us that all von Neumann algebras arise in this way: a unital ∗-subalgebra of M ⊂ B(H) is wot-closed if and only if it equals the commutant of its own commutant, M = M 00. We will denote by Z(M) = M 0 ∩ M the center of M.A von Neumann algebra is called a factor if it has trivial center Z(M) = C. Of great current research interest are the von Neumann algebras L(Γ) and L(R), which arise in a natural way from countable groups Γ and pmp equivalence relations R. The focus of this dissertation will be on the latter, deﬁned in detail below in Section 1.2.5.

1.2.2 Tracial von Neumann algebras

Much of the study of von Neumann algebras focuses on those which are tracial, i.e. which have a faithful normal positive linear functional τ : M → C which satisﬁes τ(1) = 1, and τ(xy) = τ(yx) for all x, y ∈ M, called a trace. If M is tracial and is also a factor, then the trace τ is unique. Throughout, M,N,P and Q will denote tracial von Neumann algebras. We will let τ denote the trace on each where there is no danger of confusion, and the unit ball of (say) M will be written as (M)1. The group of unitary operators in M ∗ will be denoted U(M), and if N ⊂ M, then NM (N) = {u ∈ U(M): uNu = N} will denote the normalizer of N in M. Given a tracial von Neumann algebra (M, τ), following the general Gelfand- Naimark-Segal construction we deﬁne an inner product h·, ·i on M by hx, yi = τ(y∗x), and denote by L2(M) the completion of M with respect to the induced p ∗ norm: kxk2 = τ(x x) for x ∈ M. For a von Neumann subalgebra N ⊂ M, we will 2 2 write eN ∈ B(L (M)) for the orthogonal projection onto L (N), and EN : M → N will denote the resulting faithful normal trace-preserving conditional expectation onto N. 4

1.2.3 Amenable von Neumann algebras

A tracial von Neumann algebra (M, τ) is called amenable if there exists a 2 ∗ positive linear functional φ : B(L (M)) → C such that φ|M = τ and φ(uT u ) = φ(T ) for all T ∈ B(L2(M)), u ∈ U(M). The much celebrated results of Connes in [Co75b] show that M is amenable if and only if it is hyperﬁnite, i.e. if and only if there exists an increasing sequence of ﬁnite-dimensional unital ∗-subalgebras wot ∞ S∞ {An}n=1 of M such that M = n=1 An . Amenability will be a pervasive notion in this dissertation; see Section 2.2.4 for the more general notion of relative amenability and equivalent characterizations.

1.2.4 Probability measure preserving equivalence relations

We review here the foundations of the study of measured equivalence relations as established by Feldman and Moore in [FM75a]. Throughout, let (X, µ) denote a standard probability space. A Borel equivalence relation on (X, µ) is an equivalence relation R on X such that R ⊂ X × X is a Borel set in the product space. For x ∈ X, let [x]R denote the R-equivalence class of x. R is called countable if [x]R is countable (or ﬁnite) for a.e. x ∈ X. We denote by [R] the full group of R, that is, [R] = {φ ∈ Aut(X): graph(φ) ⊂ R} where we write Aut(X) for the group of Borel automorphisms of X. R is probability measure preserving (pmp) if µ ◦ φ = µ for all φ ∈ [R]. A pmp R is ergodic if µ(E) ∈ {0, 1} for any measurable E ⊂ X satisfying µ(E \φ(E)) = 0 for all φ ∈ [R], and strongly ergodic if µ(En)(1 − µ(En)) → 0 for any sequence of measurable subsets En ⊂ X satisfying µ(En \ φ(En)) → 0 for each φ ∈ [R].

Given a positive measure subset E ⊂ X, we denote by R|E the measured equivalence relation on the probability space (E, µ/µ(E)) given by R|E = R ∩

(E × E). Measured equivalence relations R1 on (X1, µ1) and R2 on (X2, µ2) are ∼ isomorphic, written R1 = R2, if there are full measure subsets E1 ⊂ X1, E2 ⊂ X2 which admit a measure space isomorphism φ :(E1, µ1|E1 ) → (E2, µ2|E2 ) such that

(x, y) ∈ R1|E1 ⇐⇒ (φ(x), φ(y)) ∈ R2|E2 . 5

Henceforth, R will always denote a countable pmp equivalence relation on a standard probability space (X, µ). We endow R with a Borel measure m given by Z m(E) = |{y ∈ [x]R :(x, y) ∈ E}|dµ(x) for all Borel E ⊂ R X

1.2.5 Von Neumann algebras of equivalence relations

To such an equivalence relation R, we associate a von Neumann algebra L(R), first constructed and studied by Feldman and Moore in [FM75b]. Each g ∈ 2 −1 [R] gives rise to a unitary ug ∈ U(L (R, m)) defined by [ugf](x, y) = f(g x, y). Similarly, each a ∈ A = L∞(X) is identified with an operator in B(L2(R, m)) by [af](x, y) = a(x)f(x, y). The von Neumann algebra L(R) of the equivalence relation R is defined to be

∞ 00 2 L(R) = {L (X) ∪ {ug : g ∈ [R]}} ⊂ B(L (R, m))

L(R) has a faithful normal trace given by τ(x) = hx1∆, 1∆i, where 1∆ ∈ L2(R, m) is the characteristic function of the diagonal ∆ = {(x, x): x ∈ X}. We note that L2(L(R), τ) ∼= L2(R, m) as L(R) modules and we will identify these Hilbert spaces henceforth. If R is ergodic then L(R) is a factor, and if R is strongly ergodic then any sequence {an} ⊂ (A)1 with kanug − ugank2 → 0 for each g ∈ [R] must have kan − τ(an)k2 → 0. Let Z1(R,S1) denote the group of S1-valued multiplicative 1-cocycles on R, that is, the group of measurable maps c : R → S1 such that for µ-a.e. x ∈ X,

c(x, y)c(y, z) = c(x, z) for all (x, y), (y, z) ∈ R, (1.1) identifying cocycles that agree m-a.e. Given c ∈ Z1(R,S1) and g ∈ [R], let ∞ −1 fc,g ∈ U(L (X)) be given by fc,g(x) = c(x, g x). Then using (1.1), one can check that the formula

∞ ψc(aug) = fc,gaug for a ∈ L (X), g ∈ [R] (1.2)

gives rise to a well deﬁned ∗-isomorphism ψc ∈ Aut(L(R)). Note that ψc1 ◦ ψc2 = 1 1 ψc1c2 , so c 7→ ψc deﬁnes an action ψ : Z (R,S ) → Aut(L(R)). 6

1.2.6 Amenable equivalence relations

Note that L∞(R, m) acts on L2(R, m) by pointwise multiplication and that ∞ L (R, m) is normalized by each unitary ug with g ∈ [R]. Recall that R is called ∞ ∗ amenable if there is a state Φ on L (R, m) with Φ(ugfug) = Φ(f) for all f ∈ ∞ L (R), g ∈ [R] and such that Φ|L∞(X) = τ. A countable pmp equivalence relation S∞ R is amenable if and only if it is hyperﬁnite, i.e. if and only if R = n=1 Rn, for sub-equivalence relations Rn having [x]Rn is ﬁnite and [x]Rn ⊂ [x]Rn+1 for µ-a.e. x ∈ X and all n ≥ 1 [CFW81, Theorem 10]. One can show that L(R) is an amenable von Neumann algebra if and only if R is amenable. We say R has an amenable direct summand if there is a measurable subset Y ⊂ X such that µ(Y ) > 0, R|Y is amenable, and R = R|Y ∪ R|Y c . In this case, L(R) = L(R|Y ) ⊕ L(R|Y c ) has an amenable direct summand as well.

1.2.7 Orbit equivalence relations

Given a countable group Γ with a pmp action Γ y (X, µ), the orbit equivalence relation R(Γ y X) is deﬁned by

(x, y) ∈ R(Γ y X) ⇐⇒ y = gx for some g ∈ Γ, and two group actions are orbit equivalent (OE) if and only if they have isomorphic orbit equivalence relations. In the case where R = R(Γ y X) for a free3 pmp action of Γ, then ∼ ∞ L(R) = L (X) o Γ, the group-measure space von Neumann algebra, and for this reason the algebra L(R) is sometimes called the generalized group-measure space von Neumann algebra. If Γ is an amenable group then R(Γ y X) is amenable, and the converse holds if the action is free. Feldman and Moore showed in [FM75a] that any countable pmp R arises from the action of a countable group, however this action cannot always be taken to be free, a question which was settled by Furman in [Fu99].

3 Γ y (X, µ) is free if µ({x ∈ X : gx = x}) = 0 for each nonidentity g ∈ Γ. 7

1.2.8 Ampliﬁcations

In order to give the prime factorization result in Section 2.6 and the application to measure equivalence in Section 2.7, we will need the language of amplifica- ∞ 2 tions. For a II1 factor (M, τ), we consider the type II∞ factor M = M ⊗ B(` (Z)). If we denote by Tr the semifinite trace on B(`2(Z)), then τ ⊗ Tr gives a semifi- ∞ nite trace on M . For any t > 0, the amplification of M by t is the II1 factor M t = PM ∞P for a projection P ∈ M ∞ satisfying (τ ⊗ Tr)(P ) = t. Note that t such a projection exists since M is II1 and that M is well defined up to unitary ∞ conjugacy in M . If M is type In for some n ∈ Z>0 such P exists provided nt ∈ Z and in this case we define M t as above. For s, t > 0, (M s)t = M st. Now consider the tracial factor L(R) for ergodic R. If R is infinite, ergodicity implies that the space (X, µ) must be non-atomic and L(R) is a type II1 factor. For such R and t > 0, we can define as follows the amplification Rt of R in such a way that L(Rt) ∼= L(R)t. Consider the measure space (X∞, µ⊗#) = (X ×Z, µ⊗#), where # denotes the counting measure on Z. Define a countable Borel equivalence relation R∞ on ∞ ∞ t ∞ X by ((x, k), (y, m)) ∈ R iff (x, y) ∈ R. For t > 0, define R = R |E for a Borel E ⊂ X∞ with (µ⊗#)(E) = t. Such a set E exists since X is non-atomic, and using the ergodicity of R one can show that Rt is well defined up to isomorphism. From this it further follows that (Rt)s = Rts for t, s > 0. Chapter 2

Von Neumann Algebras of Equivalence Relations with Nontrivial One-Cohomology

2.1 Introduction

2.1.1 Background and statement of results.

A natural question in the classification of von Neumann algebras asks how a tracial von Neumann algebra can be written as the tensor product of subalgebras. A tracial von Neumann algebra M is called prime if whenever M = N ⊗ Q for subalgebras N,Q ⊂ M, either N or Q is of type I. For II1 factors M, this amounts to forcing either N or Q to be finite dimensional. A II1 factor is called solid if the relative commutant of any diffuse subalgebra is amenable. All non-amenable subfactors of a solid II1 factor are prime.

In [Po83], Popa proved primeness for certain II1 factors with non-separable preduals, including the group von Neumann algebra of the free group on uncountably many generators. Then in [Ge96], using free probability theory, Ge showed ∗ that the free group factors L(Fn) are prime as well. In [Oz03], Ozawa used C - algebraic methods to prove that L(Γ) is in fact solid for all icc hyperbolic groups

Γ, recovering the primeness of L(Fn) as a special case. By developing a new tech-

8 9 nique of closable derivations, Peterson showed in [Pe06] that L(Γ) is prime for non-amenable icc groups which admit an unbounded 1-cocycle into a multiple of the left regular representation. Popa then used his powerful deformation/rigidity theory to give a new proof of solidity for L(Fn), [Po06b]. Using Sinclair’s malleable deformation of L(Γ) arising from an unbounded 1-cocycle [Si10], Vaes showed in [Va10] that deformation/rigidity theory could also be used to recover Peterson’s result. In this paper, we construct an analogous deformation of L(R) and use Popa’s theory to prove the following analogue of Peterson’s primeness result in the setting of countable pmp equivalence relations:

Theorem 2A. Let R be a countable pmp equivalence relation with no amenable direct summand which admits an unbounded 1-cocycle into a mixing orthogonal representation weakly contained in the regular representation. Then L(R) N ⊗ Q for any type II von Neumann algebras N and Q and hence R R1 × R2 for any pmp Ri which have a.e. equivalence class inﬁnite. In particular, if R is ergodic then L(R) is prime.

For additional primeness results, we refer the reader to [Oz04, Po06a, CI08,

CH08, Bou12, DI12]. For a II1 factor which is not prime, it is natural to ask if it can be written uniquely as the tensor product of prime subfactors. Of course, if M = P1 ⊗ P2 for prime II1 factors P1 and P2, then any u ∈ U(P1 ⊗ P2) gives ∗ ∗ M = uP1u ⊗ uP2u as a prime factorization of M. Moreover, for any II1 factors N,Q and t > 0, there is a natural identification N ⊗ Q ∼= N t ⊗ Q1/t, where N t denotes the amplification of N by t (see Section 2.2.5). Hence prime factorization results are considered up to such amplification as well as up to unitary conjugacy. In fact, as first proved by Ozawa and Popa in [OP03] and subsequently in [Pe06, CS11, SW11, Is14, CKP14, HI15], the techniques used to prove primeness can often be used to prove unique prime factorization results. However, we find that in the setting of L(R), the presence of the Cartan subalgebra L∞(X) ⊂ L(R) can present additional obstacles to passing from a primeness result to a unique prime factorization result. These obstacles do not appear to have been encountered before; to best of our knowledge this paper gives the first unique prime factorization result for factors of the form L(R) (or L∞(X) o Λ) that do not arise also as L(Γ) 10 for some countable group Γ. The root of the difficulties in the setting of L(R) lies in the fact that our s-malleable deformation of L(R) does not deform the Cartan subalgebra L∞(X).

As an example, take any free ergodic action of a nonabelian free group Fn on a standard probability space (X, µ). Then the orbit equivalence relation R =

R(Fn y X) will satisfy the assumptions of Theorem 2A, so that P = L(R) is prime. But if we now assume that the action of Fn is not strongly ergodic, then P will have property Gamma1 and the following theorem shows that P ⊗ P admits two prime factorizations which are distinct up to unitary conjugacy and ampliﬁcation:

Theorem 2B. Let M1 and M2 be k·k2-separable II1 factors with property Gamma and set M = M1 ⊗ M2. Then there is an approximately inner automorphism φ ∈

Inn(M) such that φ(Mi) ⊀ Mj for any i, j ∈ {1, 2}.

In particular, this implies that there is no t > 0, i, j ∈ {1, 2} such that φ(Mi) t is unitarily conjugate to Mj in M. To avoid this obstruction, when considering unique factorization we will restrict to the case of strongly ergodic R and use Popa’s deformation rigidity theory to prove the following:

Proposition 2C. Let R be a strongly ergodic countable pmp equivalence relation which is non-amenable and admits an unbounded 1-cocycle into a mixing orthogonal representation weakly contained in the regular representation. Then L(R) is prime and does not have property Gamma.

Still, the presence of L∞(X) ⊂ L(R) presents additional diﬃculty in applying the techniques developed in [OP03]. Nevertheless, we are able to prove the following:

Theorem 2D. For i ∈ {1, 2, . . . , k}, let Ri be a non-amenable strongly ergodic countable pmp equivalence relation which admits an unbounded 1-cocycle into a mixing orthogonal representation weakly contained in the regular representation.

Then for each i, L(Ri) is prime and does not have property Gamma, and

1 AII1 factor M has property Gamma if there exists a sequence of unitaries {un} ⊂ M with τ(un) = 0 for all n and kunx − xunk2 → 0 for each x ∈ M. 11

(1). If M = L(R1) ⊗ L(R2) ⊗ ... ⊗ L(Rk) = N ⊗ Q for tracial factors

N,Q, there must be a partition IN ∪ IQ = {1, . . . , k} and t > 0 such that N t = N L(R ) and Q1/t = N L(R ) modulo unitary conjugacy in M. i∈IN i i∈IQ i

(2). If M = L(R1) ⊗ L(R2) ⊗ ... ⊗ L(Rk) = P1 ⊗ P2 ⊗ · · · ⊗ Pm for II1 factors P1,...,Pm and m ≥ k, then m = k, each Pi is prime, and there are t1, . . . , tk > 0 with t1t2 ··· tk = 1 such that after reordering indices and conjugating

ti by a unitary in M we have L(Ri) = Pi for all i.

(3). In (2), the assumption m ≥ k can be omitted if each Pi is assumed to be prime.

As an application, we prove the following corollary:

Corollary 2E. Let R1, R2,..., Rk be as in Theorem 2D. ∼ (1). If R1 × R2 × · · · × Rk = S1 × S2 for inﬁnite pmp equivalence relations

S1 and S2, then there is t > 0 and an integer 1 ≤ m < k such that after reordering t ∼ 1/t ∼ the indices we have S1 = R1 × R2 × · · · × Rm and S2 = Rm+1 × Rm+2 × · · · × Rk. ∼ (2). If R1 ×R2 ×· · ·×Rk = S1 ×S2 ×· · ·×Sm for inﬁnite pmp equivalence relations S1,S2,...,Sm and m ≥ k, then m = k and there are t1, . . . , tk > 0 with ∼ ti t1t2 ··· tk = 1 such that after reordering indices we have Ri = Si for all i.

Note that in Theorem 2D we assume that each Ri is strongly ergodic, but that the obstruction in Theorem 2B only applies when multiple factors have property Gamma. We leave open the case of exactly one factor with Gamma. We conclude with an application to the measure equivalence of countable groups. In [Ga01], Gaboriau showed that measure equivalent groups have propor- tional `2 Betti numbers. It follows that a countable group with positive ﬁrst `2 Betti number cannot be measure equivalent to a product of inﬁnite groups. The following theorem augments this conclusion:

Theorem 2F. Let Γ be a countable non-amenable group which admits an unbounded 1-cocycle into a mixing orthogonal representation weakly contained in the ME left regular representation. Then Γ Γ1 × Γ2 for any inﬁnite groups Γ1, Γ2. 12

2.1.2 Organization and strategy

Following the introduction, we establish the necessary preliminaries in Sec- tion 2.2. In Section 2.3 we review how an s-malleable deformation can be used to prove primeness, condensing this strategy as Theorem 2.3.2. Section 2.4 then con- structs such a deformation of L(R) by considering the Gaussian extension R˜ of R, and Section 2.5 combines this construction with Theorem 2.3.2 to prove primeness for L(R), Theorem 2A. In Section 2.6, we go on to apply this strategy in the more general context of prime factorization. We ﬁrst prove the obstruction in Theorem 2B, then con- dense the general strategy as Theorem 2.6.4. Proving Proposition 2C allows us to apply this strategy to prove Theorem 2D and subsequently Corollary 2E. The paper concludes in Section 2.7 with the application to the measure equivalence of groups, Theorem 2F.

Acknowledgements

I would like to extend my warm thanks to Adrian Ioana for proposing the topics of study in this paper, and for the many invaluable discussions without which it would not exist. I am very grateful to him for all of his shared wisdom and unfailing encouragement in every aspect of the process. I would also like to thank Stefaan Vaes and Remi Boutonnet for their very helpful remarks. Finally, I would like to express my gratitude for the detailed and insightful suggestions of the referee and to Alessandro Carderi for an illuminating discussion of them.

2.2 Preliminaries

2.2.1 Representations of equivalence relations

In analogy to group representations on Hilbert spaces, pmp equivalence relations on X are represented on measurable Hilbert bundles with base X. For 13 an excellent detailed account of measurable Hilbert bundles, we refer the reader to [Di69]. We recall here a few of the necessary facts.

Given a collection of Hilbert spaces {Hx}x∈X , we form the Hilbert bundle

X ∗ H as the set of pairs X ∗ H = {(x, ξx): x ∈ X, ξx ∈ Hx}. A section ξ of X ∗ H is a map x 7→ ξ(x) ∈ Hx. A measurable Hilbert bundle is a Hilbert bundle X ∗ H endowed with a ∞ σ-algebra generated by the maps {(x, ξx) 7→ hξx, ξn(x)i}n=1 for a fundamental ∞ sequence of sections {ξn}n=1 satisfying ∞ (i) Hx = span{ξn(x)}n=1 for each x ∈ X, and ∞ (ii) the maps {x 7→ kξn(x)k}n=1 are measurable. It is a useful fact that the σ-algebra of any measurable Hilbert bundle can be generated by an orthonormal fundamental sequence of sections, i.e. sections which moreover satisfy ∞ (iii) {ξn(x)}n=1 is an orthonormal basis of Hx for x ∈ X with dim Hx = ∞,

dim Hx and if dim Hx < ∞, the sequence {ξn(x)}n=1 is an orthonormal basis and

ξn(x) = 0 for n > dim Hx.

A measurable section of X ∗H is a section ξ such that x 7→ (x, ξ(x)) ∈ X ∗H ∞ is a measurable map, or equivalently, such that the maps {x 7→ hξ(x), ξn(x)i}n=1 ∞ are measurable for the fundamental sequence of sections {ξn}n=1. We let S(X ∗H) denote the vector space of measurable sections, identifying µ-a.e. equal sections. We then consider the direct integral Z ⊕ Z 2 Hxdµ(x) = {ξ ∈ S(X ∗ H): kξ(x)k dµ(x) < ∞} X X R which is a Hilbert space with inner product hξ, ηi = X hξ(x), η(x)idµ(x). If a ∈ ∞ R ⊕ R ⊕ A = L (X) and ξ ∈ X Hxdµ(x) we denote by aξ or ξa the element of X Hxdµ(x) ∞ given by [aξ](x) = [ξa](x) = ξ(x)a(x). If {ξn}n=1 is an orthonormal fundamental R ⊕ P∞ sequence of sections, any ξ ∈ X Hxdµ(x) has an expansion ξ = n=1 anξn where an = hξ(·), ξn(·)i ∈ A. A unitary (resp. orthogonal) representation of R on a complex (real) mea- 2 surable Hilbert bundle X ∗ H is a map (x, y) 7→ π(x, y) ∈ U(Hy, Hx) on R such 2For complex (resp. real) Hilbert spaces H, K, we write U(H, K) for the set of unitary (or- 14 that for µ-a.e. x ∈ X, we have

π(x, y)π(y, z) = π(x, z) for all (x, y), (y, z) ∈ R, and such that (x, y) 7→ hπ(x, y)ξ(y), η(x)i is a measurable map on R for all ξ, η ∈ S(X ∗ H). Given a measurable Hilbert bundle X ∗H with an orthonormal fundamental ∞ sequence of sections S = {ξn}n=1, we can always form the identity representation idS of R on X ∗ H, where idS (x, y) is determined by the formula idS (x, y)ξn(y) =

ξn(x) for each (x, y) ∈ R, ξn ∈ S. 2 To deﬁne the regular representation of R, take Hx = ` ([x]R) for each x ∈ X, and form the measurable Hilbert bundle X ∗ H with fundamental sequence of sections {ξg}g∈Γ, where ξg(x) = 1{g−1x} and Γ is a countable subgroup of [R] which generates R (which exists by [FM75a], Thm. 1). The regular representation of R is then the representation λ on X ∗ H given by λ(x, y) = id for all (x, y) ∈ R. Given representations π on X ∗ H and ρ on X ∗ K, we say that π and ρ are unitarily equivalent if there is a family of unitaries {Ux ∈ U(Hx, Kx)}x∈X with

Uxπ(x, y) = ρ(x, y)Uy for all (x, y) ∈ R, and such that x 7→ Uxξ(x) is in S(X ∗ K) for each ξ ∈ S(X ∗ H). We say that π is weakly contained in ρ, written π ≺ ρ, if for any > 0, ξ ∈ S(X ∗ H), and E ⊂ R with m(E) < ∞, there exists {η1, . . . , ηm} ⊂ S(X ∗ K) with m X m({(x, y) ∈ E : |hπ(x, y)ξ(y), ξ(x)i − hρ(x, y)ηi(y), ηi(x)i| ≥ }) < i=1 A representation π on X ∗ H is called mixing (cf. [Ki14], Def. 4.4) if for every , δ > 0 and ξ, η ∈ S(X ∗ H) with kξ(x)k = kη(x)k = 1 a.e., there is E ⊂ X with µ(X \ E) < δ such that

{y ∈ [x]R|E : |hπ(x, y)ξ(y), η(x)i| > } < ∞ for µ-a.e. x ∈ E

A 1-cocycle for a representation π on X ∗ H is a map (x, y) 7→ b(x, y) ∈ Hx on R such that for µ-a.e. x ∈ X,

b(x, z) = b(x, y) + π(x, y)b(y, z) for all (x, y), (y, z) ∈ R, (2.1) thogonal) maps from H onto K 15 and such that (x, y) 7→ (x, b(x, y)) ∈ X ∗ H is measurable. A 1-cocycle b is a coboundary if there is a measurable section ξ of X ∗ H such that b(x, y) = ξ(x)−π(x, y)ξ(y) for m-a.e. (x, y) ∈ R, and a pair of 1-cocycles b and b0 are cohomologous if b − b0 is a coboundary. We deﬁne a 1-cocycle to be bounded if there exists a sequence of measurable ∞ S∞ subsets {En}n=1 of X with µ( n=1 En) = 1 and sup{kb(x, y)k :(x, y) ∈ R|En } < ∞ for each n ≥ 1. With this deﬁnition, the analysis of Anantharaman-Delaroche [A-D03] gives the following equivalence:

Lemma 2.2.1. A 1-cocycle b for a representation π of R on X ∗H is a coboundary if and only if it is bounded.

Proof. Suppose there is ξ ∈ S(X ∗ H) such that b(x, y) = ξ(x) − π(x, y)ξ(y) for m-a.e. (x, y) ∈ R. Then for n ≥ 1 set En = {x ∈ X : kξ(x)k ≤ n}. Then S∞ n=1 En = X and for (x, y) ∈ R|En we have kb(x, y)k ≤ kξ(x)k + kπ(x, y)ξ(y)k ≤ 2n < ∞. ∞ Conversely, consider a sequence of measurable subsets {En}n=1 of X with S∞ µ( n=1 En) = 1 and sup{kb(x, y)k :(x, y) ∈ R|En } < ∞ for each n ≥ 1. Then by Lemma 3.21 of [A-D03], for each n we know that b restricted to R|En is a coboundary, i.e., there is ξn ∈ S(En ∗ H) with b(x, y) = ξn(x) − π(x, y)ξn(y) for m-a.e. (x, y) ∈ R| . We can then extend ξ to the R-saturation F = S [x] En n n x∈En R by the formula ξn(x) = b(x, y)+π(x, y)ξn(y) for some y ∈ En such that (x, y) ∈ R.

This deﬁnition does not depend on the choice of y; if z ∈ En with (x, z) ∈ R, then

[b(x, y) + π(x, y)ξn(y)] − [b(x, z) + π(x, z)ξn(z)]

= [b(x, y) − b(x, z)] + π(x, y)ξn(y) − π(x, z)ξn(z)

= − π(x, y)b(y, z) + π(x, y)ξn(y) − π(x, z)ξn(z)

= π(x, y)[−b(y, z) + ξn(y) − π(y, z)ξn(z)] = π(z, y)[0] = 0.

∞ Thus we have a sequence {ξn}n=1 such that ξn ∈ S(Fn ∗ H), b(x, y) = S∞ ξn(x) − π(x, y)ξn(y) for m-a.e. (x, y) ∈ R|Fn . Now for x ∈ F = n=1 Fn, deﬁne

ξ(x) = ξnx (x), where nx = min{n ≥ 1 : x ∈ Fn}, (2.2) 16

and let ξ(x) = 0 for x∈ / F . Note that if (x, y) ∈ R, then nx = ny since each set

Fn is R-invariant. Thus

b(x, y) = ξnx (x) − π(x, y)ξny (y) = ξ(x) − π(x, y)ξ(y) for (x, y) ∈ R|F . Since µ(F ) = 1, this is m-a.e. (x, y) ∈ R.

Finally, to see that ξ is measurable, note that ξ(x) = ξn(x) for all x ∈ Sn−1 c Fn \ k=1 Fk which (along with F ) decompose X into measurable subsets on which the restriction of ξ is measurable.

Thus a 1-cocycle that is not a coboundary must be unbounded (i.e. not bounded), for which we have another useful characterization.

Lemma 2.2.2. A 1-cocycle b for a representation π of R on X ∗ H is unbounded if and only if there is δ > 0 such that for any R > 0 there is g ∈ [R] with µ({x ∈ X : kb(x, g−1x)k > R}) ≥ δ.

Proof. First, suppose there is δ > 0 such that for any R > 0 there is g ∈ [R] with µ({kb(x, g−1x)k ≥ R}) ≥ δ. Then if b were bounded, there would be measurable δ E ⊂ X such that µ(E) > 1 − 2 and R = sup{kb(x, y)k :(x, y) ∈ R|E} < ∞. But then there would be g ∈ [R] such that F = {kb(x, g−1x)k > R} has µ(F ) ≥ δ. Noting that g−1(E ∩ F ) ⊂ Ec, we then would have

µ(E ∪ F ) = µ(E) + µ(F ) − µ(E ∩ F ) ≥ µ(E) + µ(F ) − µ(Ec) δ = 2µ(E) + µ(F ) − 1 > 2(1 − ) + δ − 1 = 1, 2 which is impossible. Conversely suppose that b is unbounded. By Feldman and Moore [FM75a], R = R(Γ y X) for a pmp action of a countable group Γ. Moreover, Γ can be chosen such that (x, y) ∈ R if and only if y = hx for some h ∈ Γ with h2 = e. ∞ Since Γ is countable, enumerate the elements of Γ of order ≤ 2 as {hn}n=1. For each n ≥ 1, we recursively deﬁne a sequence of measurable subsets n ∞ n n n {Ak }k=1. Let A1 = {x ∈ X : kb(x, h1x)k ≥ n} and given A1 ,...,Ak−1, deﬁne n hSk−1 ni Fk = X \ j=1 Aj and

n n n Ak = {x ∈ Fk ∩ hkFk : kb(x, hkx)k ≥ n} (2.3) 17

n F∞ n Set A = k=1 Ak . Note that

2 kb(hkx, hkx)k = kb(hkx, x)k = k − π(hkx, x)b(x, hkx)k = kb(x, hkx)k

n n from which it follows that hkAk = Ak for every k. We can therefore deﬁne gn ∈ [R] by the formula  n hkx, if x ∈ Ak ; gnx = (2.4) x, if x∈ / An.

2 n Note that gn = e and kb(x, gnx)k ≥ n for all x ∈ A . n T∞ n Now set En = X \ A . For x ∈ En = k=1 Fk we have kb(x, hkx)k < n for n all k such that x ∈ hkFk . Hence

n n ≥ sup{kb(x, hkx)k : k ≥ 1, x ∈ En ∩ hkFk }

≥ sup{kb(x, hkx)k : k ≥ 1, x ∈ En ∩ hkEn}

= sup{kb(x, y)k :(x, y) ∈ R|En }

S∞ Therefore setting δ = 1−µ( n=1 En), we must have δ > 0, since otherwise b would be bounded. For any R > 0, taking some integer n > R we have

−1 −1 n µ({kb(x, gn x)k ≥ R}) ≥ µ({kb(x, gn x)k ≥ n}) ≥ µ(A ) = 1 − µ(En) ≥ δ.

2.2.2 Orbit equivalence relations

Recall from Section 1.2.7 that given a countable group Γ with a pmp action Γ y (X, µ), the orbit equivalence relation R(Γ y X) is deﬁned by

(x, y) ∈ R(Γ y X) ⇐⇒ y = gx for some g ∈ Γ, and two group actions are orbit equivalent (OE) if and only if they have isomorphic orbit equivalence relations. If R = R(Γ y X) for the free pmp action of a countable group Γ, then any group representation π :Γ → U(H) of Γ on a Hilbert space H gives rise to 18

a representation πR of R, and a 1-cocycle b for π gives a 1-cocycle bR for πR as follows. We represent R on the Hilbert bundle X ∗ K where Kx = H for all x ∈ X.

Let E0 = {x ∈ X : gx = x for some nonidentity g ∈ Γ}. Then µ(E0) = 0 since Γ is countable and the action is free. Deﬁne

−1 πR(x, g x) = π(g), and (2.5) −1 bR(x, g x) = b(g), for g ∈ Γ, x∈ / E0, and since µ(E0) = 0, for x ∈ E0 take (say) π(x, y) = id and b(x, y) = 0. One can check that πR is mixing if π is mixing and bR is unbounded if b is unbounded.

Moreover if π ≺ ρ for another representation ρ of Γ, then πR ≺ ρR as well. When

π is either the left or right regular representation, then πR is unitarily equivalent to the regular representation λ.

2.2.3 Relative mixingness and weak containment of bimodules

We recall a few useful notions for bimodules over von Neumann algebras.

Let N ⊂ M be a von Neumann subalgebra. An M-M bimodule M HM is mixing relative to N if for any sequence {xn} ⊂ (M)1 with kEN (yxnz)k2 → 0 for all y, z ∈ M, we have

lim sup |hxnξy, ηi| = lim sup |hyξxn, ηi| = 0 for all ξ, η ∈ H. (2.6) n→∞ n→∞ y∈(M)1 y∈(M)1

An M-N bimodule M HN is weakly contained in a M-N bimodule M KN , written M HN ≺ M KN , if for any > 0, ﬁnite subsets F1 ⊂ M,F2 ⊂ N, and ξ ∈ H, there are η1, . . . , ηn ∈ K such that

n X |hxξy, ξi − hxηjy, ηji| < for all x ∈ F1, y ∈ F2. (2.7) j=1

Given bimodules M HN and N KP , we can form Connes’ fusion M-P bimodule M H ⊗N KP which satisﬁes ξa ⊗N η = ξ ⊗N aη for all a ∈ N, ξ ∈ H, η ∈ K (see

[PV11] for a construction). If M HN ≺ M KN , then M H ⊗N LP ≺ M K ⊗N LP for any N-P bimodule L, and P L ⊗M HN ≺ P L ⊗M KN for any P -M bimodule L. 19

The following lemma is standard and appears in Remark 3.7 of [Va10], for instance. We include the proof below for completeness.

∞ Lemma 2.2.3. Let Q ⊂ M and let H be an M-M bimodule. Suppose that {ξn}1 ⊂ H, , κ > 0 are such that

(i) kξnk ≥ for all n,

(ii) kxξnk ≤ κkxk2 for x ∈ M and all n, and

(iii) kxξn − ξnxk → 0 for each x ∈ Q. 0 2 Then there is a nonzero projection z ∈ Z(Q ∩ M) such that M [L (M)z]Qz ≺

M [Hz]Qz.

Proof. For each x ∈ M and n ≥ 1 set φn(x) = hxξn, ξni. By (ii) we have 0 ≤ 2 2 φn ≤ κ τ so there is Tn ∈ M with 0 < Tn ≤ κ such that hxξn, ξni = τ(xTn) 2 for all x ∈ M. Since Tn ≤ κ for all n, passing to a subsequence we may assume 2 that Tn → T weakly for some T ∈ M with 0 ≤ T ≤ κ . Moreover, T 6= 0 2 2 0 since τ(Tn) = kξnk ≥ for all n, and by (iii) we have T ∈ Q ∩ M. Let

δ > 0 be small enough that p = 1(δ,κ2](T ) is nonzero. Then set S = f(T ) where 1/2 2 0 f(t) = (1(δ,κ2](t)/t) for t ∈ σ(T ) so that S T = p. Since p ∈ Q ∩ M is nonzero, 0 0 0 0 1 there is p ∈ Q ∩ M with p ≤ p and EZ(Q0∩M)(p ) = m z for some m ∈ Z>0 and 0 0 ∗ 0 z ∈ Z(Q ∩M). Let v1, v2, . . . , vm ∈ Q ∩M be partial isometries with vj vj = p for Pm ∗ j 0 0 0 02 0 all 1 ≤ j ≤ m and j=1 vjvj = z. Set ηn = vjS ξn and S = Sp so that S T = p . Then for any x, a ∈ M and y ∈ Qz we have

m m ∗ X ∗ ∗ X 0 0 ∗ hxay, ai = τ(a xay) = τ(a xayvjvj ) = τ(xyvjS TS vj ) j=1 j=1 m m X 0 ∗ 0 X 0 ∗ 0 = lim τ(S vj xyvjS Tn) = lim hS vj xyvjS ξn, ξni n→∞ n→∞ j=1 j=1 m X j j = lim hxηny, ηni n→∞ j=1

2 and thus M [L (M)z]Qz ≺ M [Hz]Qz. 20

2.2.4 Relative amenability

The notion of relative amenability for von Neumann subalgebras is due to Ozawa and Popa in [OP07], from which we get the following:

Theorem 2.2.4 ([OP07]). Let N,Q be von Neumann subalgebras of (M, τ) which contain 1M . Then the following are equivalent: (1). N is amenable relative to Q inside M;

(2). There is an N-central state φ on hM, eQi such that φ|M = τ;

(3). There is a conditional expectation Φ: hM, eQi → N such that Φ|M =

EN ; 2 2 (4). There is {ξn} ⊂ L (M) ⊗Q L (M) such that for each x ∈ M, b ∈ N we have hxξn, ξni → τ(x) and kbξn − ξnbk → 0 as n → ∞; 2 2 2 (5). M L (M)N ≺ M L (M) ⊗Q L (M)N .

This generalizes the notion of amenability for subalgebras: N is amenable iﬀ it is amenable relative to C inside M for some (and hence all) M ⊃ N. We will need the following useful proposition due to Popa and Vaes:

Proposition 2.2.5 (Proposition 2.7 of [PV11]). Let N,Q1,Q2 be von Neumann 00 subalgebras of (M, τ) which contain 1M . Suppose that M = NM (Q1) and

[eQ1 , eQ2 ] = 0, with N is amenable relative to Qi for i = 1, 2.

Then N is amenable relative to Q1 ∩ Q2.

2.2.5 Ampliﬁcations

In order to give the prime factorization result in Section 2.6 and the application to measure equivalence in Section 2.7, we will need the language of ampliﬁ- cations introduced in Section 1.2.8. Let t > 0 and suppose that a pmp ergodic R has a representation π with 1-cocycle b on a Hilbert bundle X ∗ H. Then we can t t form the Hilbert bundle E ∗ H , where E is as above and where H(x,k) = Hx for each (x, k) ∈ E. Deﬁne a representation πt of Rt with 1-cocycle bt by

πt((x, k), (y, m)) = π(x, y), and (2.8) bt((x, k), (y, m)) = b(x, y), for ((x, k), (y, m)) ∈ Rt. 21

For any t > 0, π is mixing if and only if πt is mixing, b is unbounded if and only if bt is unbounded, and for another representation ρ of R, π ≺ ρ if and only if πt ≺ ρt.

2.2.6 Popa’s intertwining by bimodules

We will make essential use of the following theorem of Popa, fundamental to deformation/rigidity theory:

Theorem 2.2.6 (Popa’s Intertwining by Bimodules, Theorem 2.1 of [Po03]). Let N and P be unital subalgebras of a tracial von Neumann algebra M. The following are equivalent:

(1) There is no sequence {un} ⊂ U(N) such that kEP (xuny)k2 → 0 as n → ∞ for every x, y ∈ M; 2 (2) There is a N-P submodule H of L (M) with dimP (H) < ∞; (3) There are nonzero projections p ∈ N, f ∈ P , a unital normal ∗-homomorphism θ : pNp → fP f, and a partial isometry v ∈ M such that

θ(x)v = vx for all x ∈ pNp, v∗v ∈ (N 0 ∩ M)p, and vv∗ ∈ θ(pNp)0 ∩ fMf.

If the above equivalent conditions hold, we say that N intertwines into P inside M, written N ≺M P , or simply N ≺ P when there is no danger of confusion.

2.3 Deducing Primeness from an s-Malleable Deformation

In this section, we review how an s-malleable deformation can be used to prove primeness results using a technique introduced by Popa in [Po06b]. We deﬁne an s-malleable deformation of a tracial von Neumann algebra M as an inclusion M ⊂ M˜ into some tracial M˜ , along with a continuous action α : R → Aut(M˜ ), ˜ 2 and β ∈ Aut(M) such that β|M = id, β = id, and β ◦ αt = α−t ◦ β for all t ∈ R. To exploit an s-malleable deformation, we will use Popa’s transversality inequality from [Po06a], part (1) of the following lemma. We include as part (2) another well known inequality as we shall use the pair several times in combination. 22

Lemma 2.3.1. Let α : R → Aut(M˜ ), β ∈ Aut(M˜ ) be an s-malleable deformation ˜ of M ⊂ M. Set δt(x) = αt(x) − EM (αt(x)) for x ∈ M. Then for all x, y ∈ M and t ∈ R,

(1). kδ2t(x)k2 ≤ 2kα2t(x) − xk2 ≤ 4kδt(x)k2, and

(2). k[δt(x), y]k2 ≤ 2kxkkαt(y) − yk2 + k[x, y]k2.

Proof. For (1),

kδt(x)k2 ≤ kαt(x) − xk2 + kx − EM (αt(x))k2 = kαt(x) − xk2 + kEM (x − αt(x))k2

≤ 2kαt(x) − xk2, and since βαt = α−tβ and β|M = id, we have

kα2t(x) − xk2 = kαt(x) − α−t(x)k2

≤ kαt(x) − EM (αt(x))k2 + kα−t(x) − EM (αt(x))k2

= kδt(x)k2 + kβ(αt(x) − EM (αt(x)))k2 = 2kδt(x)k2.

For (2),

k[δt(x), y]k2 = k(1 − EM )([αt(x), y])k2 ≤ k[αt(x), y]k2

≤ kαt(x)y − αt(x)αt(y)k2 + k[αt(x), αt(y)]k2 +kαt(y)αt(x) − yαt(x)k2

≤ 2kxkkαt(y) − yk2 + k[x, y]k2

We can now show how an s-malleable deformation can be used to prove primeness.

Theorem 2.3.2 (Popa’s Spectral Gap Argument). Let M be a tracial von Neu- mann algebra with no amenable direct summand which admits an s-malleable de- ˜ ˜ formation {αt}t∈R ⊂ Aut(M) for some tracial von Neumann algebra M ⊃ M. 2 ˜ 2 Suppose that the M-M bimodule M L (M) L (M)M is weakly contained in the coarse M-M bimodule and mixing relative to some abelian subalgebra A ⊂ M. Then there is a central projection z ∈ Z(M) such that 23

1. αt → id uniformly in k · k2 on the unit ball (Mz)1 as t → 0, and

2. M(1 − z) is prime.

In particular, if the convergence αt → id as t → 0 is not uniform, then M N ⊗ Q for any N and Q of type II.

Proof. Using Zorn’s Lemma, let z ∈ Z(M) denote the maximal central projection such that αt → id uniformly in k · k2 on the unit ball (Mz)1. Then (1) is satisﬁed 0 and for any central projection z ≤ 1−z we have αt → id non-uniformly in k·k2 on 0 (Mz )1. Now suppose toward a contradiction that M(1 − z) = N ⊗ Q with N and Q not of type I. Since M has no amenable direct summand, we assume without loss of generality that Q also has no amenable direct summand. As previously, set

δt(x) = αt(x) − EM (αt(x)) for x ∈ M.

First suppose that αt → id is not uniform in k · k2 on (N)1. Then by part

(1) of Lemma 2.3.1, δt → 0 is not uniform in k · k2 on (N)1. Hence there is > 0 and sequences {an} ∈ (N)1, {tn} ⊂ R, with tn → 0 and kδtn (an)k2 > for all n.

For x ∈ Q, we have [x, an] = 0 and kαtn (x) − xk2 → 0 as n → ∞, so part (2) of Lemma 2.3.1 gives k[δtn (an), x]k2 → 0. We also have kxδtn (an)k2 ≤ kxk2, so applying Lemma 2.2.3 with H = L2(M˜ ) L2(M), and using our assumption that 2 ˜ 2 M L (M) L (M)M is weakly contained in the coarse M-M bimodule, there is a projection q ∈ Z(Q0 ∩ Q) = Z(Q) such that

2 2 ˜ 2 2 2 QL (Qq)Qq ≺ Q[(L (M) L (M))q]Qq ≺ Q[L (M) ⊗ L (M)q]Qq

2 2 ≺ QL (Q) ⊗ L (Qq)Qq

2 2 2 and hence QqL (Qq)Qq ≺ QqL (Qq) ⊗ L (Qq)Qq, which contradicts the fact that Q has no amenable direct summand. Thus we must have αt → id uniformly in k · k2 on (N)1. Next, since N is not type I, there is z0 ∈ Z(N) ⊂ Z(M)(1 − z) such that 0 0 0 Nz is type II. Then since A is abelian and Nz is type II, we have Nz ⊀M A, so 0 it follows from Theorem 2.2.6 that there is a sequence {un} ⊂ U(Nz ) such that 2 ˜ 2 for each x, y ∈ M, kEA(xuny)k2 → 0 as n → ∞. Since M L (M) L (M)M is mixing relative to A, we have that hunδt(x), δt(x)uni → 0 as n → ∞ for all x ∈ M. 24

0 Note that for any t ∈ R, x ∈ (Qz )1 we have

0 0 0 0 0 kδt(x) − z δt(x)k2 = k(1 − EM )(αt(z x) − z αt(x))k2 ≤ kαt(z ) − z k2 and so using both parts of Lemma 2.3.1, we have

0 0 0 kα2t(x) − xk2 ≤ 2kδt(x)k2 ≤ 2kαt(z ) − z k2 + 2kz δt(x)k2 1 0 0 2 2 = 2kαt(z ) − z k2 + lim inf 2k[un, δt(x)]k2 + 4Rehunδt(x), δt(x)uni n→∞ √ 0 0 h i ≤ 2kαt(z ) − z k2 + lim inf 8kαt(un) − unk2 n→∞ √ ≤ (2 + 8) sup kαt(a) − ak2 −→ 0 as t → 0. a∈(N)1 (2.9)

As this convergence is independent of x, this shows that αt → id uniformly in k·k2 0 on (Qz )1.

Now ﬁx any > 0, and let t0 > 0 be such that for |t| < t0 we have 0 0 kαt(x) − xk2 < 7 for all x ∈ (N)1 ∪ (Qz )1. Then for u ∈ U(N), v ∈ U(Qz ) we have

∗ ∗ 0 0 kαt(u)αt(v)v u − z k2 ≤ kαt(u)αt(v)z − αt(u)αt(v)k2 + kαt(u)αt(v) − uvk2 3 ≤ kz0 − α (z0)k + kα (u) − uk + kα (v) − vk < t 2 t 2 t 2 7 ∗ ∗ and so for |t| < t0, the k · k2-closed convex hull Kt of the set {αt(u)αt(v)v u : u ∈ 0 0 3 U(N), v ∈ U(Qz )} has kk − z k2 ≤ 7 for all k ∈ Kt. In particular, the unique 0 3 element kt ∈ Kt of minimal k · k2 has kkt − z k2 ≤ 7 . Since kt is unique and ∗ 0 ∗ αt(u)Ktu = Kt for all u ∈ U(N) ∪ U(Qz ), it follows that αt(u)ktu = kt for all 0 0 u ∈ U(N) ∪ U(Qz ), and hence αt(a)kt = kta for all a ∈ N ∪ Qz . Then for any a ∈ N, b ∈ Qz0, we have

αt(ab)kt = αt(a)αt(b)kt = αt(a)ktb = ktab,

0 0 0 0 and Mz = M(1 − z)z = (N ⊗ Q)z , so in fact αt(x)kt = ktx for all x ∈ Mz and 0 |t| < t0. Thus for any x ∈ (Mz )1 and |t| < t0 we have

0 0 0 kαt(x) − xk2 ≤ kαt(z x) − z αt(x)k2 + kz αt(x) − αt(x)ktk2 + kktx − xk2 3 ≤ kα (z0) − z0k + 2kk − z0k ≤ + 2 · = , t 2 t 2 7 7 25

0 0 which implies that αt → id uniformly in k · k2 on (Mz )1. But z ∈ Z(M) with z0 ≤ 1 − z, so this is a contradiction and we conclude that M(1 − z) is indeed prime.

In the particular case where the convergence αt → id is not uniform in k · k2 on (M)1, the above projection z ∈ Z(M) has 1 − z 6= 0. Suppose toward a contradiction that M ∼= N ⊗ Q with N and Q of type II. Then since M(1 − z) is prime by the above result, the decomposition M(1−z) = N(1−z) ⊗ Q(1−z) forces either N(1 − z) or Q(1 − z) to be type I. Assume without loss of generality that N(1 − z) is type I. But then taking a nonzero abelian projection p ∈ N(1 − z), we would have a type II algebra N intertwining in M into an abelian algebra Np ⊕ C(1 − p), which is impossible. Thus M N ⊗ Q with N and Q of type II.

2.4 Gaussian Extension of R and s-Malleable De- formation of L(R)

In this section we construct the s-malleable deformation that will be used to prove the main result. In [PS09] and [Si10], Peterson and Sinclair used 1-cocycles for group representations to build deformations; we follow this spirit in the setting of pmp equivalence relations. To accomplish this, we generalize Bowen’s Bernoulli shift extension of R (see [Bo12]) to the Gaussian extension R˜ of R arising from an orthogonal representation π of R. A 1-cocycle for π will then give rise to the desired s-malleable deformation of L(R) ⊂ L(R˜).

2.4.1 Gaussian extension of R

Let π be an orthogonal representation of R on a real Hilbert bundle X ∗ H, ∞ and let {ξi}i=1 be an orthonormal fundamental sequence of sections for X ∗H. Let

dim Hx Y 1 −s2/2 (Ωx, νx) = (R, √ e ds), i=1 2π 26

dim Hx ∞ and deﬁne ωx : spanR({ξi(x)}i=1 ) → U(L (Ωx)) by k ! √ k X X x ωx anξin (x) = exp(i 2 anSin ) n=1 n=1

x x dim Hx where Sj is the coordinate function Sj ((si)i=1 ) = sj for j ≤ dim Hx.

One can show that ωx extends to a k · kHx − k · k2 continuous map ωx : ∞ Hx → U(L (Ωx)) satisfying

−kξk2 ∗ τ(ωx(ξ)) = e , ωx(ξ+η) = ωx(ξ)ωx(η), ωx(ξ) = ωx(−ξ), for all ξ, η ∈ Hx. (2.10)

For x ∈ X, one can show that the linear span Dx = spanC({ωx(ξ)}ξ∈Hx ) 00 wot ∞ has Dx = Dx = L (Ωx). For (x, y) ∈ R, deﬁne a ∗-homomorphism ρ(x, y): ∞ Dy → L (Ωx) by

ρ(x, y)ωy(ξ) = ωx(π(x, y)ξ), which is well deﬁned and k · k2-isometric since (2.10) implies

∗ ∗ τ(ωy(η) ωy(ξ)) = τ(ωx(π(x, y)η) ωx(π(x, y)ξ)) for all ξ, η ∈ Hy.

In particular, ρ(x, y) is τ-preserving, and so extends to a ∗-isomorphism ρ(x, y): ∞ ∞ L (Ωy) → L (Ωx). Let θ(x,y) :Ωy → Ωx be the induced measure space isomor- −1 ∞ phism such that ρ(x, y)f = f ◦ θ(x,y) for all f ∈ L (Ωy). We now consider X ∗ Ω as measurable bundle with σ-algebra generated by P the maps (x, r) 7→ ωx( i∈I aiξi(x))(r) for I ⊂ N ﬁnite and ai ∈ R. A measure R µ ∗ ν on X ∗ Ω is then given by [µ ∗ ν](E) = X νx(Ex)dµ(x), where Ex = {s ∈ Ωx : (x, s) ∈ E}. We deﬁne the Gaussian extension of R to be the equivalence relation R˜ on (X ∗ Ω, µ ∗ ν) given by

˜ ((x, r), (y, s)) ∈ R ⇐⇒ (x, y) ∈ R and θ(y,x)(r) = s (2.11) leaving the reader to check that R˜ is a countable pmp equivalence relation. ˜ For g ∈ [R], we can deﬁneg ˜ ∈ [R] byg ˜(x, r) = (gx, θ(gx,x)(r)). One can then ∞ ˜ check that the map aug 7→ aug˜ for a ∈ L (X), g ∈ [R] imbeds L(R) into L(R) ˜ S and we identify ug and ug˜ henceforth. Moreover, noting that R = g∈[R] graph(˜g), 27 it follows that

˜ ∞ 00 ∞ 00 2 ˜ L(R) = {L (X ∗ Ω), {ug˜ : g ∈ [R]}} = {L (X ∗ Ω),L(R)} ⊂ B(L (R)) (2.12)

2.4.2 s-Malleable deformation of L(R)

Now consider a 1-cocycle b for the representation π on X ∗ H above and let M = L(R) and M˜ = L(R˜). Using the cocycle relation for b, one checks that

ct((x, r), (y, s)) = ωx(tb(x, y))(r) (2.13)

˜ deﬁnes a one-parameter family {ct}t∈R of multiplicative 1-cocyles of R, and hence ˜ as in (1.2), a one-parameter family {ψct }t∈R ⊂ Aut(M) which we will denote by

{αt}t∈R. Moreover, ct1 ct2 = ct1+t2 for all t1, t2 ∈ R, and hence t 7→ αt deﬁnes an action α : R → Aut(M˜ ). For any a ∈ L∞(X ∗ Ω), g ∈ [R],

2 2 2 2 2 kαt(aug) − augk2 = kfct,gaug − augk2 ≤ kak kfct,g − 1k2 = kak [2 − 2Re τ(fct,g)] Z 2 −1 = 2kak 1 − Re τ(ωx(tb(x, g x)))dµ(x) X

Z h 2 −1 2 i = 2kak2Re 1 − e−t kb(x,g x)k dµ(x) → 0 as t → 0, X where the convergence follows from Lebesgue’s dominated convergence theorem. When combined with (2.12), this shows that α : R → Aut(M˜ ) is a continuous ˜ action when Aut(M) is given the topology of pointwise k · k2 convergence.

Next, one can check that deﬁning βx(ωx(ξ)) = ωx(−ξ) for x ∈ X gives rise ∞ ∞ to βx ∈ Aut(L (Ωx)), which leads to β ∈ Aut(L (X ∗ Ω)) deﬁned by β(a)(x, r) = ∞ ∗ ∗ βx(a(x, ·))(r) for a ∈ L (X ∗ Ω). Then noting that ugβ(a)ug = β(ugaug) for all a ∈ L∞(X ∗ Ω), g ∈ [R˜], one can check that β extends to an ∗-automorphism of ˜ 2 M by the rule β(aug) = β(a)ug. We have β = id, β|M = id, and β ◦ αt = α−t ◦ β ˜ ˜ since one can check that β(fct,g) = fc−t,g for each g ∈ [R]. Hence α : R → Aut(M) is an s-malleable deformation of M ⊂ M˜ . 28

2.5 Primeness of L(R)

In this section, we prove the main result, Theorem 2A. Before doing so, ∞ ∞ however, we pause to further analyze the maps ρ(x, y): L (Ωy) → L (Ωx) deﬁned in Section 2.4.1. Note ﬁrst that each can be extended (and then restricted) to a unitary 2 2 ρ(x, y): L (Ωy) C → L (Ωx) C

2 Setting Kx = L (Ωx) C for x ∈ X, we now form the Hilbert bundle X ∗ K ∞ with the σ-algebra determined by fundamental sections ω0(spanQ{ξn}n=1), where ∞ −kη(x)k2 {ξn}n=1 is as in Section 2.4.1, and [ω0(η)](x) = ωx(η(x))−e for η ∈ S(X∗H). Noting that ρ(x, y)ρ(y, z) = ρ(x, z) for all (x, y), (y, z) ∈ R, we may then consider ρ as a representation of R on X ∗ K. The following lemma makes explicit the relationship between ρ and π.

ˆ L∞ n Lemma 2.5.1. For each x ∈ X, let Hx = n=1(Hx ⊗R C) . The representation ρ of R on X ∗ K is unitarily equivalent to the representation πˆ = ⊕∞ π n of R n=1 C on X ∗ Hˆ. √ ˆ −kξk2 L∞ (i 2ξ) n Proof. For x ∈ X, deﬁne Ux : Dx → C ⊕ Hx by ωx(ξ) 7→ e n=0 n! for

ξ ∈ Hx, which is well deﬁned and isometric since for any ξ, η ∈ Hx, we have √ √ * ∞ n ∞ n + ∞ n 2 M (i 2ξ) 2 M (i 2η) 2 2 X 2 e−kξk , e−kηk = e−kξk e−kηk hξ n, η ni n! n! (n!)2 n=0 n=0 n=0 ∞ n 2 2 X 2 2 2 = e−kξk e−kηk n!(hξ, ηi)n = e−kξk e−kηk e2hξ,ηi = τ(ω (η)∗ω (ξ)) (n!)2 x x n=0

Certainly C ⊆ Ux(Dx), and one can check that ξ1 · · · ξn ∈ Ux(Dx) for all

ξ1, . . . , ξn ∈ Hx by induction on n. Hence we extend this map to a unitary Ux : 2 ˆ L (Ωx) → C ⊕ Hx. Then for (x, y) ∈ R, it is immediate from the deﬁnitions ofπ ˆ and ρ that

[idC ⊕πˆ](x, y)Uyωy(ξ) = Uxωx(π(x, y)ξ) = Uxρ(x, y)ωy(ξ) for all ξ ∈ Hy, and hence

[idC ⊕πˆ](x, y)Uya = Uxρ(x, y)a 29

2 2 for all a ∈ L (Ωy) since L (Ωy) = ωy(Hy). In particular, since Uy ﬁxes C for each y ∈ X, the lemma follows.

2.5.1 L(R)-L(R) bimodules arising from representations of R

We will need one more tool before the proof of Theorem 2A. Again let M = L(R) and M˜ = L(R˜), and write A for L∞(X) ⊂ M. Note that a representation π R ⊕ of R on X ∗ H induces a group representationπ ˜ :[R] → U( X Hxdµ(x)) deﬁned −1 −1 byπ ˜g(ξ)(x) = π(x, g x)ξ(g x). Letting

Z ⊕ π 2 H := Hxdµ(x) ⊗A L (R), (2.14) X we would like to deﬁne an M-M bimodule structure on Hπ. The intuition comes from the proof of the following analogue of Fell’s absorption principle:

Lemma 2.5.2. Let π be a representation of R on a measurable Hilbert bundle X ∗

H. Then π ⊗ λ is unitarily equivalent to idS ⊗λ for any orthonormal fundamental sequence of sections S for X ∗ H.

∞ Proof. Let S = {ξn}n=1. For (x, y), (x, z) ∈ R and n, m ≥ 1, we have

hπ(x, y)ξn(y) ⊗ 1{y}, π(x, z)ξm(z) ⊗ 1{z}i = hπ(x, y)ξn(y), π(x, z)ξm(z)i · 1{y}(z)

= hπ(x, y)ξn(y), π(x, y)ξm(y)i · 1{y}(z)

= hξn(y), ξm(z)i · 1{y}(z)

= hξn(y) ⊗ 1{y}, ξm(z) ⊗ 1{z}i.

2 Since Hx ⊗ ` ([x]R) = span{ξn(x) ⊗ 1{y} :(x, y) ∈ R, n ≥ 1} for each x ∈ X, the above calculation shows that the formula

Ux(ξn(x) ⊗ 1{y}) = π(x, y)ξn(y) ⊗ 1{y} for (x, y) ∈ R, n ≥ 1

2 gives rise to a well deﬁned unitary Ux ∈ U(Hx ⊗ ` ([x]R)) (note that Ux is surjective ∞ since {π(x, y)ξn(y)}n=1 is a basis for Hx for (x, y) ∈ R) for each x ∈ X. Moreover, 30 for (x, y), (x, z) ∈ R and n ≥ 1 we have

Uz([idS ⊗λ](z, x) · ξn(x) ⊗ 1{y}) = Uz(ξn(z) ⊗ 1{y}) = π(z, y)ξn(y) ⊗ 1{y}

= [π ⊗ λ](z, x) · π(x, y)ξn(y) ⊗ 1{y} = [π ⊗ λ](z, x) · Ux(ξn(x) ⊗ 1{y}) and hence [π ⊗ λ](z, x)Ux = Uz[idS ⊗λ](z, x) for (z, x) ∈ R. For measurability, take any g, h ∈ [R], n, m ≥ 1 and note that

x 7→ hUx(ξn(x) ⊗ 1{g−1x}), ξm(x) ⊗ 1{h−1x}i −1 −1 = hπ(x, g x)ξn(g x) ⊗ 1{g−1x}, ξm(x) ⊗ 1{h−1x}i −1 −1 = hπ(x, g x)ξn(g x), ξm(x)i · 1{y∈X:g−1y=h−1y}(x) is the product of two measurable maps.

Lemma 2.5.3. The Hilbert space Hπ has an L(R)-L(R) bimodule structure which satisﬁes

aug · (ξ ⊗A η) · x =π ˜g(ξ) ⊗A augηx (2.15)

2 R ⊕ for a ∈ A, g ∈ [R], x ∈ M, η ∈ L (M), and ξ ∈ X Hxdµ.

Proof. We have already from the construction of Connes’ fusion tensor that Hπ is an A-L(R) bimodule with the right action satisfying (2.15). The proposed left and right actions certainly commute, so it is enough to show that the left action in (2.15) makes Hπ into a left Hilbert L(R)-module. For each n ≥ 1, set pn = 1{x∈X:dim Hx≥n} ∈ A. If (x, y) ∈ R, then Hx = π(x, y)Hy so dim Hx = dim Hy ∞ and therefore pn ∈ Z(L(R)). Let {ξn}n=1 be an orthonormal fundamental sequence L∞ 2 of sections for X ∗ H and note that pn = kξn(·)k. Set K = n=1 pnL (R). We L∞ 2 π wish to deﬁne a unitary U : n=1 pnL (R) → H . For any g ∈ [R], a ∈ A, and n ≥ 1, let ηn,a,g ∈ K denote the vector which is pnaug in the nth summand 2 and 0 elsewhere (note that pnaug ∈ pmL (R) for any 1 ≤ m ≤ n, so we must be careful with our notation). Then K = span{ηn,a,g : a ∈ A, g ∈ [R], n ≥ 1} and we deﬁne U(ηn,a,g) =π ˜g(ξn) ⊗A aug. Then for a, b ∈ A, g, h ∈ [R], and n ≥ 1, since 31

∗ EA(uguh) = 1{x∈X:g−1x=h−1x}, we have

∗ ∗ hπ˜g(ξn) ⊗A aug, π˜h(ξn) ⊗A buhi = τ(hπ˜g(ξn)(·), π˜h(ξn)(·)iaEA(uguh)b ) 2 ∗ ∗ ∗ = τ([ugkξn(·)k ug]aEA(uguh)b ) ∗ ∗ = τ(pnauguhb )

= hηn,a,g, ηn,b,hi

∗ and if m 6= n, then hπ˜g(ξn)(·), π˜h(ξm)(·)iEA(uguh) = 0 and therefore

hπ˜g(ξn) ⊗A aug, π˜h(ξm) ⊗A buhi = 0 = hηn,a,g, ηm,b,hi.

Thus U extends to a well deﬁned unitary U : K → Hπ (U is surjective since R ⊕ X Hxdµ = span{π˜g(ξn)a : a ∈ A, n ≥ 1} for each g ∈ [R]). L∞ 2 Now since K = n=1 pnL (R) is a left L(R)-module by left multiplication in each coordinate, Hπ becomes a left L(R)-module under the action x · η = U(x · U ∗(η)). Moreover, for a, b ∈ A, g, h ∈ [R], n ≥ 1, we have

∗ aug · (˜πh(ξn) ⊗A buh) = aug · U(ηn,b,h) = U(aug · ηn,b,h) = U(ηn,a(ugbug),gh) ∗ =π ˜gh(ξn) ⊗A a(ugbug)ugh =π ˜g(˜πh(ξn)) ⊗A augbuh. R ⊕ P∞ For any a, b ∈ A, g, h ∈ [R], ξ ∈ X Hxdµ, write ξ = n=1 π˜h(ξn)an with an ∈ A. Then using the above, ∞ ∞ X X aug · (ξ ⊗A buh) = aug · (˜πh(ξn)an ⊗A buh) = π˜g(˜πh(ξn)) ⊗A auganbuh) n=1 n=1 ∞ X ∗ = π˜g(˜πh(ξn))(uganug) ⊗A augbuh) n=1 ∞ X = π˜g(˜πh(ξn)an) ⊗A augbuh) n=1

=π ˜g(ξ) ⊗A augbuh.

2 Since elements of the form buh span a dense subspace of L (R), it follows that the left action of L(R) satisﬁes (2.15).

Given two representations π and ρ of R with π weakly contained in (resp. unitarily equivalent to) ρ, one can check that Hπ is weakly contained in (resp. unitarily equivalent to) Hρ as M-M bimodules. If a representation π is a mixing, then Hπ is mixing relative to A. 32

2.5.2 Proof of Theorem 2A

We can now prove the main primeness result.

˜ ˜ Proof. Consider the s-malleable deformation M ⊂ M, {αt}t∈R ⊂ Aut(M) constructed in Section 2.4. Note that the representationπ ˆ = ⊕∞ π n is mixing and n=1 C weakly contained in the regular representation λ of R, since π has these properties. 2 2 R ⊕ 2 By identifying L (X ∗Ω) L (X) with X [L (Ωx) C]dµ(x), one can check that L2(M˜ ) L2(M) ∼= Hρ as M-M bimodules. The latter is then unitarily equivalent to Hπˆ by Lemma 2.5.1, and sinceπ ˆ is mixing, we have that L2(M˜ ) L2(M) is mixing relative to A. Moreover, Hπˆ ≺ Hλ sinceπ ˆ ≺ λ, and one can λ ∼ 2 2 check that H = L (M) ⊗A L (M) as M-M bimodules. Since A is amenable, the latter is weakly contained in the coarse M-M bimodule. Since b is unbounded, there is δ > 0 such that for all R > 0, there is g ∈ [R] −1 with µ({kb(x, g x)k ≥ R}) ≥ δ. If we had αt → id uniformly on (M)1, there 2 δ would be t0 such that kEM (αt0 (ug))k2 > 1 − 2 for all g ∈ [R]. But taking R > 0 2 2 −2t0R δ −1 large enough that e < 2 , and g ∈ [R] with µ({kb(x, g x)k ≥ R}) ≥ δ, we would have

δ Z 2 −1 2 2 −2t0kb(x,g x)k 1 − < kEM (αt0 (ug))k2 = e dµ 2 X −1 2 −2t2R2 −1 2 ≤ µ({kb(x, g x)k < R}) + e 0 µ({kb(x, g x)k ≥ R}) (2.16) δ δ < 1 − δ + · 1 = 1 − 2 2 which is false. Hence αt → id is not uniform on (M)1, and so by Theorem 2.3.2 we conclude that M N ⊗ Q for N and Q of type II. ∼ ∼ In particular, if R = R1 × R2, then L(R) = L(R1) ⊗ L(R2), so there is j ∈ {1, 2} such that L(Rj) is not type II and hence Rj does not have a.e. 33 equivalence class inﬁnite.

2.5.3 Remark

Theorem 2A (as well as Theorem 2D) in fact holds with L(R) replaced by L(R, σ), which is constructed as L(R), but “twisted” by some 2-cocycle σ : ∞ [R] × [R] → U(L (X)), in the sense that for g, h ∈ [R] the unitaries ug, uh, ugh ∈ ˜ L(R, σ) satisfy uguh = σ(g, h)ugh. Indeed, with R and (2.13) exactly as before, the formula (1.2) now gives rise to an s-malleable deformation of L(R, σ) ⊂ L(R˜, σ). Similarly, (2.15) now deﬁnes an L(R, σ)-L(R, σ) bimodule and the necessary iden- tiﬁcations in the proof of Theorem 2A hold. There is good reason for considering the algebras L(R, σ). The subalgebra L∞(X) ⊂ L(R, σ) is a Cartan subalgebra, i.e., it is maximal abelian and its normalizer generates L(R, σ) as a von Neumann algebra. Such subalgebras have been the object of intense study (see [Io12] for a detailed survey). Feldman and Moore showed in [FM75b] that a Cartan subalgebra A ⊂ M of a tracial von Neumann algebra M always arises as L∞(X) ⊂ L(R, σ) for some 2-cocycle σ and measured equivalence relation R on a standard probability space X.

2.6 Unique Prime Factorization

In this section we obtain a unique prime factorization result for a class of type II1 factors in the spirit of [OP03]. It is important to note that for II1 factors N,Q, and any t > 0 we have N ⊗ Q ∼= N t ⊗ Q1/t, so unique factorizations are considered modulo ampliﬁcations as well as unitary conjugacy.

2.6.1 An obstruction to unique factorization

We will need two lemmas before our proof of Theorem 2B. Both are well- known, but we include their proofs for completeness. 34

Lemma 2.6.1. Let N ⊂ M be an inclusion of tracial von Neumann algebras.

For any > 0 and projection p ∈ M satisfying kp − EN (p)k2 < , there exists a √ 1/3 projection q ∈ N such that kp − qk2 < + 10 .

Proof. Note that

2 2 kEN (p) − EN (p)k2 ≤ kEN (p) − pEN (p)k2 +kp(EN (p) − p)k2 +kp − EN (p)k2 < 3 and therefore for any δ > 0, Chebyshev’s inequality gives

2 1 2 2 9 τ(1 (E (p))) ≤ τ(1 2 2 (E (p))) ≤ kE (p) − E (p)k ≤ {(δ,1−δ)} N {|t −t|>δ } N δ4 N N 2 δ4

2 92 2 1/3 so that q = 1{|t−1|≤δ}(EN (p)) satisﬁes kEN (p) − qk2 ≤ δ4 + δ . Taking δ = then gives √ 1/3 kp − qk2 ≤ kp − EN (p)k2 + kEN (p) − qk2 ≤ + 10

∞ Lemma 2.6.2. Let {pn}n=1 ⊂ M be an asymptotically central sequence of projec- ∞ tions. Then there exist commuting projections {qk}k=1 ⊂ M which are asymptoti- ∞ cally central with kpnk − qkk2 → 0 as k → ∞ for some subsequence {pnk }k=1.

Proof. Let q1 = p1. Then given q1, . . . , qk, commuting projections, we let A = 00 Lm m {q1, . . . , qk} , and note that A = i=1 Cei for projections {ei}i=1 which are mini- Pm mal in A and such that i=1 ei = 1. Let nk+1 be large enough so that kpnk+1 ei − 2 1 0 m L 0 eipnk+1 k2 < mk for each 1 ≤ i ≤ m. Then A ∩ M = i=1 eiMei and EA ∩M (x) = Pm i=1 eixei for x ∈ M and hence m m 2 X 2 X 2 0 kpnk+1 − EA ∩M (pnk+1 )k2 = kpnk+1 − eipnk+1 eik2 = keipnk+1 − eipnk+1 eik2 i=1 i=1 (2.17) m m X X 1 1 ≤ ke p − p e k2 < = (2.18) i nk+1 nk+1 i 2 mk k i=1 i=1 √ 0 1 10 Then Lemma 2.6.1 gives qk+1 ∈ A ∩ M with kpnk+1 − qk+1k2 ≤ k + k1/3 . Thus ∞ kpnk − qkk2 → 0 as k → ∞ from which it further follows that the sequence {qk}k=1 is asymptotically central. 35

Theorem 2B. Let M1 and M2 be k·k2-separable II1 factors with property Gamma and set M = M1 ⊗ M2. Then there is an approximately inner automorphism φ ∈

Inn(M) such that φ(Mi) ⊀ Mj for any i, j ∈ {1, 2}.

Proof. Since M1 and M2 have Γ, there exist asymptotically central sequences of ∞ ∞ 1 projections {pk}k=1 ⊂ M1, {qk}k=1 ⊂ M2 such that τ(pk), τ(qk) → 2 as k → ∞.

Using Lemma 2.6.2 we may assume that [pk, pj] = [qk, qj] = 0 for all k, j ≥ 1. Let ∞ {xi}i=1 ⊂ M be a k · k2-dense sequence. ∞ ∞ Claim: There are sequences {vn}n=1 ⊂ U(M1), {un}n=1 ⊂ U(M2) and a ∞ subsequence {kn}n=1 ⊂ N such that for each n ≥ 1, the asymptotically central unitaries wn = 1 − 2pkn qkn satisfy 1 1 kw v w∗ − (1 − 2q )v k ≤ and kw u w∗ − (1 − 2p )u k ≤ , n n n kn n 2 2n n n n kn n 2 2n (2.19) 1 1 kw v w∗ − v k ≤ and kw u w∗ − u k ≤ for 1 ≤ i < n, i n i n 2 2n i n i n 2 2n (2.20) 1 1 kw v w∗ − v k ≤ and kw u w∗ − u k ≤ for 1 ≤ i < n, n i n i 2 2n n i n i 2 2n (2.21) 1 kw x w∗ − x k ≤ for 1 ≤ i < n. (2.22) n i n i 2 2n Before we prove the claim, let us prove the theorem assuming it holds. For ∗ n ≥ 1, let Wn = w1w2 ··· wn. Then WnxiWn is k · k2-Cauchy for any i ≥ 1, since for any m ≥ n > i,

m X 1 1 kW x W ∗ − W x W ∗k = kw ··· w x w∗ ··· w∗ − x k ≤ < m i m n i n 2 n+1 m i m n+1 i 2 2j 2n j=n+1

∗ 3 using (2.22). Similarly, (2.22) implies that Wn xiWn is also k · k2-Cauchy for any ∗ i ≥ 1, so we may deﬁne φ ∈ Inn(M) by φ(x) = lim WnxWn for x ∈ M, where the n→∞ convergence is in the SOT. Then noting that [wn, wm] = 0 for all n, m ≥ 1, and

3 ∗ In fact Wn = Wn here, but it is useful to note this as an implication of (2.22) since the ∗ construction Wn = w1w2 ··· wn can be done without arranging wn = wn and [wn, wm] = 0 for all n, m ≥ 1. 36 using (2.20) and (2.21), for each n ≥ 1 we have

∗ ∗ ∗ ∗ ∗ kφ(vn) − wnvnwnk2 = lim kw1w2 ··· wkvnwk ··· w2w1 − wnvnwnk2 k→∞ "n−1 k # X ∗ X ∗ ≤ lim sup kwivnwi − vnk2 + kwivnwi − vnk2 k→∞ i=1 i=n+1 "n−1 k # X 1 X 1 n − 1 1 n ≤ lim sup + ≤ + = 2n 2i 2n 2n 2n k→∞ i=1 i=n+1 which, combined with (2.19), gives n + 1 kφ(v ) − (1 − 2q )v k ≤ for all n ≥ 1. (2.23) n kn n 2 2n

We then see that for any y ∈ M2,

lim sup kEM1 (yφ(vn))k2 = lim sup kEM1 (y(1 − 2qkn )vn)k2 n→∞ n→∞

= lim sup kτ(y(1 − 2qkn ))vnk2 = lim sup |τ(y)(1 − 2τ(qkn ))| n→∞ n→∞ 1 = |τ(y)(1 − 2 · )| = 0 2 where we use the fact that τ(yyn)−τ(y)τ(yn) → 0 for any y ∈ M2 and any asymptotically central sequence {yn} ⊂ M2, which follows from the uniqueness of the trace. Since M = M1 ⊗ M2, this calculation then implies that kEM1 (aφ(vn)b)k2 →

0 for all a, b ∈ M and hence φ(M1) ⊀ M1. The same argument for the sequence

{φ(un)} shows that φ(M2) ⊀ M2.

On the other hand, note that φ(pk) = pk for all k ≥ 1, so that for each x ∈ M1,

lim sup kEM2 (φ(1 − 2pk)x)k2 = lim sup kEM2 ((1 − 2pk)x)k2 k→∞ k→∞

Similarly, analyzing the sequence {φ(1 − 2qk)} shows that φ(M2) ⊀ M1. Proof of Claim: We construct the necessary sequences recursively. There- fore, suppose we are given {k1, . . . , kn−1}, {v1, . . . , vn−1}, and {u1, . . . , un−1} such 37 that (2.19), (2.20), (2.22), and (2.21) are satisﬁed (allowing these sets to be empty for the base case n = 1). We construct kn, vn, and un as follows. 00 Letting B1 = {pk1 , . . . , pkn−1 } , we know that B1 is abelian and hence of Lm m the form B1 = i=1 Cei for projections {ei}i=1 which are minimal in B1 and such Pm 0 ω Lm ω 0 ω that i=1 ei = 1. Then B1 ∩ M1 = i=1 eiM1 ei and therefore Z(B1 ∩ M1 ) = B1 0 ω since M1 is a factor. Let p denote the image of the sequence {pk} in M1 ∩ M1 , 1 0 ω noting that τ(p) = 2 = τ(1 − p). Then p ∼ (1 − p) in B1 ∩ M1 since

m m X τ(pei) X τ(p)τ(ei) EZ(B0 ∩M ω)(p) = EB (p) = ei = ei 1 1 1 τ(e ) τ(e ) i=1 i i=1 i

= τ(p) = τ(1 − p) = E 0 ω (1 − p). Z(B1∩M1 )

0 ω ∗ Thus there isv ˜ ∈ U(B1 ∩ M1 ) such thatvp ˜ v˜ = 1 − p. Setting B2 = 00 0 ω {qk1 , . . . , qkn−1 } and letting q denote the image of {qk} in M2 ∩ M2 , the same 0 ω ∗ argument shows that there isu ˜ ∈ U(B2 ∩ M2 ) such thatuq ˜ u˜ = 1 − q. Lifting ∞ ∞ v˜ andu ˜ to sequences of unitaries {v˜k}k=1 ⊂ U(M1), {u˜k}k=1 ⊂ U(M2) which asymptotically commute with B1 and B2, we can then ﬁnd kn large enough that vn =v ˜kn and un =u ˜kn have 1 1 kv p v∗ − (1 − p )k ≤ , ku q u∗ − (1 − q )k ≤ , (2.24) n kn n kn 2 2n+1 n kn n kn 2 2n+1 1 1 kv p v∗ − p k ≤ , and ku q u∗ − q k ≤ for 1 ≤ i < n, n ki n ki 2 2n+1 n ki n ki 2 2n+1 (2.25) and we further assume that kn is large enough that (2.22) and (2.21) are satisﬁed ∞ (which can be done since {(1 − 2pkqk)}k=1 is asymptotically central). Noting that

[1 − 2pkn qkn ][1 − 2(1 − pkn )qkn ] = 1 − 2qkn , from (2.24) we get

∗ ∗ kwnvnwn − [1 − 2qkn ]vnk2 = k[1 − 2pkn qkn ][1 − 2(vnpkn vn)qkn ] − [1 − 2qkn ]k2 1 1 ≤ k2(1 − p )q − 2(v p v∗)q k ≤ 2 · = , kn kn n kn n kn 2 2n+1 2n

∗ 1 and similarly kwnunwn − [1 − 2pkn ]unk2 ≤ 2n so that (2.19) is satisﬁed. For 38

1 ≤ i < n, we use (2.24) to estimate

∗ ∗ ∗ ∗ kvnwi vn − wi k2 = k[1 − 2(vnpki vn)qki ] − [1 − 2pki qki ]k2 1 1 ≤ 2kv p v∗ − p k ≤ 2 · = , n ki n ki 2 2n+1 2n

∗ ∗ ∗ 1 and similarly kunwi un − wi k2 ≤ 2n so that (2.20) holds.

2.6.2 Unique prime factorization via s-malleable deformation

The principle challenge in the proof of the unique prime factorization in Theorem 2.6.4 is controlling the Cartan subalgebras of each factor. The following proposition will be critical for this reason:

Proposition 2.6.3. Let M = N ⊗ Q = M1 ⊗ M2 be a II1 factor without property

Gamma, and suppose that N ≺M A ⊗ M2 for some Cartan subalgebra A of M1. t ∗ Then there is t > 0 and u ∈ U(M) such that uN u ⊂ M2 under the identiﬁcation N ⊗ Q = N t ⊗ Q1/t.

Proof. By Theorem 2.2.6 there are projections p ∈ N, f ∈ A ⊗ M2, a unital normal

∗-homomorphism θ : pNp → f(A ⊗ M2)f, and a nonzero partial isometry v ∈ M, such that

θ(x)v = vx for all x ∈ pNp, v∗v ∈ (N 0 ∩ M)p, and vv∗ ∈ θ(pNp)0 ∩ fMf (2.26)

Let L = θ(pNp)0 ∩fMf, Z = Z(L), and e = vv∗. Note that Af ⊂ L and therefore 0 Z ⊂ (Af) ∩ fMf = f(A ⊗ M2)f. From (2.26) it follows that

∗ ∗ 0 ∗ ∗ ∗ v Zv ⊂ Z(v v(N ∩ M)v v) = Z(Q)v v = Cv v and hence Ze = v(Cv∗v)v∗ = Ce. Therefore setting z = C(e) (the support of e in Z), and taking any z0 ∈ Z, z0 ≤ z, we have z0e ∈ Ce and hence z0e ∈ {0, e} which implies that z0 ∈ {0, z}. Thus Lz is a ﬁnite factor. Hence there is e0 ∈ Lz, e0 ≤ e 39

0 0 1 0 with τLz(e ) = τ(e )/τ(z) = n for some integer n. Let v1 = e v and note that for any x ∈ pNp we have

∗ ∗ 0 ∗ 0 ∗ 0 ∗ 0 ∗ v1v1x = v e vx = v e θ(x)v = v θ(x)e v = xv e v = xv1v1 (2.27)

∗ 0 and hence v1v1 ∈ (pNp) ∩ pMp = Qp, so let q ∈ Q be a projection such that ∗ v1v1 = q ⊗ p. Let s = τ(q)τ(z)/τ(e0) and identify Q ⊗ N = Qs ⊗ N 1/s such that Cq ⊗ pNp = Cq0 ⊗ p0N 1/sp0 and qQq ⊗ Cp = q0Qsq0 ⊗ Cp0 for projections q0 ∈ Qs, 0 1/s 0 0 1 0 p ∈ N with τ(q ) = τ(q)/s = τ(e )/τ(z) = n and τ(p ) = τ(p)s = τ(z). s t Since Q and Lz are factors, let w1, . . . , wn ∈ Q and u1, . . . , un ∈ Lz be Pn ∗ Pn ∗ ∗ 0 ∗ 0 partial isometries with j=1 wjwj = 1, j=1 ujuj = z and wj wj = q , uj uj = e Pn ∗ ∗ 0 ∗ for all 1 ≤ j ≤ n. Then setting w = j=1 ujv1wj we have w w = p and ww = z, 1/s ∗ and wN w ⊂ z(A ⊗ M2)z. Cutting w to the right by a projection in N under p0, we may assume 0 1 that τ(p ) = τ(z) = m for some integer m. By [Po81, Theorem 3.2 and Remark

3.5.2], we can ﬁnd a copy of the hyperﬁnite II1 factor R, with A ⊂ R ⊂ M1 and 0 0 R ∩ M1 = C. Note that Af ⊂ L =⇒ Z ⊂ (Af) ∩ fMf = f(A ⊗ M2)f =⇒ z ∈ A ⊗ M2 ⊂ R ⊗ M2. Since R ⊗ M2 is a factor, there are partial isometries ∗ Pm ∗ u˜1,..., u˜m ∈ R ⊗ M2 withu ˜j u˜j = z for each j and j=1 u˜ju˜j = 1. Taking partial ∗ Pm ∗ isometriesw ˜1,..., w˜m ∈ N withw ˜j w˜j = z for each j and j=1 w˜jw˜j = 1, and Pm ∗ 1/s ∗ setting u0 = j=1 u˜jww˜j we have u0 ∈ U(M) and u0N u0 ⊂ R ⊗ M2. N∞ N∞ Now write R = j=1 M2(C) and set Rk = j=k+1 M2(C), so that R = hNk i j=1 M2(C) ⊗ Rk for any k ≥ 1. Then for any > 0, there is k ≥ 1 such that kb − E s ∗ (b)k < for all b ∈ U(R ). Indeed if not, there would be u0Q u0 2 k

> 0 and {b } ⊂ U(R) with b ∈ U(R ) and kb − E s ∗ (b )k ≥ for all k k k k u0Q u0 k 2 k. Then {bk} would be an asymptotically central sequence in R and hence in 1/s R ⊗ M2. In particular, {bk} would asymptotically commute with u0N u0 which does have property Gamma since M is non-Gamma. But this would imply that kb −E s ∗ (b )k → 0 by Connes characterization of property Gamma in [Co75a], k u0Q u0 k 2 a contradiction. 1 1 In particular, taking = we ﬁnd k ≥ 1 such that kb − E s ∗ (b)k < 2 u0Q u0 2 2 s ∗ for all b ∈ U(Rk) which implies that Rk ≺ u0Q u0. It follows that R = M2k (Rk) 40

s ∗ has R ≺ u0Q u0. Using Lemma 3.5 of [Va07], we pass to relative commutants to 1/s ∗ 0 s ∗ ﬁnd that u0N u0 ≺ R ∩ M = M2 and then M1 ≺ u0Q u0. 0 Then by Proposition 12 of [OP03], since M1 ∩ M = M2 is a factor, there is ∗ sr ∗ r > 0 andu ˜0 ∈ U(M) such thatu ˜0M1u˜0 ⊂ u0Q u0 after identifying 1/s s ∗ ∼ 1/sr sr ∗ ∗ u0(N ⊗ Q )u0 = u0(N ⊗ Q )u0. Setting t = 1/sr and u =u ˜0u0, we have

t ∗ ∗ sr ∗ 0 0 uN u = (˜u0u0Q u0u˜0) ∩ M ⊂ M1 ∩ M = M2.

Theorem 2.6.4. Let M1,...,Mk be II1 factors without property Gamma, each with i ˜ an s-malleable deformation {αt}t∈R ⊂ Aut(Mi) for some tracial von Neumann ˜ 2 ˜ algebras Mi ⊃ Mi. Suppose that for each i, the Mi-Mi bimodule Mi L (Mi) 2 L (Mi)Mi is weakly contained in the coarse Mi-Mi bimodule and mixing relative to i some abelian subalgebra Ai ⊂ Mi. Assume that the convergence αt → id is not uniform in k · k2 on (Mi)1 for any i. Then Mi is prime for each i, and

(1). If M = M1 ⊗ M2 ⊗ ... ⊗ Mk = N ⊗ Q for tracial factors N,Q, there must be a partition I ∪ I = {1, . . . , k} and t > 0 such that N t = N M and N Q i∈IN i Q1/t = N M modulo unitary conjugacy in M. i∈IQ i

(2). If M = M1 ⊗ M2 ⊗ ... ⊗ Mk = P1 ⊗ P2 ⊗ · · · ⊗ Pm for II1 factors

P1,...,Pm and m ≥ k, then m = k, each Pi is prime, and there are t1, . . . , tk > 0 with t1t2 ··· tk = 1 such that after reordering indices and conjugating by a unitary

ti in M we have Mi = Pi for all i.

(3). In (2), the assumption m ≥ k can be omitted if each Pi is assumed to be prime.

Proof. We prove (1) by induction on k. Note that by Theorem 2.3.2, we know that each Mi is prime, so the case k = 1 can only occur if either Q or N is ﬁnite dimensional. Without loss of generality, assume N = Mn(C) for some n ∈ Z>0.

Then t = 1/n does the job with IN = ∅. Now suppose that k ≥ 2 and for convenience set M = N ⊗ Q. Since M is non-amenable, we assume without loss of generality that Q is non-amenable. For i ˜ ˜ i ˜ each i, we extend αt ∈ Aut(Mi) to M = M1 ⊗ · · · ⊗ Mi ⊗ · · · ⊗ Mk by the rule 41

i i αt|Mj = id for j 6= i. Thus for each i we obtain an s-malleable deformation {αt}t∈R ˜ i i i i of M ⊂ M . For x ∈ M, set δt(x) = αt(x)−EM (αt(x)). For each I ⊂ {1, 2, . . . , k}, N ˆ N ˆ let MI = j∈I Mj and MI = j∈ /I Mj so that M = MI ⊗ MI . i We claim that there must be i such that αt → id uniformly in k·k2 on (N)1.

Suppose not. Then using Lemma 2.3.1, for each i we ﬁnd i > 0 and sequences i i i i i i 2 ˜ i 2 {x } ⊂ (N)1, {t } ⊂ with t → 0 as n → ∞ and ξ = δ i (x ) ∈ L (M ) L (M) n n R n n tn n i i i i satisfying kξnk ≥ i, kxξnk ≤ kxk2, and kxξn −ξnxk → 0 as n → ∞ for each x ∈ Q. Since N = Q0 ∩ M is a factor, applying Lemma 2.2.3 gives

2 2 ˜ i 2 M L (M)Q ≺ M L (M ) L (M)Q (2.28)

2 ˜ 2 2 2 But since Mi [L (Mi) L (Mi)]Mi ≺ Mi L (Mi) ⊗ L (Mi)Mi for each i, we also have

[L2(M˜ i) L2(M)] ≺ L2(M) ⊗ L2(M) (2.29) M M M Mˆ i M for each i. Then combining (2.28) and (2.29) we have

L2(M) ≺ L2(M) ⊗ L2(M) , (2.30) M Q M Mˆ i Q

ˆ so that Q is amenable relative to Mi in M for each i. But note that for any 00 I,J ⊂ {1, 2, . . . , n}, the subalgebras MI and MJ satisfy M = NM (MI ) and

[eMI , eMJ ] = 0, so that after k − 1 applications of 2.2.5 we ﬁnd that Q is amenable Tk ˆ relative to i=1 Mi = C, which contradicts the nonamenability of Q. Thus there i must indeed be some i ∈ {1, 2, . . . , k} such that αt → id uniformly in k · k2 on

(N)1. 2 ˜ i 2 ˆ 2 ˜ We have that L (M ) L (M) is mixing relative to Ai ⊗ Mi since L (Mi) 2 L (Mi) is mixing relative to Ai. It follows that there can be no sequence {un} ⊂ (N) with kE (xu y)k → 0 for each x, y ∈ M. If there were, we would 1 Ai ⊗ Mˆ i n 2 i conclude, just as in (2.9), that αt → id uniformly on (Q)1, and then on all of (M)1 as in the proof of Theorem 2.3.2. This would then contradict the assumption that i the convergence αt → id is not uniform on (Mi)1. ˆ Thus N ≺M Ai ⊗ Mi by Theorem 2.2.6. Then by Proposition 2.6.3, there is t > 0 such that after decomposing M = N t ⊗ Q1/t and conjugating by a unitary, t ˆ t 0 ˆ 1/t ˆ ˆ t we have N ⊂ Mi. Set P = (N ) ∩ Mi = Q ∩ Mi so that Mi = N ⊗ P . If P 42

ˆ nt 1/nt is type In for some n, it follows that Mi = N and Mi = Q and the proof is done. Otherwise, P is type II1 and by the inductive hypothesis, there is a partition st 1/s IN ∪ IP = {1, . . . , k}\{i} and s > 0 such that N = MIN and P = MIP modulo unitary conjugation. Then since Q1/st = (N st)0 ∩ M = M 0 ∩ M = M ⊗ M , IN i IP setting IQ = IP ∪ {i} concludes the proof of (1). We also prove (2) by induction on k. The case k = 1 follows immediately from the primeness of M1. For k ≥ 2, we apply (1) with N = P1 ⊗ · · · ⊗ Pm−1 and

Q = Pm, to ﬁnd a partition IN ∪ IQ = {1, . . . , k}, t > 0 such that after conjugating t t 1/t by a unitary in M we have P1 ⊗ · · · ⊗ Pm−1 = MIN and Pm = MIQ . Then m − 1 ≥ |IN | so we apply the inductive hypothesis to conclude that |IN | = m − 1 and ﬁnd s1, . . . , sm−1 with s1s2 ··· sm−1 = 1 such that after reordering and unitary

t tsi conjugation (in N ) we have Mi = Pi for 1 ≤ i ≤ m − 1. But

m ≥ k = |IN | + |IQ| = m − 1 + |IQ| =⇒ |IQ| = 1 and m = k, (2.31) so setting tm = 1/t and ti = tsi for 1 ≤ i ≤ m − 1 ﬁnishes the proof of (2). For (3), we proceed just as for (2), except that we replace (2.31) by the 1/t observation that Pm = MIQ implies |IQ| = 1 when Pm is assumed to be prime.

2.6.3 Unique prime factorization for equivalence relations

In order to deduce Theorem 2D from Theorem 2.6.4, we prove the following proposition:

Proof. That M = L(R) is prime is simply a special case of Theorem 2A. Again, ˜ ˜ consider the s-malleable deformation M ⊂ M, {αt}t∈R ⊂ Aut(M) constructed in Section 2.4, and suppose toward a contradiction that M has property Gamma. 43

Then there is a sequence {un} ∈ U(M) with τ(un) = 0 for all n and kunx−xunk2 →

0 as n → ∞ for each x ∈ M. Then for any u ∈ NM (A) we have

∗ ∗ kuEA(un)u − EA(un)k2 = kEA(uunu ) − EA(un)k2

∗ ≤ kuunu − unk2 → 0 as n → ∞

00 Since the sequence EA(un) is bounded in norm and M = NM (A) , it follows that kxEA(un) − EA(un)xk2 → 0 for each x ∈ M. Since R is strongly ergodic, it follows that kEA(un)k2 = kEA(un) − τ(EA(un))k2 → 0 as n → ∞. 2 Fix any g ∈ [R] with g = e. Note zg = EA(ug) is a projection given ∗ 0 ∗ by zg = 1{s∈X:gs=s} = zg−1 . Moreover, ugEA(ug) ∈ A ∩ M = A =⇒ ugzg = ∗ ∗ EA(ugzg) = EA(ug)zg = zg. Hence

∗ ∗ kEA(unug)zgk2 = kEA(unugzg)k2 = kEA(un)zgk2 ≤ kEA(un)k2 → 0 as n → ∞ (2.32)

∗ Moreover, since 1−zg = 1{s∈X:gs6=s}, for any nonzero z ≤ 1−zg with ugzug = z, we 0 0 ∗ 0 0 ∗ 0 can ﬁnd nonzero z ≤ z such that ugz ug ≤ z −z (if not we would have ugz ug = z 0 2 for all z ≤ z and then z ≤ zg). Then because g = e, it follows that we can ﬁnd a ∗ projection z ∈ A such that 1 − zg = z + ugzug, so that

∗ 2 ∗ ∗ 2 kEA(unug)(1 − zg)k2 = kEA(unug)(z + ugzug)k2 ∗ 2 ∗ ∗ 2 = kEA(unug)zk2 + kEA(unug)ugzugk2 ∗ ∗ 2 = kEA(unug)(z − ugzug)k2 ∗ ∗ ∗ 2 = kzEA(unug) − EA(unugugzug)k2 ∗ ∗ 2 2 = kEA(zunug − unzug)k2 ≤ kzun − unzk2 → 0 as n → ∞. (2.33)

∗ Combining (2.32) and (2.33) we see that kEA(unug)k2 → 0 as n → ∞ for each g ∈ [R] with g2 = e. By Feldman and Moore [FM75a], we know that (x, y) ∈ R if and only if y = gx for some g ∈ [R] with g2 = e, so that L(R) = 2 00 {aug : a ∈ A, g ∈ [R], g = e} . It therefore follows that kEA(xuny)k2 → 0 as n → ∞ for any x, y ∈ M. 2 ˜ 2 From the proof of Theorem 2A, we know that M L (M) L (M)M is mixing relative to A, so this implies that hunδt(x), δt(x)uni → 0 as n → ∞ for each x ∈ M. 44

We also know that αt → id is not uniform on (M)1, and hence by part (1) of

Lemma 2.3.1 there is > 0 and sequences {xk} ⊂ (M)1, {tk} ⊂ R, tk → 0, such that kδtk (xk)k2 ≥ for all k. Then using Lemma 2.3.1, for any k we get 2 2 1 2 ≤ kδtk (xk)k2 = lim inf k[un, δtk (xk)]k2 + Rehunδtk (xk), δtk (xk)uni n→∞ 2

1 2 2 ≤ lim inf [2kαtk (un) − unk2 + k[un, xk]k2] ≤ 8 lim inf kδtk/2(un)k2 2 n→∞ n→∞

tk Thus setting sk = 2 for each k we can ﬁnd nk ≥ k such that kδsk (unk )k2 ≥ 4 . Then using Lemma 2.3.1 again, for any x ∈ M,

k[δsk (unk ), x]k2 ≤ 2kαsk (x) − xk2 + k[unk , x]k2 → 0 as k → ∞.

Since we also have kxδtk/2(unk )k2 ≤ kxk2 for all k, we apply Lemma 2.2.3 to ﬁnd 2 2 ˜ 2 that M L (M)M ≺ M L (M) L (M)M . But since we know from the proof of 2 ˜ 2 2 2 Theorem 2A that M L (M) L (M)M ≺ M L (M) ⊗ L (M)M , this implies that M is amenable, a contradiction.

Combining Theorem 2.6.4 with Proposition 2C and the proof of Theorem 2A, we get Theorem 2D immediately. We prove Corollary 2E below:

Proof of Corollary 2E. For (1), let Ai ⊂ L(Ri) and Bi ⊂ L(Si) denote the canonical Cartan algebras of the factors. By [FM75b], the hypothesis leads to a nor- ∼ mal ∗-isomorphism M = L(R1) ⊗ L(R2) ⊗ ... ⊗ L(Rk) = L(S1) ⊗ L(S2) which identiﬁes A1 ⊗ A2 ⊗ · · · ⊗ Ak = B1 ⊗ B2. Applying Theorem 2D, we ﬁnd t > 0, u ∈ U(M), and an integer 1 ≤ m < k such that after reordering the in- t ∗ 1/t ∗ dices we have uL(S1) u = L(R1) ⊗ L(R2) ⊗ ... ⊗ L(Rm) and uL(S2) u =

L(Rm+1) ⊗ L(Rm+2) ⊗ ... ⊗ L(Rk). Setting A = A1 ⊗ · · · ⊗ Ak, we have t ∗ ∗ t ∗ uB1u ≺M A (as u (uB1u )u ⊂ A) which implies that t ∗ t ∗ t ∗ uB1u ≺uL(S1) u A1 ⊗ · · · ⊗ Am. Indeed, if there were {un} ⊂ U(uB1u ) with t ∗ kEA1 ⊗ ··· ⊗ Am (xuny)k2 → 0 for all x, y ∈ uL(S1) u , one can check that it would give t ∗ kEA(xuny)k2 → 0 for all x, y ∈ M as well. Then since uB1u and A1 ⊗ · · · ⊗ Am t ∗ are both Cartan subalgebras of uL(S1) u , we know as in [Po01b] that there t ∗ t ∗ ∗ is v1 ∈ U(uL(S1) u ) such that v1uB1u v1 = A1 ⊗ · · · ⊗ Am. Thus ad v1u is 45

t ∼ t an isomorphism of L(S1) = L(S1) onto L(R1 × R2 × · · · × Rm) which identi- t t ∼ fies B1 and A1 ⊗ · · · ⊗ Am. A second application of [FM75b] then gives S1 = 1/t R1 × R2 × · · · × Rm. Similarly, one identifies B2 and Am+1 ⊗ · · · ⊗ Ak to con- 1/t ∼ clude that S2 = Rm+1 × · · · × Rk. We prove (2) by induction on k. The case k = 1 follows immediately from Theorem 2A. For k ≥ 2, we apply (1) to find t > 0 and an integer 1 ≤ t t ∼ j < k such that after reordering indices, S1 × · · · × Sm−1 = R1 × · · · × Rj and 1/t ∼ Sm = Rj+1 × · · · × Rk. Then m − 1 ≥ j so we apply the inductive hypothesis to conclude that j = m − 1 and find s1, . . . , sm−1 with s1s2 ··· sm−1 = 1 such

tsi that after reordering we have Ri = Si for 1 ≤ i ≤ m − 1. Finally, we have 1/t ∼ 0 < k − j ≤ m − (m − 1) = 1, and so k = j + 1, m = k, and Sm = Rm.

2.7 Application to Measure Equivalent Groups

The tools developed in the previous sections lend themselves easily to the measure equivalence of groups, a notion ﬁrst introduced by Gromov [Gr91]. Count- ME able groups Γ1 and Γ2 are called measure equivalent (ME), written Γ1 ∼ Γ2, if there is a Lebesgue measure space (Y, ν) and commuting free measure preserving actions Γi y (Y, ν), i ∈ {1, 2}, which each admit a ﬁnite measure fundamental domain. Measure equivalence is closely related to stable orbit equivalence. Recall that two probability measure preserving actions Γi y (Xi, µi), i ∈ {1, 2}, on standard probability spaces (Xi, µi) are stably orbit equivalent (SOE) if for each i ∈ {1, 2} we can choose a measurable subset Ei ⊂ Xi meeting the orbit of a.e. ∼ x ∈ Xi, such that the restricted equivalence relations are isomorphic, i.e. R1|E1 = ME R2|E2 where Ri = R(Γi y Xi) for i ∈ {1, 2}. Then Γ ∼ Λ if and only if Γ and Λ admit SOE free actions. This equivalence was proved by Furman in [Fu99] where it is attributed to Zimmer and Gromov, and the form stated here (that the actions can be taken to be free) was proved in [Ga00]. Gaboriau showed in [Ga01] that measure equivalent groups have propor- 2 ME tional ` Betti numbers, i.e., if Γ ∼ Λ there is λ > 0 such that βn(Γ) = λβn(Λ) for 46

all n. In particular, if β1(Γ) > 0 then Γ cannot be measure equivalent to a product of inﬁnite groups (as β1 = 0 for a product of inﬁnite groups). The following theorem strengthens this conclusion since we know from [PT07] that if β1(Γ) > 0 then Γ is non-amenable and admits an unbounded 1-cocycle for the left regular representation (which is mixing).

ME Proof. Suppose that Γ ∼ Γ1 × Γ2 for groups Γ1, Γ2. Then there are SOE free 0 actions Γ y (X, µ) and Γ1 × Γ2 y (Y, ν). Letting R = R(Γ y X) and R =

R(Γ1 × Γ2 y Y ), this means there are measurable E ⊂ X, F ⊂ Y meeting a.e. ∼ 0 0 orbit and such that R|E = R |F . We may assume that R and R are ergodic, ∼ since if not, we replace µ|E = ν|F by a measure in the ergodic decomposition of ∼ 0 R|E = R |F and then extend this measure toµ ˜ on X andν ˜ on Y using the fact that E ⊂ X, F ⊂ Y meet a.e. orbit. Then for t1 = µ(E), t2 = ν(F ), we have Rt1 ∼= (R0)t2 , and hence Rt1/t2 ∼= R0. t Set t = t1/t2, M = L(R ), and let (Xt, µt) denote the underlying probability space of Rt. Since Γ y X is free, we see from (2.5) that R admits an unbounded 1-cocycle b into a mixing orthogonal representation π weakly contained in the regular representation. Let πt and bt be the amplifications as in (2.8). Then, as in Section 2.4, we construct from πt and bt an imbedding M ⊂ M˜ and s-malleable ˜ ˜ deformation {αs}s∈R ⊂ Aut(M), β ∈ Aut(M). As in the proof of Theorem 2A, we know that L2(M˜ ) L2(M) is weakly contained in the coarse M-M bimodule ∞ and mixing relative to the abelian subalgebra A = L (Xt). Thus M satisfies the assumptions of Theorem 2.3.2, and we need only modify its proof slightly. We know that R (and hence Rt) is non-amenable since Γ is non-amenable and Γ y X is free. It follows that either Γ1 or Γ2 must be non-amenable, so assume t ∼ 0 without loss of generality that Γ2 is non-amenable. Since R = R , we have an ∼ ∞ ∞ ∞ isomorphism M = L (Y ) o (Γ1 × Γ2) which identifies A = L (Xt) and L (Y ).

We therefore consider the commuting subalgebras L(Γ1),L(Γ2) ⊂ M. Then just as in the proof of Theorem 2.3.2, we must have αs → id uniformly in k · k2 on the 47

unit ball of L(Γ1), since otherwise we would have

2 2 2 2 2 L(Γ2)L (L(Γ2))L(Γ2) ≺ L(Γ2)L (Mf) L (M)L(Γ2) ≺ L(Γ2)L (L(Γ2)) ⊗ L (L(Γ2))L(Γ2) contradicting the nonamenability of Γ2.

Assuming toward a contradiction that Γ1 is also inﬁnite, take a sequence ∞ {ugn }n=1 ⊂ Γ1. From the freeness of the action it follows that limn→∞ kEA(xugn y)k2 = 0 for each x, y ∈ M. Then just as in (2.9), combining 2 t 2 t the sequence {ugn } with the mixingness of L (Mf ) L (M ) relative to A gives

αs → id uniformly in k · k2 on the unit ball of L(Γ2).

Therefore for any > 0, we can ﬁnd s0 > 0 such that for |s| < s0 we have kαs(x) − xk2 < 4 for all x ∈ L(Γ1) ∪ L(Γ2) with kxk ≤ 1. Then for |s| < s0, the ∗ ∗ k · k2-closed convex hull Ks of the set {αs(ug)αs(uh)uguh : g ∈ Γ1, h ∈ Γ2} has a unique element ks ∈ Ks of minimal k · k2 satisfying kks − 1k ≤ 2 .

For a ∈ U(A) and (g, h) ∈ Γ1 × Γ2 using the facts that αs(a) = a,[ug, uh] = ∗ 0 and ug, uh ∈ NM (A), one can check that αs(auguh)Ks(auguh) = Ks. From ∗ the uniqueness of ks it then follows that αs(auguh)ks(auguh) = ks and hence

αs(x)ks = ksx for all x ∈ M. Then

kαs(x) − xk2 ≤ kαs(x) − αs(x)ksk2 + kksx − xk2 ≤ 2kks − 1k2 ≤ for all x ∈ (M)1, |s| < s0. Thus αs → id uniformly on (M)1, which contradicts the unboundedness of bt just as in (2.16).

Note

Chapter 2 is, in part, a reprint of the material as it appears in [Ho15] Daniel J. Hoﬀ, Von Neumann algebras of equivalence relations with nontrivial one-cohomology, J. Funct. Anal. 270 (2016), no. 4, 1501–1536. MR 3447718. of which the dissertation author was the primary investigator and author. Chapter 3

Von Neumann’s Problem and Extensions of Non-amenable Equivalence Relations

by Lewis Bowen1, Daniel Hoﬀ2, and Adrian Ioana3

3.1 Introduction and statement of main results

3.1.1 Background

The notion of amenability for groups was introduced by J. von Neumann in order to explain the Banach-Tarski paradox [vN29]. He showed that any countable group that contains the free group F2 on two generators is non-amenable. The question of whether any non-amenable group contains F2, became known as von Neumann’s problem, and was eventually settled in the negative by A. Ol’shanskii [Ol80]. Remarkably, D. Gaboriau and R. Lyons proved that von Neumann’s problem has a positive solution in the context of measurable group theory [GL07] (see

1L.B. was partially supported by NSF grant DMS-1500389 and NSF CAREER Award DMS- 0954606 2D.H. was partially supported by the NSF-GRFP under Grant No. DGE-1144086 3A.I. was partially supported by NSF Grant DMS 1161047, NSF CAREER Award DMS #1253402, and a Sloan Foundation Fellowship

48 49 also the survey [Ho11]). More precisely, they showed that any countable non- amenable group Γ admits F2 as a “measurable subgroup”: there exists a free ergodic probability measure preserving (pmp) action Γ y (X, µ) whose associated orbit equivalence relation R(Γ y X) contains the orbit equivalence relation of a free ergodic pmp action F2 y (X, µ). Moreover, the Bernoulli action of Γ on ([0, 1]Γ, λΓ) has this property, where λ denotes the Lebesgue measure on [0, 1]. At the end of [GL07], the authors posed the following analogue of von Neumann’s problem for equivalence relations: does every ergodic non-amenable countable pmp equivalence relation R contain R(F2 y X) for some free ergodic pmp action of F2? The main result of [GL07] shows that this is indeed the case if R arises from the Bernoulli action with base ([0, 1], λ) of a non-amenable group. Our ﬁrst goal is to show that, more generally, von Neumann’s problem has a positive answer for the Bernoulli extension with base ([0, 1], λ) of any ergodic non-amenable equivalence relation.

3.1.2 Von Neumann’s problem for Bernoulli extensions

To state our main result, let us introduce some notation. Let R be a countable pmp equivalence relation on a probability space (X, µ), and let (K, κ) be another probability space, both always assumed to be standard. Given x ∈ X,

[x]R denotes the equivalence class of x.

[x]R We denote by XK the set of pairs (x, ω) with x ∈ X and ω ∈ K . We endow XK with the smallest σ-algebra of sets which makes the maps (x, ω) 7→ x and (x, ω) 7→ ω(θ(x)) measurable, for every θ ∈ [R] (where [R] denotes the full group of R). Also we endow XK with the probability measure µκ given by

[x]R dµκ(x, ω) = dκ (ω) dµ(x).

We denote by RK the equivalence relation on XK given by (x, ω)RK (y, ξ) iﬀ xRy and ω = ξ, and call it the Bernoulli extension of R with base space (K, κ) (see [Bo12, Section 11]). The following is the main result of this paper: 50

Theorem 3A. Let R be an ergodic non-amenable countable pmp equivalence relation on a probability space (X, µ). Let (K, κ) be a probability space.

If (K, κ) = ([0, 1], λ), then there exists a free ergodic pmp action F2 y

(XK , µκ) such that R(F2 y XK ) ⊂ RK , almost everywhere. Moreover, if R has an ergodic subequivalence relation of inﬁnite index which is non-amenable or normal, or if R has an inﬁnite fundamental group, then the above conclusion holds for any non-trivial probability space (K, κ).

Here and after, we say that a pmp action Γ y (X, µ) is essentially free (in short, free) if the stabilizer Γx = {g ∈ Γ|g · x = x} is trivial, for almost every x ∈ X.

Remark 3.1.1. If R is the orbit equivalence relation of a free pmp action Γ y X, then RK is isomorphic to the orbit equivalence relation of the product action Γ y X × KΓ (Proposition 3.3.2). Thus, if R = R(Γ y [0, 1]Γ) and (K, κ) = ([0, 1], λ), then RK is isomorphic to R. Theorem 3A implies that R contains the orbits of a free ergodic pmp action of F2, for any non-amenable Γ, and therefore recovers [GL07, Theorem 1].

The pioneering work [GL07] has recently spawned a lot of interest in measure theoretic versions of von Neumann’s problem. Thus, the main result of [GL07] was strengthened in [Ku13], and its proof was simpliﬁed in [Th13]. Very recently, von Neumann’s problem was shown to have a positive solution for non-amenable equivalence relations that act on hyperbolic bundles [Bo15], while a measure theoretic version of it for locally compact groups was formulated and proven in [GM15]. Theorem 3A leads to a characterization of non-amenability for ergodic equivalence relations in terms of actions of F2, which can be viewed as a weak version of von Neumann’s problem for equivalence relations. Before stating this result, we introduce the notion of extensions of equivalence relations, which will be key in the rest of the paper.

Deﬁnition 1. For countable pmp equivalence relations R on (X, µ) and R˜ on (X,˜ µ˜), we say that R˜ is a class-bijective extension (in short, an extension) of R if there is a Borel map p : X˜ → X satisfying 51

1. µ(E) =µ ˜(p−1(E)), for all Borel sets E ⊂ X,

2. p| is injective, for almost every x ∈ X˜, and [x]R˜ ˜ 3. p([x]R˜ ) = [p(x)]R, for almost every x ∈ X.

Remark 3.1.2. A map p : X˜ → X which satisfies conditions (1)-(3) in the above definition is called a local OE (or local isomorphism) of R˜, R in [Po05, Definition 1.4.2].

Corollary 3B. An ergodic countable pmp equivalence relation R is non-amenable if and only if it admits an extension which contains almost every orbit of a free ergodic pmp action of F2.

3.1.3 Uncountably many non-isomorphic extensions

In the rest of the introduction, we discuss several applications of Theorem 3A to the classiﬁcation of orbit equivalence relations. To this end, we need to recall some terminology.

Deﬁnition 2. Recall that two equivalence relations R and S on probability spaces (X, µ) and (Y, ν) are called isomorphic (resp. stably isomorphic) if there exist co- null (resp. non-null) Borel subsets X0 ⊂ X,Y0 ⊂ Y and a measure preserving 0 0 0 Borel isomorphism θ : X0 → Y0 such that xRx iﬀ θ(x)Sθ(x ), for all x, x ∈ X0. −1 Here, we endow X0 ⊂ X with the probability measure µ(X0) (µ|X0). Moreover, if R and S are countable pmp, then they are called von Neumann equivalent (resp. stably von Neumann equivalent) if their von Neumann algebras L(R) and L(S) are isomorphic (resp. pL(R)p ∼= qL(S)q, for some non-zero projections p ∈ L(R), q ∈ L(S)). Two pmp actions Γ y (X, µ) and Λ y (Y, ν) of two locally compact second countable (lcsc) groups Γ and Λ are called orbit equivalent (resp. stably orbit equivalent) if their orbit equivalence relations are isomorphic (resp. stably isomorphic). Finally, the actions are called von Neumann equivalent if the associated crossed product von Neumann algebras are isomorphic. 52

In the early 1980s, D. Ornstein and B. Weiss [OW80], extending work of H. Dye [Dy59], showed that any two ergodic pmp actions of countable inﬁnite amenable groups are orbit equivalent. Over the next two decades, several families of non-amenable countable groups, including property (T) groups [Hj02] and nonabelian free groups [GP03], were shown to admit uncountably many actions which are pairwise not orbit equivalent. Unifying many of these results, it was shown in [Io06] that any countable group Γ containing a copy of F2 has uncountably many free ergodic actions which are pairwise not orbit equivalent. In [Ep07], I. Epstein gave a new co-induction construction for group actions and combined it with the methods of [Io06, GL07] to show that any countable non-amenable group Γ admits uncountably many non orbit equivalent actions. In [Io06], this result was strengthened to such Γ admitting uncountably many actions which are pairwise not von Neumann equivalent. Here we combine Theorem 3A with the co-induction construction of [Ep07] and the methods of [Io06] to prove the following:

Theorem 3C. Let R be a non-amenable ergodic countable pmp equivalence relation on a standard probability space. Then R admits uncountably many ergodic extensions which are pairwise not stably von Neumann equivalent (hence, pairwise not stably isomorphic).

Remark 3.1.3. Theorem 3C implies the following dichotomy: any ergodic countable pmp equivalence relation has either only one or uncountably many ergodic extensions, up to isomorphism. Indeed, if R is an amenable countable pmp equivalence relation, then R is hyperﬁnite by [CFW81]. As a consequence, any two ergodic extensions of R are hyperﬁnite, and thus isomorphic by [Dy59].

Remark 3.1.4. Let Γ be a countable non-amenable group. If R is the orbit equivalence relation of some free pmp action of Γ, then any extension of R is the orbit equivalence relation of some other free pmp action of Γ. Theorem 3C implies that Γ admits uncountably many actions which are pairwise not stably von Neumann equivalent, thereby strengthening the results of [Io06, Ep07]. As we explain next, Theorem 3C also allows us to derive a new application to the orbit equivalence theory of locally compact group actions. 53

3.1.4 Actions of locally compact groups

As a consequence of [CFW81], all free properly ergodic pmp actions of a unimodular amenable lcsc group G are pairwise orbit equivalent. On the other hand, it was shown in [Zi84, Example 5.2.13] (see also [GG88, Corollary A. 10]) that any connected semisimple Lie group G with R-rank(G) ≥ 2, ﬁnite center, and no compact factors has uncountably many mutually non orbit equivalent free ergodic pmp actions. Moreover, by combining [Ep07] with an induction argument it follows that, more generally, any unimodular non-amenable lcsc group G possessing a lattice has uncountably many non orbit equivalent free ergodic pmp actions. However, in spite of these advances, the situation for general non-amenable unimodular lcsc groups G remained unclear. Theorem 3C allows us to settle this question by showing that any such G has uncountably many non orbit equivalent actions. Moreover, we prove:

Corollary 3D. Let G be a unimodular non-amenable lcsc group. Then G admits uncountably many free ergodic pmp actions on standard probability spaces which are pairwise not von Neumann equivalent (hence, pairwise not orbit equivalent).

Our approach to deducing this result from Theorem 3C is based on the notion of cross section equivalence relations, and inspired by [KPV13]. Specifically, we rely on the following elementary observation: if R is a cross section equivalence relation of some free ergodic pmp action of a unimodular lcsc group G, then any ergodic extension of R can be realized as a cross section equivalence relation of some other free ergodic pmp action of G (see Proposition 3.8.2). This observation turns out to also be useful in a different context. Very recently, M. Gheysens and N. Monod introduced a measure-theoretic analogue of closed subgroup embeddings for locally compact groups, called tychomorphism [GM15, Definition 14]. Using this notion, they formulated and proved a generalization of the Gaboriau-Lyons theorem for lcsc groups G: if G is non-amenable, then there is a tychomorphism from F2 to G (see [GM15, Theorem B]). Combining Theorem 3A and Proposition 3.8.2 leads to a proof of this result which bypasses 54 the usage of the structure theory of locally compact groups as in [GM15] (see Subsection 3.8.2).

3.1.5 Outline of the proof of Theorem 3A

We end the introduction by outlining the proof of the main assertion of Theorem 3A. This relies on an extension of techniques from [GL07]. To fix notation, let R be an ergodic non-amenable countable pmp equivalence relation on a ˜ ˜ probability space (X, µ). Let (K, κ) = ([0, 1], λ), and put X := XK and R := RK . ˜ Our goal is to show that R contains the orbits of a free ergodic pmp action of F2. To this end, denote by u :[R] → U(L2(R, m)) the canonical representation of the full group [R]. Since R is ergodic and non-amenable, after replacing R with a subequivalence relation, we may assume that R is generated by finitely many Pn −1 automorphisms θ1, ..., θn ∈ [R] such that the operator T = i=1(u(θi) + u(θi )) satisfies kT k < 2n. Moreover, after replacing the set S = {θ1, ..., θn} with a power k S = {θi1 ...θik |1 6 i1, ..., ik 6 n}, for large enough k, we may assume that kT k 6 n.

For x ∈ X, we denote by Gx = ([x]R,Ex) the graph on [x]R associated to ˜ the graphing {θ1, ..., θn}. Then we can identify the X with the set of pairs (x, ω), with x ∈ X and ω ∈ [0, 1]Ex , such that R˜ is identiﬁed with the equivalence relation given by: ((x, ω), (y, ξ)) ∈ R˜ iﬀ (x, y) ∈ R and ω = ξ.

Ex Ex For p ∈ [0, 1] and x ∈ X, we denote by πp : [0, 1] → {0, 1} the map

Ex πp(ω) = (1[0,p](ωe))e, and view πp(ω) ∈ {0, 1} as a subgraph of Gx, for every

Ex ω = (ωe)e ∈ {0, 1} . In the first part of the proof, we use results from percolation theory, notably 1 1 [NS81] and [BS96], to show that if p is in the interval 2n−kT k+1 , kT k , then the graph πp(ω) has infinitely many infinite clusters (i.e. connected components), for almost every (x, ω) ∈ X˜. In the second part of the proof, we consider the cluster equivalence relation ˜ ˜ ˜ ˜ Rcl on X given by: ((x, ω), (y, ξ)) ∈ Rcl iff ((x, ω), (y, ξ)) ∈ R and x, y belong to the same cluster of πp(ω) = πp(ξ)[Ga05]. By combining the first part of the proof ˜ with results from [LS99, AL06] and [Ga99] we conclude that the restriction Rcl to its infinite locus is ergodic and has normalized cost > 1. 55

˜ ˜ Finally, since Rcl ⊂ R, a combination of results from [Hj06] and [KM04, ˜ Pi05] implies that R contains the orbits of a free ergodic pmp action of F2.

3.1.6 Organization

Besides the introduction, this paper has seven other sections. In Section 3.2, we collect several facts about equivalence relation. In particular, we prove that we may assume kT k 6 n and show that the isoperimetric constant of the graph Gx satisﬁes ι(Gx) > 2n−kT k. Section 3.3 contains various general results on Bernoulli extensions of equivalence relations. Sections 3.4 and 3.5 are devoted to the ﬁrst and second part of the proof of the main assertion of Theorem 3A described above. In Section 3.6, we complete the proof of Theorem 3A and deduce Corollary 3B. Finally, in Sections 3.7 and 3.8, we present the proofs of Theorem 3C and Corollary 3D, respectively.

Acknowledgements

We are grateful to Nicolas Monod for helpful comments.

3.2 Preliminaries

In this section we recall several general notions and results regarding equivalence relations.

3.2.1 Amenable equivalence relations

Recall from Section 1.2.6, that a countable pmp equivalence relation R on (X, µ) is called amenable if there exists a state Φ : L∞(R) → C such that ∗ ∞ R Φ(u(θ)fu(θ) ) = Φ(f), for all f ∈ L (R), θ ∈ [R], and Φ(a) = X a dµ, for all a ∈ L∞(X)[CFW81, Deﬁnition 6]. Next, we record the well-known fact that an ergodic equivalence relation R is non-amenable if and only if the unitary representation u :[R] → U(L2(R, m)) has spectral gap: 56

Lemma 3.2.1. Let R be a non-amenable ergodic countable pmp equivalence relation. 1 Pn Then we can ﬁnd n ≥ 1 and θ1, ..., θn ∈ [R] such that k n i=1 u(θi)k <

1. Moreover, if c > 0, then we can ﬁnd n ≥ 1 and θ1, ..., θn ∈ [R] such that 1 Pn k n i=1 u(θi)k < c.

1 Pn Proof. Assume by contradiction that k n i=1 u(θi)k = 1, for all θ1, ..., θn ∈ [R]. Then by arguing as in the proof of [Ha83, Lemma 2.2] it follows that there exists a state Φ : L∞(R) → C such that Φ(u(θ)fu(θ)∗) = Φ(f), for all f ∈ L∞(R) and every θ ∈ [R]. Since R is ergodic, any such state Φ also satisﬁes that Φ(g) = R ∞ X g dµ, for all g ∈ L (X) (see e.g. the proof of [HV12, Lemma 4.2]). This implies that R is amenable, which is a contradiction. 1 Pn For the moreover assertion, let θ1, ..., θn ∈ [R] with δ := k n i=1 u(θi)k < 1. Let c > 0 and choose m ≥ 1 such that δm < c. Then we have that

n !m 1 X 1 X u(θ ...θ ) = u(θ ) ≤ δm < c. nm i1 im n i 1≤i1,...,im≤n i=1

Thus, the set {θi1 ...θim |1 ≤ i1, ..., im ≤ n} ⊂ [R] satisﬁes the second assertion of the lemma.

3.2.2 Extensions and expansions of equivalence relations

Let R and R˜ be countable pmp equivalence relations on probability space (X, µ) and on (X,˜ µ˜), respectively.

Deﬁnition 3. We say that R˜ is a class-bijective extension (in short, an extension) of R if there is a Borel map p : X˜ → X satisfying

1. µ(E) =µ ˜(p−1(E)), for all Borel E ⊂ X,

2. p| is injective, for almost every x ∈ X˜, and [x]R˜ ˜ 3. p([x]R˜ ) = [p(x)]R, for almost every x ∈ X.

We say that R˜ is an expansion of R if condition (3) is weakened to 57

˜ (3’) p([x]R˜ ) ⊃ [p(x)]R for almost every x ∈ X.

Notation. Below we use the notation R˜ → R to mean that R˜ is an extension of R.

Remark 3.2.2. Assume that R˜ is an extension of R and let S ≤ R be a subequivalence relation. Then S˜ := {(x, y) ∈ R|˜ (p(x), p(y)) ∈ S} is an extension of S, which we call the lift of S to R˜.

Remark 3.2.3. Assume that R˜ is an expansion of R. Then R˜ contains an exten- ˜ ˜ ˜ ˜ sion R0 ≤ R of R deﬁned by R0 = {(x, y) ∈ R|(p(x), p(y)) ∈ R}. Note, however, that containing an extension of R is not equivalent to being an expansion of R.

Suppose that R˜ is an expansion of R and let p : X˜ → X as in the above ˜ −1 deﬁnition. If θ ∈ [R], then for almost every x ∈ X, the set p (θ(p(x))) ∩ [x]R˜ contains exactly one point x0 ∈ X˜. We may therefore deﬁne θ˜ ∈ [R˜] by θ˜(x) = x0. Note that θ◦p = p◦θ˜, for all θ ∈ [R]. One can check that θ 7→ θ˜ is a homomorphism from [R] into [R˜]. For a ∈ L∞(X), we leta ˜ = a ◦ p ∈ L∞(X˜). The next result is due to S. Popa [Po05, Proposition 1.4.3]. For completeness, we include a proof.

Lemma 3.2.4. [Po05] There is a trace preserving ∗-homomorphism π : L(R) → L(R˜) satisfying

1. π(a) =a ˜, for every a ∈ L∞(X),

2. π(u(θ)) = u(θ˜), for every θ ∈ [R], and

3. π(L∞(X))0 ∩ L(R˜) = L∞(X˜).

Moreover, if R˜ is an extension of R, then the linear span of {bu(θ˜)|b ∈ L∞(X˜), θ ∈ [R]} is dense in L(R˜), in the strong operator topology.

Proof. We denote by τ and h., .i the canonical trace and inner product on L(R) ˜ (resp. L(R)) and by EL∞(X) (resp. EL∞(X˜)) the conditional expectations onto 58

L∞(X) (resp. L∞(X˜)). Note ﬁrst that the map π : L∞(X) → L∞(X˜) given by π(a) =a ˜ deﬁnes a trace preserving ∗-homomorphism. Moreover, if θ ∈ [R], then

π(EL∞(X)(u(θ))) = 1{x∈X|θ(x)=x} ◦ p = 1{x∈X˜|θ(p(x))=p(x)} = 1{x∈X˜:θ˜(x)=x} ˜ = EL∞(X˜)(u(θ)).

P Let D ⊂ L(R) be the ∗-subalgebra consisting of ﬁnite sums of the form θ aθu(θ). Then D ⊂ L(R) is dense in the strong operator topology, and for every a, b ∈ L∞(X), θ, ρ ∈ [R], we have

˜ ∗ ˜ −1 ∗ −1 hπ(a)u(θ), π(b)u(˜ρ)i = τ(π(b a)u(θρ˜ )) = τ(π(b a)EL∞(X˜)(u(θρg ))) ∗ −1 ∗ −1 = τ(π(b a)π(EL∞(X)(u(θρ )))) = τ(b aEL∞(X)(u(θρ ))) = hau(θ), bu(ρ)i.

˜ P P ˜ Therefore, the map π : D → L(R) deﬁned by θ aθu(θ) 7→ θ a˜θu(θ) is well- deﬁned and trace-preserving. Moreover, since π(u(θ)au(θ)∗) = π(a ◦ θ−1) = π(a) ◦ θ˜−1 = u(θ˜)au(θ˜)∗ and the maps a 7→ a˜ and u(θ) 7→ u(θ˜) are ∗-homomorphisms, π is a ∗-homomorphism. Since π is trace-preserving, it extends to a trace-preserving ∗-homomorphism π : L(R) → L(R˜) satisfying (1) and (2). ∞ 0 ˜ ˜ To prove (3), let y ∈ π(L (X)) ∩ L(R). Fix θ ∈ [R] and set bθ = ∗ ∞ EL∞(X˜)(yu(θ) ). Then for any a ∈ π(L (X)) we have

∗ ∗ ∗ bθa = abθ = EL∞(X˜)(ayu(θ) ) = EL∞(X˜)(yau(θ) ) = bθ(u(θ)au(θ) ).

Thus, for almost every x ∈ supp(b ) ⊂ X˜, we have p(x) = p(θ−1(x)). Since p| θ [x]R˜ −1 is injective, we derive that x = θ (x), for almost every x ∈ supp(bθ). Hence, ∞ ˜ ˜ ∗ for any b ∈ L (X) and almost every x ∈ X, we have bθ(x)[u(θ)bu(θ) ](x) = −1 ∞ ˜ 0 ˜ bθ(x)b(θ x) = bθ(x)b(x). Therefore, we get that bθu(θ) ∈ L (X) ∩ L(R) = L∞(X˜). Since this holds for any θ ∈ [R˜], we conclude that y ∈ L∞(X˜). Finally, assume that R˜ is an extension of R, and let θ ∈ [R˜]. Then for ˜ ˜ S almost every x ∈ X we have (p(x), p(θ(x))) ∈ R. Hence X = ρ∈[R]{x ∈ ˜ S ˜ X|p(θ(x)) = ρ(p(x))} = ρ∈[R]{x ∈ X|θ(x) =ρ ˜(x)}. Thus, we can write uθ = P∞ ∞ ˜ P∞ n=1 znu(˜ρn), for some {ρn} ⊂ [R] and projections {zn} ⊂ L (X) with n=1 zn = 1. In particular, this gives the moreover conclusion. 59

3.2.3 Graphed equivalence relations and isoperimetric constants

Let R be a countable pmp equivalence relation on a probability space (X, µ).

A graphing of R is an at most countable family {θi}i≥1 ⊂ [[R]]. A graphing {θi}i≥1 is generating if R is the smallest equivalence relation which contains the graph of

θi for all i ≥ 1.

Any graphing {θi : Ai → Bi}i≥1 gives rise to a graph structure on R (see [Ga99]). More precisely, for x ∈ X, we deﬁne an unoriented (multi-)graph

Gx = ([x]R,Ex) whose vertex set is the equivalence class [x]R and whose edge set

Ex consists of the pairs (y, θi(y)), for every i ≥ 1 and y ∈ [x]R ∩ Ai. Note that we allow multiple edges between two given vertices. Therefore, if the graphing is n finite and given by {θi}i=1, with θ1, ..., θn ∈ [R], then Gx is a 2n-regular graph. Let G = (V,E) be an unoriented infinite (multi-)graph with vertex set V and edge set E. Given a non-empty finite set F ⊂ V , let ∂EF be the set of edges which have exactly one endpoint in F . The edge-isoperimetric constant of G is defined as |∂ F | ι(G) = inf E ∅= 6 F ⊂ V finite subset . |F |

If R is an amenable countable pmp equivalence relation, then ι(Gx) = 0, for n almost every x ∈ X, for any finite graphing {θi}i=1 (see [Ka97, Theorem 2]). On the other hand, the converse is false. More precisely, [Ka97, Section 3] provides an example of a non-amenable equivalence relation R which admits a finite generating n graphing {θi}i=1 such that ι(Gx) = 0, for almost every x ∈ X. Nevertheless, the combination of Lemma 3.2.1 and Lemma 3.2.5 below n shows that if R is non-amenable and ergodic, then we can find a graphing {θi}i=1 with θ1, ..., θn ∈ [R] such that the associated graphs satisfy ι(Gx) > 0, for almost every x ∈ X.

Lemma 3.2.5. Let R be a countable pmp equivalence relation on a probability space (X, µ) and θ1, ..., θn ∈ [R]. For every x ∈ X, consider the unoriented graph

Gx = ([x]R,Ex) deﬁned as above. Pn −1 Then ι(Gx) ≥ 2n − k i=1(u(θi) + u(θi ))k, for almost every x ∈ X. 60

Pn −1 Proof. Denote δ = 2n − k i=1(u(θi) + u(θi ))k. Let S be the set of y ∈ X such that ι(Gy) < δ. Assume by contradiction that µ(S) > 0. For all y ∈ S we can ﬁnd a ﬁnite set Ay ⊂ [y]R satisfying |∂Ey (Ay)| < δ|Ay| in such a way that the set A := {(x, y) ∈ R|y ∈ S, x ∈ Ay} is Borel. Moreover, after replacing S with a non-null Borel subset, we may assume that supy∈S |Ay| < ∞. 2 R −1 If we view 1A ∈ L (R, m), then hu(θ)(1A), 1Ai = S |{x ∈ Ay|θ (x) ∈

Ay}| dµ(y), for all θ ∈ [R]. By using this identity we derive that

* n ! + X −1 (u(θi) + u(θi )) (1A), 1A i=1 Z X −1 = |{1 ≤ i ≤ n|θi (x) ∈ Ay}| + |{1 ≤ i ≤ n|θi(x) ∈ Ay}| dµ(y) S x∈Ay Z Z = 2n|Ay| − |∂Ey (Ay)| dµ(y) > (2n − δ) |Ay| dµ(y) S S

= (2n − δ) m(A) = (2n − δ)h1A, 1Ai.

Pn −1 This contradicts the fact that k i=1(u(θi) + u(θi ))k = 2n − δ.

3.2.4 Cost of equivalence relations

Let R be a countable pmp equivalence relation on a probability space (X, µ).

The cost of a graphing {θi : Ai → Bi}i≥1 is the sum of the measures of the domains: P i≥1 µ(Ai). The cost of R is defined as the infimum of the cost of all generating graphings of R [Ga99, Defintion I.5].

Let A ⊂ X be a Borel set of positive measure and denote by R|A := R∩(A×

A) the restriction of R to A. Then the normalized cost of R|A is deﬁned as the cost of R|A with respect to the probability measure on A given by µA(B) = µ(B)/µ(A), for any Borel set B ⊂ A. In the proof of our main result we will use the following theorem.

Theorem 3.2.6. Assume that R is ergodic and has cost in (1, ∞). Then there exists a free ergodic pmp action F2 y (X, µ) such that R(F2 y X) ⊂ R, almost everywhere. 61

This theorem is the combination of Propositions 13 and 14 from [GL07]. Its proof relies on a theorem due to G. Hjorth [Hj06] and on a result from [KM04, Pi05] (see [GL07] for details).

3.3 Bernoulli extensions of equivalence relations

In this section, we ﬁrst recall the construction of Bernoulli extensions and prove that Bernoulli extensions preserve ergodicity. We then study isomorphisms of Bernoulli extensions and their behavior with respect to restrictions to subequivalence relations and compressions.

3.3.1 Bernoulli extensions and ergodicity

Let R be a countable pmp equivalence relation on a probability space (X, µ). Let (K, κ) be a probability space.

[x]R We denote by XK the set of pairs (x, ω) with x ∈ X and ω ∈ K . We endow XK with the smallest σ-algebra of sets which makes the maps (x, ω) 7→ x and (x, ω) 7→ ω(θ(x)) measurable, for every θ ∈ [R]. We also endow XK with the probability measure µκ given by

[x]R dµκ(x, ω) = dκ (ω) dµ(x).

Lastly, we denote by RK the equivalence relation on XK given by (x, ω)RK (y, ξ) iﬀ xRy and ω = ξ, and call it the Bernoulli extension of R with base space (K, κ) (see [Bo12, Section 11]).

Lemma 3.3.1. If R is ergodic, then RK is ergodic.

Proof. Assume that R is ergodic. Then we can find θ ∈ [R] which acts ˜ ergodically on (X, µ) (see [Ke10, Theorem 3.5]). We define θ ∈ [RK ] by letting ˜ θ(x, ω) = (θ(x), ω), for (x, ω) ∈ XK . Then, in order to conclude that RK is ergodic, ˜ it suffices to prove that θ acts ergodically on XK . Let S ≤ R be the subequivalence relation generated by θ. Since θ and N hence S is ergodic, we can find {θi}i=1 ∈ [R] such that for almost every x ∈ X 62

N we have θi([x]S ) ∩ θj([x]S ) = ∅, for all i 6= j, and [x]R = ∪i=1θi([x]S ) (see [Io09, Lemma 1.1]). Here, N ∈ N ∪ {∞} is the index of S in R. {1,...,N}×Z Now deﬁne σ : X × K → XK by letting σ(x, (ki,j)i∈{1,...,N},j∈Z) =

[x]R j (x, ω), where ω ∈ K is given by ω(θiθ (x)) = ki,j, for all i ∈ {1, ..., N} and j ∈ Z. Further, we endow X ×K{1,...,N}×Z with the probability measure µ×κ{1,...,N}×Z. Then it is clear that σ is an isomorphism of probability spaces and that

−1 ˜ {1,...,N}×Z (σ ◦θ◦σ)(x, k) = (θ(x), (ki,j+1)), for x ∈ X, k = (ki,j)i∈{1,...,N},j∈Z ∈K .

Thus, θ˜ is conjugate to the product θ × τ between θ and the Bernoulli shift τ of Z on (K{1,...,N})Z. Since θ is ergodic and τ is weakly mixing, we conclude that ˜ θ is ergodic. Let us also note that if R is the orbit equivalence relation of some free action, then the Bernoulli extensions of R can be described explicitly.

Proposition 3.3.2. Assume that R = R(Γ y X), for some essentially free pmp action Γ y (X, µ). Γ Then RK is isomorphic to R(Γ y X × K ).

Γ Proof. Let θ : XK → X × K be given by θ(x, ω) = (x, η), where η(g) = ω(g−1x), for every g ∈ Γ. It is immediate to see that θ is an isomorphism of probability spaces which implements an isomorphism between RK and R(Γ y Γ X × K ).

3.3.2 Isomorphisms of Bernoulli extensions

Let R be a countable pmp equivalence relation on a probability space (X, µ). Next, we study the isomorphism problem for Bernoulli extensions. For this we need the following deﬁnition:

Deﬁnition 4 (Isomorphism of extensions). Let R˜, S˜ be countable pmp equivalence ˜ ˜ ˜ relations on probability spaces (X, µe) and (Y, νe). Suppose that π : X → X, φ : Y˜ → X are Borel maps which give extensions of R˜ and S˜ over R, respectively. We say that the extensions R˜ → R and S˜ → R are isomorphic if there is an isomorphism ψ : X˜ → Y˜ of R˜ with S˜ such that π = φ ◦ ψ. 63

Let (K, κ) be a standard probability space. If (K, κ) is purely atomic then we deﬁne its Shannon entropy by X H(K, κ) := −µ({k}) log(µ({k})). k∈K By convention 0 · log(0) = 0. Otherwise, we set H(K, κ) := +∞.

Theorem 3.3.3. Let (K, κ), (L, λ) be probability spaces with the same Shannon entropy. Then the corresponding Bernoulli extensions of R are isomorphic.

Proof. By Ornstein’s Isomorphism Theorem [Or70a, Or70b] the Bernoulli shifts Z y (K, κ)Z and Z y (L, λ)Z are isomorphic. Let Φ : KZ → LZ be such an isomorphism. Fix an ergodic element θ ∈ [R] (see [Ke10, Theorem 3.5] for the proof of

[x]R y Z existence). Let x ∈ X and ω ∈ K . For y ∈ [x]R, let ω ∈ K be the map given by ωy(n) = ω(θny).

0 Also, deﬁne ω :[x]R → L by

0 y ω (y) = Φ(ω )0.

0 y That is, ω is the time 0 coordinate of Φ(ω ). Finally, deﬁne Ψ : XK → XL by

Ψ(x, ω) = (x, ω0).

It is an exercise to check that Ψ gives the desired isomorphism.

Deﬁnition 5. Theorem 3.3.3 allows us to deﬁne the Bernoulli extension of R with base entropy t ∈ (0, ∞] to be any Bernoulli extension of R with base space (K, κ) satisfying H(K, κ) = t.

3.3.3 Bernoulli extensions restricted to subequivalence relations

Theorem 3.3.4. Let R be a countable ergodic pmp equivalence relation on a probability space (X, µ). Let (K, κ) be a probability space and RK → R the corresponding 64

Bernoulli extension of R. Let S ≤ R be an ergodic subequivalence relation and ˜ S ≤ RK the lift of S to RK . Then the extension S˜ → S is isomorphic to the Bernoulli extension of S with base space entropy equal to H(K, κ)[R : S].

N Proof. Let N = [R : S]. Since S is ergodic, we can ﬁnd {θi}i=1 ∈ [R] such that for almost every x ∈ X we have θi([x]S ) ∩ θj([x]S ) = ∅, for all i 6= j, and N [x]R = ∪i=1θi([x]S ) (see [Io09, Lemma 1.1]). N Let (L, λ) = (K, κ) . We denote by (YL, νλ) the underlying space of the

Bernoulli extension SL the Bernoulli extension of S with base space (L, λ). Specif-

0 0 [x]S ically, YL = {(x, ω )|x ∈ X, ω ∈ L }. 0 Deﬁne the isomorphism Φ : XK → YL by letting Φ(x, ω) = (x, ω ), where 0 0 ω :[x]S → L is deﬁned by ω (y)n = ω(θn(y)). It is straightforward to verify ˜ that Φ is an isomorphism between the extensions S → S and SL → S. Since H(L, λ) = NH(K, κ) = [R : S] H(K, κ), the conclusion follows.

3.3.4 Compressions of Bernoulli extensions

Theorem 3.3.5. Let R be a countable ergodic pmp equivalence relation on a probability space (X, µ). Let (K, κ) be a probability space and RK → R the corresponding ˜ Bernoulli extension of R. Let Y ⊂ X be a non-null Borel set and let Y ⊂ XK be the corresponding lift.

Then the extension RK |Y˜ → R|Y is isomorphic to the Bernoulli extension of R|Y with base space entropy equal to H(K, κ)/µ(Y ).

To prove this we ﬁrst study the classiﬁcation of inhomogeneous Bernoulli shifts. Let T ∈ Aut(X, µ) be an automorphism of a probability space. Let ∆ be a complete metric space and Prob(∆) the set of probability measures on ∆ endowed with the weak* topology. Suppose that φ : X → Prob(∆) is a Borel map. For

Z x ∈ X, let κx be the probability measure on ∆ obtained as the direct product of the measures φ(T nx)(n ∈ Z). Z Deﬁne the measure µe on X × ∆ by

dµe(x, ω) = dκx(ω) dµ(x). 65

Z We let σ denote the shift map from ∆ to itself given by σ(ω)n = ωn+1. Then µe is T × σ-invariant. The automorphism T × σ is called the inhomogeneous Bernoulli shift over T with data φ.

Lemma 3.3.6. The inhomogeneous Bernoulli shift deﬁned above is measurably conjugate to a direct product T × U, where U is a Bernoulli shift with h(U) = R H(φ(x)) dµ(x). Moreover, the conjugacy can be chosen to be the identity on the X coordinate.

Proof. This is a straightforward consequence of Thouvenot’s Relative Isomorphism Theorem [Th75]. We provide some details here, guided by the formulation of Thouvenot’s Theorem presented in [Ki84].

Let (B, ρ) be a complete separable metric space. If ν1, ν2 are probability measures on Bm (for some integer m > 0), then we deﬁne the d¯-distance between them by: Z m ¯ −1 X d(ν1, ν2) = inf m ρ(xi, yi) dJ(x, y) J i=1 where the inﬁmum is over all probability measures J on Bm × Bm with marginals

ν1 and ν2. Let (Y, C, ν) be a standard probability space and S : Y → Y a measure- preserving automorphism. Also F ⊂ C be an S-invariant sub-sigma-algeba and m m ψ : Y → B a measurable map. Let ψ1 : Y → B be the map

m m ψ1 (y) = (ψ(Sy), . . . , ψ(S y)).

0 N Also let ψ−∞ : Y → B be the map

0 −1 −2 ψ−∞(y) = (ψ(y), ψ(S y), ψ(S y),...).

Then (S, ψ, ν) is F-conditionally very weak Bernoulli (VWB) if whenever y ∈ Y is random with Law(y) = ν then

¯ m m 0 lim [d(Law(ψ1 (y)|F), Law(ψ1 (y)|F, ψ−∞)] = 0. m→∞ E

m A word about this expression is in order. Law(ψ1 (y)) is just the pushforward m m m measure (ψ1 )∗µ (since y has law µ). Law(ψ1 (y)|F) is the distribution of ψ1 (y) 66

m conditioned on F. In other words, it is the conditional expectation of (ψ1 )∗µ m 0 relative to F. Similarly, Law(ψ1 (y)|F, ψ−∞) is the conditional expectation of m 0 (ψ1 )∗µ relative to F and the sigma-algebra generated by ψ−∞. The expected value in the expression above is over y. In the special case in which B is a ﬁnite set, Thouvenot proved that if (S, ψ, ν) is F-conditionally VWB then (S, ψ, ν) is F-relatively Bernoulli. The latter means: if G is the sigma-algebra generated by the functions {y 7→ ψ(Sny): n ∈ Z} then:

•F is independent of G (so for any A ∈ F,A0 ∈ G, µ(A ∩ A0) = µ(A)µ(A0)),

• the factor corresponding to G is isomorphic to a Bernoulli shift. The entropy rate of this factor is necessarily equal to h(S, G) = h(S, G ∨ F|F).

It is straightforward to generalize the proof to the case in which B is an arbitrary complete metric space. Alternatively, one can use the fact that inverse limits of Bernoulli shifts are Bernoulli [Or74]. Now let T ∈ Aut(X, µ), φ : X → Prob(∆) andµ ˜ be as before this lemma. We set Y = X × ∆Z, S = T × σ, ν =µ ˜ and let ψ : X × ∆Z → ∆ be the map

ψ(x, y) = y0 (where y = (yi)i∈Z). Also let F be the sigma-algebra generated by projection to the X-coordinate. It is straightforward to check that (S, ψ, ν) is F-conditionally VWB. Indeed, in this case,

m m 0 Law(ψ1 (y)|F) = Law(ψ1 (y)|F, ψ−∞).

This implies the lemma except for the entropy assertion. To see that, observe that Z h(σ|F) = H(φ(x)) dµ(x) = h(S|F).

Since measure-conjugacies that preserve a sigma-algebra preserves entropy relative to that sigma-algebra, it follows that h(U) must equal R H(φ(x)) dµ(x).

Next, we extend the previous result to equivalence relations. Let R be a countable ergodic pmp equivalence relation on (X, µ). Let φ : X → Prob(∆) be a

[x]R Borel map. For x ∈ X, let κx denote the probability measure on ∆ given by:

κx is the direct product of φ(y) over all y ∈ [x]R. 67

[x]R Let Xφ be the set of all pairs (x, ω) with x ∈ X and ω ∈ ∆ . We endow

Xφ with the smallest σ-algebra of sets which makes the maps (x, ω) 7→ x and

(x, ω) 7→ ω(θ(x)) measurable, for every θ ∈ [R]. Also we endow Xφ with the probability measure µφ deﬁned by

µφ(x, ω) = dκx(ω) dµ(x).

0 0 Let Rφ be the equivalence relation on R given by (x, ω)Rφ(x , ω ) if and only if 0 0 xRx and ω = ω . We call Rφ the inhomogeneous Bernoulli extension over R with data φ.

Lemma 3.3.7. The inhomogeneous Bernoulli extension Rφ → R is isomorphic to the Bernoulli extension of R with base space entropy equal to R H(φ(x)) dµ(x).

Proof. Let (K, κ) be a probability space with entropy equal to R H(φ(x)) dµ(x). Let θ ∈ [R] be an ergodic element (see [Ke10, Theorem 3.5]). By Lemma 3.3.6, the inhomogeneous Bernoulli shift over θ with data φ is isomorphic to θ ×S, where S is the Bernoulli shift with base space (K, κ). Let Φ : X × ∆Z → X × KZ be such an isomorphism.

x Z Given x ∈ X and ω :[x]R → ∆, let ω ∈ ∆ be the map

ωx(n) = ω(θnx).

0 Also, deﬁne ω :[x]R → K by

0 y ω (y) = Φ(y, ω )0.

0 y That is, ω is the time 0 coordinate of Φ(y, ω ). Finally, deﬁne Ψ : Xφ → XK by

Ψ(x, ω) = (x, ω0).

It is an exercise to check that Ψ gives the desired isomorphism.

Proof of Theorem 3.3.5. Without loss of generality, we may assume K is a compact metrizable space. Let ∗ be an element not contained in K. Let K∗ = K ∪ {∗} be N S the disjoint union and ∆ = K∗ . For S ⊂ N, we identify the product space K with the set of sequences α = (α1, α2, ...) ∈ ∆ such that αi ∈ K, if i ∈ S, and 68

S αi = ∗, if i∈ / S. We also view the product measure κ as a measure on ∆ be letting κS(∆ \ KS) = 0.

Since R is ergodic, we can ﬁnd θ1, θ2, ... ∈ [[R]] such that dom(θi) ⊂ Y , for all i, θi(Y ) ∩ θj(Y ) = ∅, for all i 6= j, and ∪iθi(Y ) is co-null in X. Then P i µ(dom(θi)) = 1. Moreover, for almost every x ∈ X, we have that [x]R is the disjoint union of {θi(y)| y ∈ dom(θi)}, over all y ∈ [x]R ∩ Y .

For x ∈ Y , let S(x) be the set of all i with x ∈ dom(θi). Define φ : X → S(x) Prob(∆) by φ(x) = κ . Let (XK , µκ) be the underlying space of the Bernoulli ˜ extension RK . We denote by Y the lift of Y to XK . Let (Xφ, µφ) be the underlying space of the Bernoulli extension of R|Y by φ. ˜ 0 0 Define the isomorphism Φ : Y → Xφ by Φ(x, ω) = (x, ω ) where ω : N 0 0 [x]R|Y → ∆ = K∗ is defined by ω (y)n = ∗, if n∈ / S(y), and ω (y)n = ω(θn(y)), otherwise. It is straightforward to verify that Φ is an isomorphism between the extension RK |Y˜ → R|Y and the extension (R|Y )φ → R|Y .

Finally, Lemma 3.3.7 implies that the extension (R|Y )φ → R|Y is isomorphic to the Bernoulli extension of R|Y with base space entropy equal to Z ∞ −1 −1 X µ(Y ) H(φ(x)) dµ(x) = µ(Y ) H(K, κ) µ(dom(θi)) = H(K, κ)/µ(Y ), Y i=1 where the µ(Y )−1 terms appear because we have to renormalize the measure on Y .

3.4 Bernoulli percolation on graphed equivalence relations

This section is devoted to the ﬁrst part of the proof of main assertion of Theorem 3A. We start by recalling several concepts and results regarding Bernoulli percolation on graphs. 69

3.4.1 Bernoulli percolation on graphs

Let G = (V,E) be an infinite (multi-)graph with vertex set V and symmetric set of edges E. That is, we allow multiple edges between two given vertices. A connected component of G is called a cluster. We identify points in the standard Borel space {0, 1}E with subsets of the edge set E. This allows us to view {0, 1}E as the Borel space of all subgraphs of G with the same set of vertices V . A simple cycle in G is a cycle that does not use any vertex or edge more than once. A simple bi-infinite path in G is a bi-infinite path that does not use any vertex or edge more than once.

An infinite set of vertices V0 ⊂ V is end convergent if for every finite K ⊂ V , there is a connected component of G\K that contains all but finitely many vertices of V0. Two end-convergent sets V0,V1 are equivalent if V0 ∪ V1 is end-convergent. An end of G is an equivalence class of end-convergent sets. The Bernoulli(p) bond percolation on G is the process of independently keep- ing edges with probability p and deleting them with probability 1 − p. Concretely, E E we endow {0, 1} with the probability measure λp , where λp is the probability measure on {0, 1} with weights 1 − p and p. Since the event that ω ∈ {0, 1}E has an infinite cluster is a tail event, Kolmogorov’s 0-1 law implies that the probability

E E α(p) := λp ({ω ∈ {0, 1} | ω has an inﬁnite cluster}) is equal to 0 or 1.

The critical value pc(G) ∈ [0, 1] is deﬁned as pc(G) = sup{p ∈ [0, 1]| α(p) =

0}. It is easy to see that if p ≥ pc(G), then

E E pc(ω) = pc(G)/p, for λp -almost every ω ∈ {0, 1} . (3.1)

One also defines pu(G) as the infimum of the set of p ∈ [0, 1] such that ω E E has a unique infinite cluster, for λp -almost every ω ∈ {0, 1} . The following result due to I. Benjamini and O. Schramm (see [BS96, The- orem 4]) provides an upper bound for pc.

1 Theorem 3.4.1. If G = (V,E) is a graph then pc(G) ≤ ι(G)+1 . 70

E E For any subset A ⊂ {0, 1} and edge e ∈ E, we denote by ΠeA ⊂ {0, 1} the set {ω ∪ {e}|ω ∈ A}. We also denote by Π¬eA the set {ω \{e}|ω ∈ A}. The Bernoulli(p) percolation with p ∈ (0, 1] is insertion tolerant: if A ⊂ E E E {0, 1} is a Borel subset with λp (A) > 0, then λp (ΠeA) > 0, for any edge e ∈ E. If p ∈ [0, 1) then it is deletion tolerant: if A ⊂ {0, 1}E is a Borel subset with E E λp (A) > 0, then λp (Π¬eA) > 0, for any edge e ∈ E. Moreover, we have that E E E E λp (ΠeA) ≥ pλp (A) and λp (Π¬eA) ≥ (1 − p)λp (A). We end this subsection with two well-known consequences of insertion and deletion tolerance:

Lemma 3.4.2. Let G = (V,E) be a multi-graph and p ∈ (0, 1). Assume that E E ω has Np inﬁnite clusters, for λp -almost every ω ∈ {0, 1} , for some constant

Np ∈ N ∪ {∞}. Then we have

(1) If G is connected, then Np ∈ {0, 1, ∞}. E (2) If Np = 1, then the inﬁnite cluster of ω has one end, for λp -almost every ω ∈ {0, 1}E.

Proof. Part (1) is a direct consequence of insertion tolerance and is due to Newmann and Schulman (see [NS81] and the second part of the proof of[LP13, Theorem 7.6]). For part (2), we reproduce the argument given in the proof of [LP13, Theorem 7.33]. If ω has a unique infinite cluster for almost every ω ∈ {0, 1}E, then that cluster has one end. Otherwise, by removing a finite number of edges and using deletion tolerance, we would get that ω has at least two infinite clusters with positive probability.

3.4.2 Inﬁnitely many inﬁnite clusters

Before stating the main result of this section, we need to introduce some notation that we will use throughout this and the next section.

Notation. Let R be an ergodic countable pmp equivalence relation on a probability space (X, µ). Suppose that R is generated by ﬁnitely many automorphisms

θ1, ..., θn ∈ [R]. 71

• For x ∈ X, we deﬁne an unoriented connected (multi-)graph Gx = ([x]R,Ex)

whose edge set Ex consists of the pairs (y, θi(y)) with y ∈ [x]R and i ∈ {1, ..., n} (see Section 3.2.3).

• Fix p ∈ (0, 1) and endow {0, 1} with the probability measure λp with weights 1 − p and p.

• Let X˜ be the set of pairs (x, ω) with x ∈ X and ω ∈ {0, 1}Ex , and endow X˜

Ex with the probability measureµ ˜ given by dµ˜(x, ω) = dλp (ω)dµ(x).

• Let R˜ be the equivalence relation on X˜ given by ((x, ω), (y, ξ)) ∈ R˜ iﬀ (x, y) ∈ R and ω = ξ.

Let u :[R] → U(L2(R, m)) be the unitary representation defined in section Pn −1 ??. Consider the self-adjoint operator T = i=1(u(θi) + u(θi )) and note that kT k ≤ 2n. The main goal of this section is to show that if kT k ≤ n, then there is a non-trivial interval of p ∈ (0, 1) such that ω has infinitely many infinite clusters, for almost every (x, ω) ∈ X˜.

Ex Here, we view every ω ∈ {0, 1} as a subgraph of Gx. Recall that we allow

Gx (and therefore ω) to have multiple edges joining the same two points.

1 1 Theorem 3.4.3. In the setting from above, assume that (2n−kT k)+1 < p < kT k . Then ω has inﬁnitely many inﬁnite clusters, for µ˜-almost every (x, ω) ∈ X˜.

Remark 3.4.4. Let G be the Cayley graph of a countable group Γ with respect to a ﬁnite symmetric set of generators S. Let λ :Γ → U(`2Γ) be the left regular P representation of Γ. Put T = g∈S λ(g). I. Pak and T. Smirnova-Nagnibeda |S| showed that if kT k ≤ 2 , then pc(G) < pu(G) (see [PS-N00]). Theorem 3.4.3 is an analogue of their result for equivalence relations.

Towards Theorem 3.4.3, we ﬁrst prove three lemmas:

Lemma 3.4.5. R˜ is isomorphic to the Bernoulli extension of R with base space n n ({0, 1} , λp ). 72

Proof. For x ∈ X, the map βx :[x]R × {1, ..., n} → Ex given by βx(y, i) =

(y, θi(y)) is a bijection. Moreover, if [x]R = [y]R and we identify Ex and Ey in the ˜ natural way, then βx ≡ βy. It follows that R is indeed isomorphic to the Bernoulli n n shift over R with base space ({0, 1} , λp ). Lemma 3.4.6. For (x, ω) ∈ X˜, let N(x, ω) be the number of inﬁnite clusters of ω ∈ {0, 1}Ex .

Then there exists Np ∈ {0, 1, ∞} such that N(x, ω) = Np, for µ˜-almost every (x, ω) ∈ X˜.

Proof. Combining lemmas 3.3.1 and 3.4.5 yields that R˜ is ergodic. Since the ˜ ˜ measurable function N : X → N ∪ {∞} is R-invariant, we can ﬁnd Np ∈ N ∪ {∞} ˜ such that N(x, ω) = Np, for almost every (x, ω) ∈ X. Hence, we can ﬁnd x ∈ X

Ex Ex such that ω has Np inﬁnite clusters, for λp -almost every ω ∈ {0, 1} . Since Gx is connected, Lemma 3.4.2 (1) implies that Np ∈ {0, 1, ∞}.

1 Lemma 3.4.7. If (2n−kT k)+1 < p ≤ 1, then Np ∈ {1, ∞}. Proof. By combining Lemma 3.2.5 and Theorem 3.4.1 we get that 1 1 pc(Gx) ≤ ≤ < p, for µ-almost every x ∈ X. (3.2) ι(Gx) + 1 (2n − kT k) + 1 Therefore, for almost every x ∈ X, we have that ω has at least one inﬁnite

Ex Ex cluster, for λp -almost every ω ∈ {0, 1} . Thus, N(x, ω) ≥ 1, for almost every ˜ (x, ω) ∈ X. Together with Lemma 3.4.6 this gives that Np ∈ {1, ∞}. We are now ready to prove Theorem 3.4.3. The proof is an adaptation of an argument due to O. Schramm showing that pu(G) ≥ 1/γ(G), for any transitive 1/n graph G (see [LP13, Theorem 7.33]). Here, γ(G) := lim supn→∞ an(G) , where an(G) is the number of simple cycles of length n in G.

3.4.3 Proof of Theorem 3.4.3

By Lemma 3.4.7 we have that Np ∈ {1, ∞}. To show that Np = ∞, assume by contradiction that Np = 1. Thus, ω has a unique inﬁnite cluster, for almost every (x, ω) ∈ X˜. Denote by C(x, ω) this unique inﬁnite cluster. Lemma 3.4.2 (2) then implies that C(x, ω) has one end, for almost every (x, ω) ∈ X˜. 73

Let A be the set of (x, ω) ∈ X˜ such that ω (viewed again as a subgraph of

Gx = ([x]R,Ex)) contains an inﬁnite number of simple cycles through the vertex x. We continue with the following: Claim. µ˜(A) > 0.

Proof of the claim. By inequality 3.2 we have that pc(Gx) < p, for almost every x ∈ X. In combination with formula 3.1 we get that pc(ω) = pc(Gx)/p < 1, for almost every (x, ω) ∈ X˜. On the other hand, if a graph G of bounded degree does not contain a simple bi-inﬁnite path, then pc(G) = 1 (see [LPS06, Lemma 3.19]). Altogether, we deduce that ω contains a simple bi-inﬁnite path, for almost every (x, ω) ∈ X˜.

Recall that we view ω as a graph with vertex set [x]R. It follows that there is a measurable map θ : X → X such that for almost every x ∈ X, we have that

Ex θ(x) ∈ [x]R and that the set of ω ∈ {0, 1} for which there is a simple bi-infinite path in ω containing θ(x) has positive measure. Since µ(θ(X)) > 0 and Gθ(x) is ˜ naturally identified with Gx, the set B of (x, ω) ∈ X for which there exists a simple bi-infinite path in ω containing x must also have positive measure. Since ω has a unique infinite cluster, we derive that there is a simple bi- infinite path in C(x, ω) containing x, for almost every (x, ω) ∈ B. Now, we view such an infinite path as the union of two disjoint infinite simple paths starting at x. Since C(x, ω) has only one end, these two paths can by connected by paths in C(x, ω) that do not intersect any given finite subset of C(x, ω). This implies that there are an infinite number of simple cycles in C(x, ω) (and hence in ω) through x, for almost every (x, ω) ∈ B. We conclude thatµ ˜(A) ≥ µ˜(B) > 0, which proves the claim. −1 −1 Next, let m = 2n and enumerate {ψ1, ..., ψm} = {θ1, ..., θn, θ1 , ..., θn }.

Note that for every y ∈ [x]R there are exactly m edges having y as an endpoint, namely (y, ψi(y)), for i ∈ {1, ..., m}.

For k ≥ 1 and i1, ..., ik ∈ {1, ..., m}, we deﬁne Ai1,...,ik to be the set of x ∈ X such that ψik ...ψi2 ψi1 (x) = x and x 6= ψia ...ψi1 (x) 6= ψib ...ψi1 (x) 6= x, for all 1 ≤ a < b < k. In this case, x, ψi1 (x), ..., ψik ...ψi1 (x) is a simple cycle in Gx.

Conversely, any simple cycle in Gx containing x is of this form. Further, we deﬁne 74

˜ ˜ Ai1,...,ik to be the measurable set of (x, ω) ∈ X such that x ∈ Ai1,...,ik and the cycle x, ψi1 (x), ..., ψik ...ψi1 (x) belongs to ω. Then A consists of the points (x, ω) ∈ X˜ which belong to inﬁnitely many ˜ sets of the form Ai1,...,ik . Sinceµ ˜(A) > 0 by the claim, we derive that P∞ P µ˜(A˜ ) = ∞. Sinceµ ˜(A˜ ) = pkµ(A ) we conclude k=1 i1,...,ik∈{1,...,m} i1,...,ik i1,...,ik i1,...,ik that ∞   X k X p  µ(Ai1,...,ik ) = ∞. (3.3)

k=1 i1,...,ik∈{1,...,m} 2 Let ∆ = {(x, x)|x ∈ X} and view 1∆ ∈ L (R, m). Then hu(ψ)(1∆), 1∆i = Pm µ({x ∈ X|ψ(x) = x}), for every ψ ∈ [R]. Hence, since T = i=1 u(ψi), for every k ≥ 1 we have that

k X X hT (1∆), 1∆i = µ({(x ∈ X|ψik ...ψi1 (x) = x}) ≥ µ(Ai1,...,ik ).

i1,...,ik∈{1,...,m} i1,...,ik∈{1,...,m} (3.4) P∞ k k By combining equations 3.3 and 3.4 we deduce that k=1 p hT (1∆), 1∆i = ∞. This implies that pkT k ≥ 1 which leads to the desired contradiction.

3.5 Ergodicity of the cluster equivalence relation

This section is devoted to the second part of the proof of the main assertion of Theorem 3A. Consider the setting from 3.4.2. In particular, p ∈ (0, 1) is ﬁxed, and X˜ is the set of pairs (x, ω), with x ∈ X and ω ∈ {0, 1}Ex , endowed with the probability

Ex ˜ ˜ measure given by dµ˜(x, ω) = dλp (ω)dµ(x). Two points (x, ω), (y, ξ) ∈ X are R- equivalent if xRy and ω = ξ. Recall that we view every ω ∈ {0, 1}Ex as a subgraph of Gx = ([x]R,Ex). Following D. Gaboriau [Ga05, Section 1.2] we deﬁne a subequivalence rela- ˜ ˜ tion Rcl of R, called the cluster equivalence relation. Thus, we say that two points ˜ ˜ ˜ (x, ω), (y, ξ) ∈ X are Rcl-equivalent if they are R-equivalent and x, y belong to the same cluster of ω = ξ. For (x, ω) ∈ X˜, we let C(x, ω) be the cluster of x in ω. We denote by U ∞ the set of points (x, ω) ∈ X˜ such that C(x, ω) is inﬁnite. Then U ∞ is an 75

˜ ˜ Rcl-invariant set and the restriction Rcl|U ∞ has infinite classes. In this section we show that if ω has infinitely many infinite clusters, for ˜ ˜ almost every (x, ω) ∈ X, then Rcl|U ∞ is ergodic and has cost > 1.

Theorem 3.5.1. Assume that ω has inﬁnitely many inﬁnite clusters, for µ˜-almost every (x, ω) ∈ X˜. ˜ Then the restriction Rcl|U ∞ is ergodic.

R. Lyons and O. Schramm proved that the infinite clusters that may appear in Bernoulli(p) bond percolation on a transitive graph are indistinguishable (see [LS99, Theorem 1.1] for the precise statement). D. Gaboriau and R. Lyons then showed that indistinguishability of infinite clusters is equivalent to ergodicity of the restriction of the cluster equivalence relation to its infinite locus (see [GL07, Proposition 5]). Theorem 3.5.1 is a generalization of these results. Its proof is an immediate consequence of work of D. Aldous and R. Lyons [AL06] who noted that the results from [LS99] extend to the more general context of unimodular random networks. More precisely, we will show that the following result, stated implicitly in [AL06], implies Theorem 3.5.1.

Theorem 3.5.2. Let A be a Borel subset of the set {(A, x)|A ∈ {0, 1}[x]R ×

Ex {0, 1} , x ∈ X}. Assume that if (A, x) ∈ A and y ∈ [x]R, then (A, y) ∈ A. ˜ Then the set of (x, ω) ∈ X, for which there exist two inﬁnite clusters C1,C2 of ω such that ((C1, ω), x) ∈ A and ((C2, ω), x) ∈/ A, has µ˜-measure zero.

Before deducing Theorem 3.5.1 from Theorem 3.5.2, let us explain how the latter follows from [AL06]. Recall from [AL06, Section 2] that a network is a (multi- )graph G = (V,E) together with a complete separable metric space Ξ and maps from V and E to Ξ. A rooted network (G, o) is a network with a distinguished vertex o. Then G∗ denotes the set of isomorphism classes of rooted connected locally ﬁnite networks.

By [AL06, Example 9.9] the graphs (Gx)x∈X give rise to a unimodular random rooted network. More precisely, consider the map Φ : X → G∗ given by

Φ(x) = (Gx, x). Then the push-forward Φ∗µ is a unimodular probability measure on G∗ (see [AL06, Deﬁnition 2.1]). Moreover, the measureµ ˜ corresponds to 76

Bernoulli(p) percolation on Φ∗µ. Since p ∈ (0, 1], we have thatµ ˜ is insertion tolerant in the sense of [AL06, Definition 6.4]. Therefore, by [AL06, Theorem 6.15],µ ˜ has indistinguishable infinite clusters. Finally, translating this fact leads to Theorem 3.5.2. ∞ ˜ Proof of Theorem 3.5.1. Let Y ⊂ U be a Rcl-invariant Borel subset. We define A as the set of ((C, ω), x) with x ∈ X, ω ∈ {0, 1}Ex and C infinite cluster of ω such that (y, ω) ∈ Y , for all y ∈ C. Let x ∈ X, ω ∈ {0, 1}Ex and C infinite cluster of ω such that ((C, ω), x) 6∈ A. Then (y, ω) ∈/ Y , for some y ∈ C. But then for all z ∈ C we have that

(z, ω) ∼ ˜ (y, ω) and since Y is R˜ -invariant, we deduce that (z, ω) ∈/ Y . Rcl cl Since A is clearly invariant under changing the “root” x, Theorem 3.5.2 implies that for almost every (x, ω) ∈ X˜ we have that either (y, ω) ∈ Y , for all y contained in some inﬁnite cluster of ω, or (y, ω) ∈/ Y , for all y contained in some inﬁnite cluster of ω. ˜ This implies that Y is invariant under R|U ∞ . Since by lemmas 3.3.1 and 3.4.5 we have that R˜ is ergodic, it follows thatµ ˜(Y ) ∈ {0, µ˜(U ∞)}, which proves ˜ that Rcl|U ∞ is ergodic.

Proposition 3.5.3. Assume ω has inﬁnitely many inﬁnite clusters, for µ˜-almost ˜ ˜ every (x, ω) ∈ X. Then the normalized cost of Rcl|U ∞ is > 1.

Proof. We begin with the following claim: Claim. Each infinite cluster of ω has infinitely many ends, forµ ˜-almost every (x, ω) ∈ X˜. Proof of the claim. The proof is a straightforward adaptation of the proofs of Propositions 3.9 and 3.10 in [LS99]. By the discussion following [BS96, Conjecture 4.1] it is enough to show that no infinite cluster of ω has an isolated end. Assume that some cluster of ω has an isolated end, with positive probability. Then insertion tolerance guarantees that, with positive probability, a cluster of ω will have at least 3 ends with one of them being isolated. ˜ Let An be the set of (x, ω) ∈ X with the property that C(x, ω) \{y ∈ C(x, ω)| d(x, y) ≤ n} has at least 3 infinite components, where d is the cluster 77

metric. Our assumption implies that the set of (x, ω) ∈ An for which C(x, ω) has an isolated end, has positive probability, for some n ≥ 1.

If C(x, ω) ∩ An 6= ∅, then we let K(x, ω) be the set of y ∈ C(x, ω) ∩ An that are closest to x. Next, we letm ˜ be the usual inﬁnite measure of R˜ and deﬁne F : R˜ → [0, 1] by letting  −1 |K(x, ω)| if C(x, ω) ∩ An 6= ∅ and y ∈ K(x, ω) F ((x, ω), (y, ω)) = 0 otherwise

Since P F ((x, ω), (y, ω)) ∈ {0, 1}, for all (x, ω) ∈ X˜, we get that (y,ω)∈[(x,ω)]R˜ R R˜ F dm ˜ ≤ 1.

On the other hand, let (x, ω) ∈ An and η be an isolated end of C(x, ω). Then we can find B ⊂ C(x, ω) finite and a neighborhood D of η such that the points in C(x, ω) ∩ An that are closest to any given point y ∈ D lie in B. Thus, we have that K(y, ω) ⊂ B, for all (y, ω) ∈ [(x, ω)]R˜ with y ∈ D. In particular, |K(y, ω)| ≤ |B|, for all such y. Since D is infinite, it follows that P F ((y, ω), (z, ω)) = (y,ω),(z,ω)∈[(x,ω)]R˜ ,z∈B ∞. Since B is finite, we derive that P F ((y, ω), (z, ω)) = ∞, for some (y,ω)∈[(x,ω)]R˜ R z ∈ B. This clearly implies that R˜ F dm ˜ = ∞, which gives a contradiction. ∞ For i ∈ {1, ..., n}, let Ai be the set of (x, ω) ∈ U such that x and θi(x) lie ˜ ˜ ˜ in the same cluster of ω. We define θi ∈ [[Rcl]] by letting θi(x, ω) = (θi(x), ω), for ˜ n ˜ all (x, ω) ∈ Ai. Then {θi}i=1 is a generating graphing of Rcl|U ∞ . Moreover, for all ∞ ˜ (x, ω) ∈ U , the graph of the equivalence class of (x, ω) in Rcl|U ∞ associated to ˜ n {θi}i=1 is isomorphic to the cluster C(x, ω). By the claim, the latter has infinitely many ends, for almost every (x, ω) ∈ ∞ ˜ U . Since Rcl|U ∞ is ergodic by Theorem 3.5.1,[Ga99, Corollaire IV.24] gives that ˜ Rcl|U ∞ has normalized cost > 1. 78

3.6 Proofs of Theorem 3A and Corollary 3B

3.6.1 A generalization of Theorem 3A

The main goal of this section is to prove Theorem 3A. Let R be a non- amenable countable ergodic pmp equivalence relation on a probability space (X, µ). We would like to understand for which probability spaces (K, κ) there exist a free ergodic pmp action F2 y (XK , µκ) such that R(F2 y XK ) ≤ RK , almost everywhere. While we expect that this should be the case for any non-trivial (K, κ), at this point we only have partial answers. The next theorem, which clearly generalizes Theorem 3A, summarizes our main results. Recall the deﬁnition of the Shannon entropy H(K, κ) from §3.3.2.

Theorem 3.6.1. Let R be a non-amenable countable ergodic pmp equivalence relation on (X, µ).

1. There is a number β(R) ∈ [0, ∞] such that if H(K, κ) > β(R), then there

exists a free ergodic pmp action F2 y (XK , µκ) such that R(F2 y XK ) ≤

RK , almost everywhere. If H(K, κ) < β(R), then no such action exists.

2. β(R) is ﬁnite. In particular, if (K, κ) is non-atomic, then there exists a free

ergodic pmp action F2 y (XK , µκ) such that R(F2 y XK ) ≤ RK , almost everywhere.

3. For any ergodic non-amenable subequivalence relation S ≤ R, β(R) ≤ [R : S]−1β(S). In particular, if S has inﬁnite index, then β(R) = 0.

4. For any non-null Borel set Y ⊂ X, β(R) ≤ µ(Y )β(R|Y ). In particular, if R has inﬁnite fundamental group, then β(R) = 0.

5. If R contains a normal ergodic subequivalence relation S C R with inﬁnite index, then β(R) = 0.

3.6.2 Proof of Theorem 3.6.1

We begin by deﬁning β(R) and showing that it is ﬁnite. 79

Definition 6. We define β(R) ∈ [0, ∞] to be the infimum of all numbers of the form H(K, κ) where (K, κ) is a probability space satisfying: there exist a free ergodic pmp action F2 y (XK , µκ) such that R(F2 y XK ) ≤ RK , almost everywhere.

Proposition 3.6.2. If (L, λ) is any probability space with H(L, λ) > β(R), then there exists a free ergodic pmp action F2 y (XL, µλ) such that R(F2 y XL) ≤ RL, almost everywhere.

Proof. By hypothesis, there exists a probability space (K, κ) with H(K, κ) <

H(L, λ) and a free ergodic pmp action F2 y (XK , µκ) such that S := R(F2 y

XK ) ≤ RK , almost everywhere. Let (N, η) be a probability space such that H(N, η) = H(L, λ) − H(K, κ). The Shannon entropy of (N × K, η × κ) equals the

Shannon entropy of (L, λ). Theorem 3.3.3 implies that the extension RL → R is isomorphic to RN×K → R. The latter extension has RK as an intermediate factor.

Next, we lift the action F2 y XK to a free pmp action F2 y XN×K so that ˜ S := R(F2 y XN×K ) is the lift of S through the extension RN×K → RK . Since the extension RN×K → RK is isomorphic to the Bernoulli extension (RK )N → RK , Theorem 3.3.4 implies that the extension S˜ → S is isomorphic to a Bernoulli extension. Since S is ergodic, Lemma 3.3.1 implies that S˜ is ergodic, hence the ∼ action F2 y XN×K is ergodic. Since RL = RN×K by Theorem 3.3.3, we are done.

Next, we obtain a nontrivial upper bound on β. Deﬁne α(R) = log(n), where n ≥ 3 is the smallest natural number such that there exist θ1, . . . , θn ∈ [R] with n 1 X u(θ ) < 1/4. n i i=1 Proposition 3.6.3. There is a universal constant C > 0 such that

β(R) ≤ α(R) + C.

In particular, β(R) is ﬁnite. 80

Proof. By Lemma 3.2.1, non-amenability of R implies α(R) is ﬁnite. Let n ≥ 3 with log(n) = α(R). Let θ0 ∈ [R] be ergodic and θ1, . . . , θn ∈ [R] such that 1 Pn n i=1 u(θi) < 1/4. Then

n 1 X u(θ ) < 1/2 n + 1 i i=0 and the subequivalence relation R0 generated by θ0, . . . , θn is ergodic. Pn −1 Let T = i=0(u(θi) + u(θi )). Then kT k < n + 1. Let p = 1/(n + 2). Note that 1 1 < p < . (3.5) 2(n + 1) − kT k + 1 kT k

Consider the notation from 3.4.2, for the ergodic equivalence relation R0 and its generating graphing θ0, ..., θn (instead of R and θ1, ..., θn) and for the parameter p defined above. By inequality 3.5, Theorem 3.4.3 implies that ω has infinitely many infinite ˜ ˜ ˜ clusters, for almost every (x, ω) ∈ X. Let Rcl ⊂ R be the cluster equivalence relation and U ∞ ⊂ X˜ as defined in the beginning of Section 3.5. By combining ˜ ˜ Theorem 3.5.1 and Proposition 3.5.3 we conclude that Rcl ⊂ R|U ∞ is ergodic and ˜ ˜ has normalized cost > 1. Moreover, the cost of Rcl ⊂ R|U ∞ is clearly finite. By Lemma 3.4.5, R˜ is isomorphic to the Bernoulli extension with base space n+1 n+1 (K, κ) := ({0, 1} , λp ). In particular, since R is ergodic, Lemma 3.3.1 gives that R˜ is ergodic. Therefore, we can find an ergodic subequivalence relation S ⊂ R˜ ∞ ˜ whose restriction to U coincides with Rcl|U ∞ . Then the induction formula [Ga99, Proposition II.6 (2)] implies that S has cost in (1, ∞). By applying Theorem 3.2.6 to S, it follows that there exists a free ergodic ˜ ˜ pmp action F2 y (X, µ˜) such that S0 := R(F2 y X) ≤ S. In particular, S0 ≤ ˜ ∼ R = RK . Thus, we deduce that

β(R) ≤ H(K, κ) = −(n + 1)(p log(p) + (1 − p) log(1 − p)) log(n + 2) log(1 − 1/(n + 2)) = (n + 1) − (n + 1)2 n + 2 n + 2 ≤ log(n + 2) + 1 ≤ log(n) + C = α(R) + C, where C = 1 + log(5/3). 81

Proposition 3.6.4. If S ≤ R is an ergodic non-amenable subequivalence relation, then we have β(R) ≤ β(S)[R : S]−1.

Proof. Let (K, κ) be a probability space with H(K, κ) > β(S)[R : S]−1. By ˜ ˜ Theorem 3.3.4, if S is the lift of S to RK then S → S is isomorphic to the Bernoulli extension of S with base entropy H(K, κ)[R : S] > β(S). By the deﬁnition of β, ˜ there is a free ergodic pmp action F2 y (XK , µκ) whose orbits are contained in S. ˜ Since S ≤ RK , these orbits are also contained in RK . Therefore, β(R) ≤ H(K, κ), and the inequality follows by taking the inﬁmum over all such H(K, κ).

Proposition 3.6.5. Let Y ⊂X be a non-null Borel set. Then β(R) ≤ β(R|Y )µ(Y ).

Proof. Let (K, κ) be a probability space and suppose H(K, κ) > β(R|Y )µ(Y ). ˜ By Theorem 3.3.5, if Y is the lift of Y to XK , then RK |Y˜ → R|Y is isomorphic −1 to the Bernoulli extension of R|Y with base entropy H(K, κ)µ(Y ) > β(R|Y ). ˜ So by the deﬁnition of β, there is a free ergodic pmp action F2 y Y such that ˜ S = R(F2 y Y ) satisﬁes S ≤ RK |Y˜ , almost everywhere.

Since RK is ergodic by Lemma 3.3.1, we can ﬁnd an ergodic subequivalence equivalence relation T ≤ RK such that T |Y˜ = S. Then [Ga99, Theorem IV.15] and [Ga99, Proposition II.6 (2)] together imply that the cost of T belongs to (1, +∞).

Theorem 3.2.6 further implies that T and thus RK contains almost every orbit of a free ergodic pmp action F2 y XK . Therefore, β(R) ≤ H(K, κ), and the conclusion follows by taking the inﬁmum over all such H(K, κ).

Proposition 3.6.6. If R contains an ergodic normal subequivalence relation N C R such that R/N is non-amenable, then β(R) = 0.

Recall from [FSZ89] that there exists a countable group, denoted R/N , and a cocycle c : R → R/N , such that N is the kernel of c. Moreover, for any θ ∈ R/N there is an element θe ∈ [R] such that c(θx,e x) = θ for a.e. x. The element θe is called a lift of θ. These are all the facts we will need about normal subequivalence relations. We will prove Proposition 3.6.6 by lifting an appropriate set of elements from R/N and using the bound in Proposition 3.6.3. 82

Lemma 3.6.7. Let θ1, . . . , θn ∈ R/N and let θe1,..., θen ∈ [R] be lifts. Then

n n 1 X 1 X u(θei) ≤ λ(θi) n n i=1 i=1 where λ : R/N → U(`2(R/N )) is the left-regular representation.

2 2 Proof. Let ∆ = {(x, x)|x ∈ X} and view 1∆ ∈ L (R, m). Let δe ∈ ` (R/N ) denote the Dirac function at the identity e ∈ R/N . Then we have

* n ! + n 1 X 1 X 1 X u(θei) 1∆, 1∆ = µ({x ∈ X|θei(x) = x}) ≤ δθ ,e n n n i i=1 i=1 i=1 * n ! + 1 X = u(θ ) δ , δ . n i e e i=1 The conclusion follows immediately by combining this inequality with the following three facts:

−1 −1 • If θe1, θe2 ∈ [R] are lifts of θ1, θ2 ∈ R/N , then θe1 is a lift of θ1 , and θe1θe2 is

a lift of θ1θ2.

1 ∗ m 2m •k T k = lim (T T ) 1∆, 1∆ , for every T ∈ L(R). m→∞

1 ∗ m 2m • (T T ) δe, δe ≤ kT k, for every T ∈ L(R/N ) and all m ≥ 1.

Let F ≤ R be a ﬁnite subequivalence relation. We denote by X/F the quotient space and by R/F the quotient equivalence relation on X/F. More precisely, the elements of X/F are the F-classes of X. Let π : X → X/F be the natural projection map and endow X/F with the push forward measure µF := π∗µ.

Note that ([x]F , [y]F ) ∈ R/F if and only if xRy. We leave the proof of the following easy lemmas as exercises.

Lemma 3.6.8. If N C R is a normal subequivalence relation and F ≤ N is a ﬁnite subequivalence relation, then N /F is normal in R/F. Moreover R/N ∼= (R/F)/(N /F). 83

Lemma 3.6.9. There exists a Borel set Y ⊂ X such that every F-class contains ∼ exactly one element of Y . Moreover R|Y = R/F. If each F class contains exactly m ∈ N elements, then µ(Y ) = 1/m.

Proof of Proposition 3.6.6. By Kesten’s Theorem [Ke59] non-amenability of the group R/N implies the existence of elements θ1, . . . , θn ∈ R/N with n ≥ 3 such that n 1 X θ < 1/4. n i i=1 Let m > 1 be a natural number. Let F ≤ N be a ﬁnite subequivalence relation such that every F-class contains m elements. By Lemma 3.6.8, R/N ∼= 0 0 (R/F)/(N /F). So there exist elements θ1, . . . , θn ∈ (R/F)/(N /F) such that

n 1 X θ0 < 1/4. n i i=1 By Lemma 3.6.7 we get that α(R/F) ≤ log(n). Lemma 3.6.9 implies that

α(R|Ym ) ≤ log(n), where Ym ⊂ X is any Borel subset with µ(Ym) = 1/m. By Propositions 3.6.3 and 3.6.5,

β(R) ≤ β(R|Ym )/m ≤ log(n)/m + C/m where C > 0 is a universal constant. Taking m → ∞, we obtain β(R) = 0.

By collecting the above results, we are now ready to prove Theorem 3.6.1.

Proof of Theorem 3.6.1. Items (1-4) are proven in Propositions 3.6.2, 3.6.3, 3.6.4, 3.6.5 respectively. To prove item (5), suppose N ≤ R is ergodic and normal, and R/N is inﬁnite. If R/N is amenable, then since R is non-amenable, N must also be non-amenable. In this case, the conclusion follows from Proposition 3.6.4. On the other hand, if R/N is non-amenable, the conclusion follows from Proposition 3.6.6.

3.6.3 Proof of Corollary 3B

Let R be an ergodic countable pmp equivalence relation on a probability space (X, µ). Assume that R is non-amenable. By Theorem 3A, the Bernoulli 84 extension of R with non-atomic base contains almost every orbit of a free ergodic pmp action of F2. Conversely, assume that R admits an extension R˜ on a probability space ˜ ˜ (X, µ˜) which contains the orbit equivalence relation S = R(F2 y X) of a free ˜ ˜ ergodic pmp action F2 y (X, µ˜). Let p : X → X denote the quotient map. Assume by contradiction that R is amenable. By Connes-Feldman-Weiss’ theorem, R must be hyperfinite [CFW81]. So there is an increasing sequence {Rn}n≥1 of finite subequivalence relations of R with R = ∪n≥1Rn, almost everywhere. Then Sn = {(x, y) ∈ S|(p(x), p(y)) ∈ Rn} is a finite subequivalence relation of S such that S = ∪n≥1Sn, almost everywhere. In other words, S would be hyperfinite, which is a contradiction.

3.7 Uncountably many ergodic extensions of nonamenable R

The goal of this section is to prove Theorem 3C. To this end, we will make use of I. Epstein’s co-induction construction [Ep07].

3.7.1 Co-induced equivalence relation

Let Γ0 y β(X, µ) be a free ergodic pmp action and R an ergodic pmp equivalence relation on (X, µ) such that R0 = R(Γ0 y βX) ≤ R. Since R is ergodic, there is N0 ∈ Z>0 ∪ {∞} such that [x]R contains exactly N0 R0-classes for almost every x ∈ X. Let N = [0,N0) ∩ Z.

Then for any pmp action Γ0 y α(Y, ν), there is a pmp countable equivalence R N N relation Rα = CIndβ (α) on (X × Y , µ × ν ) called the coinduced equivalence relation, whose construction we will brieﬂy recall (see also [IKT08, Section 3]).

Let {Cj}j∈N ⊂ [R] with C0 = id and such that for almost every x ∈ X, the sequence {Cj(x)}j∈N contains exactly one member of each R0-class contained in

[x]R. These are called choice functions (see [IKT08, Remark 2.1] for proof of their existence). For almost every x ∈ X, this gives us a way to number the R0-classes 85

0 0 contained in [x]R. If (x, x ) ∈ R, then x will give rise to a new numbering of the 0 0 R0-classes in [x ]R = [x]R and hence a permutation π(x, x ) ∈ SN deﬁned by

0 0 [Ck(x)]R0 = [Cπ(x,x )(k)(x )]R0 (3.6) which satisﬁes π(x, x0)π(x0, x00) = π(x, x00) for almost every (x, x0), (x0, x00) ∈ R. N Since β is free, we can then deﬁne δ(x,x0) ∈ (Γ0) by

0 0 Cn(x ) = δ(x,x0)(n) · Ck(x) where n = π(x, x )(k). (3.7)

N For y ∈ Y , let yn ∈ Y denote the nth component of y. Then we can then deﬁne N N the co-induced equivalence relation Rα on (X × Y , µ × ν ) by

0 0 0 0 (x, y)Rα(x , y ) ⇐⇒ xRx and yn = δ(x,x0)(n) · yπ(x,x0)−1(n) for all n ∈ N

Proposition 3.7.2 below gives some important properties that this construc- tions satisﬁes. For clarity in its proof, we ﬁrst isolate the following basic fact as a lemma:

Lemma 3.7.1. Let H1 and H2 be Hilbert spaces, H = H1 ⊗ H2, {ξn} ⊂ H1,

{ηn} ⊂ H2 such that supn kξnk < ∞ and ηn → 0 weakly in H2. Then ξn ⊗ ηn → 0 weakly in H.

Proof. Note that supn kηnk < ∞ by the uniform boundedness principle, so {ξn⊗ηn} is bounded and it is enough to check that |hξn ⊗ ηn, ξ ⊗ ηi| ≤ kξk · supk kξkk ·

|hηn, ηi| → 0 as n → ∞ for each ξ ∈ H1, η ∈ H2.

Proposition 3.7.2. Let Γ0 y β(X, µ) be a free ergodic pmp action and R an ergodic pmp equivalence relation on (X, µ) such that R(Γ0 y βX) ≤ R. Then for any pmp action Γ0 y α(Y, ν) with ν nonatomic, the coinduced equivalence relation R Rα = CIndβ (α) satisﬁes:

1. Rα is an extension of R.

2. If α is weakly mixing, then Rα is ergodic.

3. If α is free, then Rα is an expansion of R(Γ0 y αY ). 86

Remark 3.7.3. Assume that R is the orbit equivalence relation of some free pmp action Γ yσ (X, µ) of a countable group Γ. Let Γ yτ (X × Y N , µ × νN ) be the co-induced action of α, modulo (β, σ) (see [Ep07] and also [IKT08, Section 3

(A)], where this terminology is deﬁned). Then Rα is precisely the orbit equivalence relation of τ. In particular, if α is weakly mixing, then Proposition 3.7.2 (2) implies that τ is ergodic. This fact allows to simplify the proof of [Ep07, Lemma 2.6]. Indeed, in the context from [Ep07, Lemma 2.6], it follows that the action c of Γ obtained by coinducing the weakly mixing action a × aπ of F2 modulo (a0, b0) is ergodic, hence the use of the ergodic decomposition of c is redundant.

Proof. (1). Consider the measurable map p : X ×Y N → X defined by p(x, y) = x. N −1 N Then µ = [µ × ν ] ◦ p and for (x, y) ∈ X × Y we have p([(x, y)]Rα ) = [x]R N injectively since π(x,x) = id, δ(x,x) = id . N (2). Let E ⊂ X ×Y be an Rα-invariant Borel subset and let 1E denote the 2 N 2 ∼ 2 characteristic function viewed as an element of L (X) ⊗ k∈N L (Y ) = L (X × N −1 Y ). Then σθ(1E) = 1E for all θ ∈ [Rα], where we define σθ(ξ) = ξ ◦ θ for 2 N ˜ ξ ∈ L (X × Y ). For θ ∈ [R], let θ ∈ [Rα] be its lift, i.e. the unique element in ˜ [Rα] such that p ◦ θ = θ ◦ p. ∞ 2 Let {ηi}i=0 be an orthonormal basis of L (Y ) with η0 = 1, and for i = N N P (ij)j∈N ∈ ( ≥0) , let ηi = ηi . Then expanding 1E = N ξi ⊗ ηi with Z j∈N j i∈(Z≥0) 2 2 ξi ∈ L (X), we will show that 1E ∈ L (X) ⊗ C by showing that ξi = 0 for any N i ∈ (Z≥0) which has ik 6= 0 for some k ∈ N. This will finish the proof. Indeed, 2 since 1E ∈ L (X) ⊗ C is Rα-invariant, it follows that 1E is R-invariant. Since R is ergodic, this will then force 1E ∈ C, i.e., µ(E) ∈ {0, 1}.

Fix such i with ik 6= 0. Since α is weakly mixing and ηik ⊥ C, there is a ∞ 2 sequence {gn}n=1 ⊂ Γ0 such that αgn (ηik ) → 0 weakly in L (Y ) as n → ∞. Let N {Cj}j=0 ⊂ [R] be the choice functions used to construct Rα, and for each n ≥ 1, −1 set θn = Ck ◦ gn ◦ Ck ∈ [R]. Then Ck(θnx) = gn · Ck(x) and so

π(x,θnx)(k) = k and δ(x,θnx)(k) = gn for all n ≥ 1.

0 −1 0 0 N ˜−1 Hence yk = gn yk for n ≥ 1 and x, x ∈ X, y, y ∈ Y with θn · (x, y) = 87

0 0 2 N 2 (x , y ). Therefore deﬁning ζn ∈ L (X) ⊗ j∈N\{k} L (Y ) by

−1 O 0 0 0 ˜ −1 ζn(x, y0,..., yˆk,... ) = ξi(θn x) ⊗ ηij (yj) where (x , y ) = θn · (x, y) j∈N\{k} we have σ (ξ ⊗ η ) = ζ ⊗ α (η ) with kζ k = kξ k. Then for any n, θ˜n i i n gn ik n i

2 2 −1 kξik = kξi ⊗ ηik = h1E, ξi ⊗ ηii = hσ (1E), ξi ⊗ ηii = h1E, σ˜ (ξi ⊗ ηi)i θ˜n θn

= h1E, ζn ⊗ αgn (ηik )i

and h1E, ζn ⊗ αgn (ηik )i → 0 as n → ∞ by Lemma 3.7.1, so we indeed have ξi = 0. N (3). Consider the surjection p : X × Y → Y by p(x, y) = y0. Then N −1 ν = µ × ν ◦ p and p([(x, y)]Rα ) ⊃ [p(x, y)]R(Γ0yαY ). Take any (x, y) ∈ X ×Y N and suppose that (x0, y0) 6= (x00, y00) are members 0 00 0 −1 00 −1 of [(x, y)]Rα with y0 = y0 . Then for k = π(x, x ) (0) and m = π(x, x ) (0) we have

−1 0 −1 00 −1 yk = δ(x,x0)(0) y0 = δ(x,x0)(0) y0 = δ(x,x0)(0) δ(x,x00)(0)ym.

−1 If k = m and g = δ(x,x0)(0) δ(x,x00)(0) = e (the identity of Γ0), then

0 0 00 00 x = C0(x ) = δ(x,x0)(0)Ck(x) = δ(x,x00)(0)Ck(x) = C0(x ) = x which would contradict (x0, y0) 6= (x00, y00). On the other hand, if k = m and g 6= e, then by the freeness of Γ0 y α(Y, ν),

N N (µ × ν )({(x, y) ∈ X × Y : yk = gyk}) = ν({y ∈ Y : y = gy}) = 0.

Hence

N N (µ × ν )({(x, y) ∈ X × Y : p is not injective on [(x, y)]Rα }) X X N N ≤ (µ × ν )({(x, y) ∈ X × Y : yk = gym})

k6=m∈N g∈Γ0 Z X X N−1 = ν({gym})d(µ × ν )(x, (y0,..., yˆk,... )) = 0 X×Y N−1 k6=m∈N g∈Γ0 since ν is non-atomic. 88

3.7.2 A separability argument

2 2 Let λ denote the Haar measure on T. Let SL2(Z) y (T , λ ) be the pmp action given by matrix multiplication. Consider a fixed embedding of F2 as a finite 0 2 2 index subgroup of SL2(Z). Then the restricted action F2 y α (T , λ ) is free, weakly mixing, and rigid, in the sense of S. Popa [Po01b, Corollary 5.2]. The ∞ 2 ∞ 2 latter means that the inclusion of von Neumann algebras L (T ) ⊂ L (T ) o F2 has relative property (T), as defined in [Po01b, Definition 4.2]. 0 2 If an equivalence relation R on (X, µ) is an expansion of R(F2 y α T ), σ 0 then there is a canonical way to define an extension F2 y X of α whose orbit equivalence relation is contained in R. Specifically, if p : X → T2 denotes the quotient map, then σ is the unique such action satisfying p ◦ σ(g) = α0(g) ◦ p, for every g ∈ F2.

Lemma 3.7.4. Let {Ri}i∈I on {(Xi, µi)}i∈I be an uncountable collection of stably von Neumann equivalent ergodic pmp countable equivalence relations, each an 0 2 i expansion of R(F2 y α T ). For each i ∈ I, let F2 y σ Xi denote the canonical 0 i extension of α with R(F2 y σ Xi) ≤ Ri. Then there exists an uncountable set J ⊂ I such that for any i, j ∈ J there i j is a σ -invariant (resp. σ -invariant) non-null Borel set Ei ⊂ Xi (resp. Ej ⊂ Xj) i j with the restricted actions σ |Ei and σ |Ej conjugate.

Lemma 3.7.4 is an analogue of [Io06, Theorems 1.3 and 4.7] for equivalence relations. Its proof combines relative property (T) with a separability argument. This idea, originally due to A. Connes [Co80], has also been successfully used by D. Gaboriau and S. Popa in [Po01b, GP03].

Proof. Since {Ri}i∈I are stably von Neumann equivalent, after replacing I with an uncountable subset, we may ﬁnd a separable II1 factor M and non-zero projections ∼ pi ∈ L(Ri) such that M = piL(Ri)pi, for all i ∈ I. We denote by τ and k.k2 the trace and 2-norm on M, and by τi the trace on L(Ri). For each i ∈ I, let ∞ ∞ Bi = L (Xi) and Ni = L (Xi) oσi F2, regarded as subalgebras of L(Ri). Let ∞ 2 ∞ 2 ∼ ∼ A = L (T ) and Q = L (T ) oα0 F2. We have copies Ai = A, Qi = Q with 0 Ai ⊂ Bi, Qi ⊂ Ni, and by Lemma 3.2.4, Ai ∩ L(Ri) = Bi for each i ∈ I. 89

2 Since I is uncountable, we can find t ∈ (0, 1] such that I = {i ∈ I|1 − ≤ t/τi(pi) ≤ 1} is uncountable, for all > 0. As A is diffuse, there is a projection q ∈ A such that τ(q) = t. Since Aq ⊂ qQq has relative property (T), there is a finite set F ⊂ (qQq)1 and δ > 0 such that for any qQq-qQq bimodule H with nonzero ξ0 ∈ H satisfying kxξ0 − ξ0xk < δkξ0k for all x ∈ F , there is nonzero 3 ξ ∈ H with aξ = ξa for all a ∈ Aq. Let > 0 small enough that 1−2 < δ and set

I1 = I.

For x ∈ Q, we let xi ∈ Qi denote the image in Qi. Each L(Ri) is a factor, so by conjugating by a unitary in each, we may assume that qi ≤ pi, for all i ∈ I1. 2 Then identifying piL(Ri)pi with M, we have qi ∈ M and τ(qi) ≥ 1 − so that k1M − qik2 ≤ , for each i ∈ I1. 2 Then for any i, j ∈ I1, endow qiL (M)qj with a qQq-qQq bimodule structure 2 given by deﬁning x · ξ · y = xiξyj, for all x, y ∈ qQq and ξ ∈ qiL (M)qj. Let 2 ξi,j = qiqj ∈ qiL (M)qj and note that kξi,j − 1M k2 ≤ k1M − qik2 + k1M − qjk2 ≤ 2 and hence kξi,jk2 ≥ 1 − 2.

Since M is k · k2-separable, there is an uncountable set J ⊂ I1 such that kxi − xjk2 < for all i, j ∈ J and x ∈ F . Fix any i, j ∈ J. Then for any x ∈ F ,

kxiξi,j − ξi,jxjk2 ≤ kxi − xjk2 + k1M − qik2 + k1M − qjk2 ≤ 3 < δ(1 − 2)

≤ δkξi,jk2,

2 and so by relative property (T) there is nonzero ξ ∈ qiL (M)qj with aiξ =

ξaj for all a ∈ Aq. Then the polar decomposition ξ = v|ξ| has v ∈ M with ∗ ∗ aiv = vaj for all a ∈ Aq. Set ei = vv and ej = v v. ∗ ∗ ∗ ∗ For any b ∈ Bjqj and a ∈ Aq we have aivbv = vajbv = vbajv = vbv ai, ∗ 0 ∗ so vBjv ⊂ (Aiqi) ∩ qiMqi = Biqi and similarly v Biv ⊂ Bjqj, and in particular, ei ∈ Bi, ej ∈ Bj. We thus deﬁne a trace preserving ∗-isomorphism Ψ : Bjej → Biei by b 7→ vbv∗.

Then for positive measure sets Fi ⊂ Xi and Fj ⊂ Xj with ei = 1Fi and ej = 1Fj , there is a measure space isomorphism Θ : (Fi, µi) → (Fj, µj) such that Ψ(b) = b ◦ Θ for all b ∈ B . Let E = S σi (F ) and E = S σj(F ). Then j i g∈F2 g i j g∈F2 g j i j Ei is σ -invariant, Ej is σ -invariant and we will show that Θ can be extended to 90

a measure space isomorphism Θ : (Ei, µi) → (Ej, µj) by the formula

j i i Θ(x) = [σg ◦ Θ ◦ σg−1 ](x) for x ∈ σg(Fi), g ∈ F2 (3.8)

j i which will then satisfy [σg ◦ Θ](x) = [Θ ◦ σg](x) for x ∈ Ei, g ∈ F2, showing that i j σ |Ei and σ |Ej are conjugate. Toward showing that (3.8) is well deﬁned, for g ∈ F2 let ugi ∈ Qi and ugj ∈ Qj denote respectively the canonical unitaries implementing σi and σj. Viewing v∗u∗ v ∈ e L(R )e ⊂ L(R ), for a ∈ A we have gi j j j j

∗ ∗ j ∗ ∗ ∗ i ∗ ∗ ∗ ∗ ∗ a u v u v = u σ (a )v u v = u v σ −1 (a )u v = u v u a v = u v u va j gj gi gj g−1 j gi gj g i gi gj gi i gj gi j so that u v∗u∗ v ∈ A0 ∩ L(R ) = B . Therefore for any b ∈ B , we have gj gi j j j j

u vu∗ b(u v∗u∗ v)v∗ = u vu∗ (u v∗u∗ v)bv∗ = (u e u∗ )vbv∗ gi gj gj gi gi gj gj gi gi i gi and hence

i j ∗ ∗ i σg(vσg−1 (b)v )ei = vbv σg(ei) for all b ∈ Bj, g ∈ F2,

−1 which when applied to h g ∈ F2 for g, h ∈ F2 gives

i j ∗ i i j ∗ i j σg(vσg−1 (b)v )σh(ei) = σh(vσh−1 (b)v )σg(ei) for all b ∈ σh(Bj) = Bj, which translates to

j i j i i i [σg ◦ Θ ◦ σg−1 ](x) = [σh ◦ Θ ◦ σh−1 ](x) for all x ∈ σg(Fi) ∩ σh(Fi) showing that (3.8) is well deﬁned.

3.7.3 Proof of Theorem 3C

Let R be a non-amenable ergodic countable pmp equivalence relation on a probability space (X, µ). Our goal is to show that R has uncountably many ergodic extensions which are pairwise not stably von Neumann equivalent. Below, for a α 0 pmp action F2 y (Y, ν), we denote by πα and πα the Koopman representations 2 2 of F2 on L (Y ) C1 and L (Y ), respectively. Let R˜ on (X,˜ µ˜) denote the Bernoulli extension of R with base space ˜ ([0, 1], λ). By Theorem 3A, there is a free ergodic pmp action F2 y βX such ˜ ˜ that R0 := R(F2 y βX) ≤ R. 91

By [Sz88] there is an uncountable family {πi : F2 → U(Hi)}i∈I of nonequivalent irreducible representations who are mixing, i.e. hπi(g)ξ, ηi → 0 as g → ∞ for any ξ, η ∈ Hi. By considering the Gaussian action corresponding to the realiﬁcation of πi (as in [Ke10], for example), we obtain an uncountable family i 0 of actions {F2 y α (Yi, µi)}i∈I such that πi ⊂ παi , for each i ∈ I. For each i ∈ I, note that αi × α0 is weakly mixing since αi is mixing and 0 ˜ R˜ i 0 ˜ α is weakly mixing. By Proposition 3.7.2, Ri = CIndβ (α × α ) on (Yi, ν˜i) is an ergodic extension of R˜ and hence of R. Thus, we are done, unless uncountably ˜ many of the Ri are stably von Neumann equivalent. Therefore, assume toward a ˜ contradiction that there is an uncountable subset I0 ⊂ I such that the {Ri}i∈I0 are stably von Neumann equivalent. ˜ αi×α0 2 By Proposition 3.7.2, each Ri is an expansion of R(F2 y Yi × T ) α0 2 i ˜ 0 and hence of R(F2 y T ). Let F2 y σ Yi denote the canonical extension of α .

Then by Lemma 3.7.4, there is an uncountable subset J ⊂ I0 such that for each i j ˜ i, j ∈ J there is a σ -invariant (resp. σ -invariant) positive measure set Ei ⊂ Yi ˜ i j (resp. Ej ⊂ Yj) with the restricted actions σ |Ei and σ |Ej conjugate. i i 0 i Since σ is an extension of the ergodic action α × α of F2, σ |Ei is also an extension thereof. Hence, for all i, j ∈ J,

0 0 0 ∼ 0 πi ⊂ π i ⊂ π i 0 ⊂ π i = π j ⊂ πσj α α ×α σ |Ei σ |Ej so that πσj has uncountably many nonequivalent irreducible sub-representations, 2 contradicting the separability of L (Ej).

3.8 Actions of locally compact groups

In this section we prove Corollary 3D and explain how Theorem 3A implies [GM15, Theorem B]. We begin by recalling the notion of cross section of actions of lcsc groups (see [KPV13, Deﬁnition 4.1]).

Deﬁnition 7. Let G be a lcsc group and G y (X, µ) a free nonsingular action on a standard probability space (X, µ). A Borel set Y ⊂ X is called a cross section of G y (X, µ) if there exists a neighborhood U of the identity in G such that the 92 map U × Y → X given (g, y) 7→ gy is injective, and µ(X \ G · Y ) = 0. A cross section Y ⊂ X is called co-compact if there is a compact set K ⊂ G such that K · Y is a G-invariant Borel set and µ(X \ K · Y ) = 0.

Remark 3.8.1. Assume that G is a lcsc unimodular group. Let G y (X, µ) be a free pmp action and Y ⊂ X a cross section. Then R = {(y, y0) ∈ Y ×Y |Gy = Gy0} deﬁnes a countable Borel equivalence relation, called the cross section equivalence relation. Moreover, if λ is a ﬁxed Haar measure of G, then there exist a unique R-invariant probability measure ν on Y and constant c ∈ (0, +∞) such that for every neighborhood U of the identity in G such that the map ζ : U ×Y → X given by ζ(g, y) = gy is injective, we have ζ∗(λ|U ×ν) = c µ|U·Y (see [KPV13, Proposition 4.3]). Hereafter, we refer to ν as the canonical R-invariant probability measure on Y .

We continue with an elementary result which gives a construction of actions of locally compact groups with prescribed cross section equivalence relations.

Proposition 3.8.2. Let G be a lcsc unimodular group and G y (X, µ) a free pmp action. Let Y ⊂ X be a co-compact cross section of G y (X, µ), R be the cross section equivalence relation, and ν be the canonical R-invariant probability measure on Y . Let R¯ be a countable pmp extension of R on a standard probability space (Y,¯ ν¯). Then there exist a free pmp action G y (X,˜ µ˜), and a co-compact cross section Y˜ ⊂ X˜ such that the following holds. Denote by R˜ the cross section equivalence relation on Y˜ , and endow Y˜ with the canonical R˜-invariant probability measure ν˜. Then R˜ is isomorphic to R¯.

Proof. Let X1 ⊂ X be the set of points with trivial stabilizer. Then X1 is a G-invariant Borel set (see e.g. [MRV11, Lemma 10]). Moreover, the freeness assumption implies that X1 ⊂ X is co-null. Let K ⊂ G be a compact set such that X2 = K · Y is a co-null G-invariant Borel subset of X.

Put X0 := X1 ∩X2 and Y0 := Y ∩X0. Then X0 ⊂ X is a co-null G-invariant

Borel subset, Y0 ⊂ Y is an R-invariant Borel subset, and K · Y0 = X0. Let U be a neighborhood of the identity in G such that the map U ×Y → X given (g, y) 7→ gy 93

is injective. Since U · (Y \ Y0) is contained in X \ X0, it is a null set. Let λ be a

Haar measure of G. Since λ(U)ν(Y \ Y0) = c µ(U · (Y \ Y0)) = 0, for some c > 0, and λ(U) > 0, we get that Y0 is co-null in Y .

Altogether, we have that G y (X0, µ|X0 ) is a pmp action such that every point has trivial stabilizer, Y0 ⊂ X0 is a co-compact cross section with K ·Y0 = X0,

R|Y0 is the associated cross section equivalence relation, and ν|Y0 is the canonical

R|Y0 -invariant probability measure on Y0. Moreover, since Y0 ⊂ Y is co-null, R|Y0 is isomorphic to R. Thus, after replacing X, Y with X0, Y0, we may assume that the stabilizer of every point in X is trivial, and K · Y = X, for a compact set K ⊂ G. Let U be a neighborhood of the identity in G such that the map U ×Y → X given (g, y) 7→ gy is injective. Deﬁne π : U · Y → Y by letting π(gy) = y. n Since K is compact, we can ﬁnd g1, ..., gn ∈ G such that K ⊂ ∪i=1giU. Hence n X = K · Y ⊂ ∪i=1giU · Y . It follows that we can extend π to a Borel map π : X → Y in such a way that π(x) ∈ Gx, for every x ∈ X. Let p : Y¯ → Y be the quotient map. After replacing Y¯ with a co-null R¯-

¯ invariant Borel subset, we may assume that p|[¯y]R¯ is injective and p([¯y]R) = [p(¯y)]R, for ally ¯ ∈ Y¯ . ˜ ¯ ˜ Let X = X ×Y Y be the “ﬁbered product” Borel space given by X = {(x, y¯) ∈ X × Y¯ |π(x) = p(¯y)}. We deﬁne a free Borel action G y X˜ as follows. ˜ Let g ∈ G and (x, y¯) ∈ X. Since π(gx) ∈ Gx ∩ Y , we get that π(gx) ∈ [π(x)]R =

[p(¯y)]R. Thus, there is a uniquey ˆ ∈ [¯y]R¯ such that p(ˆy) = π(gx). Finally, we let g(x, y¯) = (gx, yˆ). It is easy to check that this indeed defines a Borel action of G. Next, let Y˜ = {(p(¯y), y¯)|y¯ ∈ Y¯ }. Then Y˜ is a Borel subset of X˜. Let ¯ (g1, y¯1), (g2, y¯2) ∈ U × Y such that g1(p(¯y1), y¯1) = g2(p(¯y2), y¯2). Then g1p(¯y1) = g2p(¯y2) and since p(¯y1), p(¯y2) ∈ Y , we deduce that g1 = g2, which implies that ˜ ˜ y1 = y2. Thus, the map U × Y → X given by (g, y) 7→ gy is injective. Let ˜ −1 (x, y¯) ∈ X. Since K · Y = X, we can find g ∈ K such that g x ∈ Y . Lety ˆ ∈ [¯y]R¯ such that g−1(x, y¯) = (g−1x, yˆ). Since p(ˆy) = π(g−1x) = g−1x, we deduce that g−1(x, y¯) ∈ Y˜ . Thus, K · Y˜ = X˜. This proves that Y˜ is a co-compact cross section for the Borel action G y X˜. In particular, the first paragraph of the proof implies 94 that there exists a Borel mapπ ˜ : X˜ → Y˜ such thatπ ˜(y) = y, for every y ∈ Y˜ , and π˜(x) ∈ Gx, for every x ∈ X˜. Further, consider the cross section equivalence relation R˜ = {(y, y0) ∈ Y˜ × Y˜ |Gy = Gy0}. Let θ : Y¯ → Y˜ be the Borel isomorphism given by θ(¯y) = (p(¯y), y¯). It is easy to see that (θ × θ)(R¯) = R˜. We endow Y˜ with the probability measure ¯ ˜ ν˜ = θ∗ν¯. Sinceν ¯ is R-invariant,ν ˜ is R-invariant. By [Sl15, Section 4.2], the R˜-invariant probability measureν ˜ on the co- compact cross section Y˜ can be “lifted” to a G-invariant finite measureµ ˜ on X˜. Specifically, for a Borel set A ⊂ X˜, we have

µ˜(A) = (λ × ν˜)({(g, y˜) ∈ G × Y˜ |π˜(gy˜) =y ˜ and gy˜ ∈ A}).

Since the map U × Y˜ → U · Y˜ given by (g, y) 7→ gy is a bijection, it follows that under this identiﬁcation we have thatµ ˜|U·Y˜ = λ|U × ν˜. By using [KPV13, Proposition 4.3] we conclude thatν ˜ is the canonical R˜-invariant probability mea- ˜ ˜ 1 sure on the cross section Y for the free pmp action G y (X, µ˜(X) µ˜). This concludes the proof of the proposition.

3.8.1 Proof of Corollary 3D

Let G y (X, µ) be a free ergodic pmp action (see [KPV13, Remark 1.1] for a proof of existence). By [KPV13, Theorem 4.2] we can find a co-compact cross section Y of G y (X, µ). Denote by R the associated cross section equivalence relation, and endow Y with the canonical R-invariant probability measure ν. Since G is non-amenable and G y (X, µ) is ergodic, [KPV13, Proposition 4.3] gives that R is non-amenable and ergodic. ˜ By Theorem 3C, we can find an uncountable family {Ri}i∈I of countable ergodic pmp extensions of R which are pairwise not stably von Neumann equivalent. By Proposition 3.8.2, for every i ∈ I which can find a free ergodic pmp action ˜ ˜ ˜ G y (Xi, µ˜i) and a co-compact cross section Yi ⊂ Xi such that the associated cross ˜ section equivalence relation is isomorphic to Ri. ˜ We claim that the actions G y (Xi, µ˜i), i ∈ I, are pairwise not von Neu- ∞ ˜ ∼ ∞ ˜ mann equivalent. Indeed, assume that L (Xi) o G = L (Xj) o G, for some i 6= j. 95

∞ ˜ ∞ ˜ On the other hand, L (Xi) o G and L (Xj) o G are ampliﬁcations of the II1 ˜ ˜ factors L(Ri) and L(Rj) by [KPV13, Lemma 4.5]. It follows that we can ﬁnd ˜ ˜ ˜ ∼ ˜ non-zero projections pi ∈ L(Ri) and pj ∈ L(Rj) such that piL(Ri)pi = pjL(Rj)pj. ˜ ˜ This contradicts the fact that Ri and Rj are not stably von Neumann equivalent.

3.8.2 Deducing [GM15, Theorem B] from Theorem 3A

Let G be a non-amenable lcsc group. Let λ be a Haar measure of G. To show the existence of a tychomorphism from F2 to G, in the sense of [GM15, Deﬁnition 14], we ﬁrst reduce to the case when G is unimodular.

Denote by G0 the kernel of the modular homomorphism of G. Then G0 is non-amenable. Moreover, G0 is unimodular. Indeed, since G0 < G is a closed normal subgroup, G/G0 is a locally compact group, thus it admits a G-invariant

Borel measure. [BdHV08, Corollary B.1.7.] now implies that G0 is unimodular. Thus, by [GM15, Proposition 18], we may assume that G is unimodular. Since the conclusion follows from the Gaboriau-Lyons theorem in the discrete case, we may additionally assume that G is not discrete. Let G y (X, µ) be a free ergodic pmp action, Y a co-compact cross section, R the cross section equivalence relation, and ν the canonical R-invariant probability measure on Y . Since G is non-amenable and G y (X, µ) is ergodic, R is non-amenable and ergodic. By Theorem 3A there exist a countable ergodic pmp extension R˜ of R on ˜ ˜ a probability space (Y, ν˜) and a free ergodic pmp action F2 y (Y, ν˜) such that ˜ ˜ F2y ⊂ [y]R˜ , for all y ∈ Y . By Proposition 3.8.2, we can be realize Y as a co- compact cross section of some free ergodic pmp action G y (X,˜ µ˜), such that R˜ is precisely the associated cross section equivalence relation. Moreover, the proof of Proposition 3.8.2 gives that any point in X˜ has trivial stabilizer and K · Y˜ = X˜, for K ⊂ G compact. Let U ⊂ G be a neighborhood of the identity such that the map ζ : U ×Y˜ → X˜ given by ζ(h, y) = hy is injective. Deﬁne

˜ ˜ 0 ˜ ˜ 0 X0 := U · Y and D := {(x, x ) ∈ X × X0)|Gx = Gx }. 96

Consider the obvious action of G on D on the first coordinate. As in the end of [GM15, Section 5], we endow D with a G-invariant measure m by pushing forward λ × µ through the identification G × X˜ → D given by (g, x) 7→ (gx, x). |X˜0 0 Then (D, m) is a finite amplification of the G-space (G, λ), in the sense of [GM15, Definition 11].

Next, we define an m-preserving action F2 y D, as follows. Fix θ ∈ F2. If 0 0 ˜ 0 ˜ (x, x ) ∈ D, then x ∈ X0, hence we can write x = hy, for some h ∈ U and y ∈ Y . ˜ 0 ˜ We define θ(x, x ) = (x, hθ(y)). Let α : Y0 → G be given by θ(y) = α(y)y, for ˜ ˜ ˜ every y ∈ Y0. Then in the above identification G × X0 ≡ D, θ corresponds to the ˜ −1 −1 Borel automorphism of G × X0 given by (g, hy) 7→ (ghα(y) h , hθ(y)), for all ˜ h ∈ U, y ∈ Y0. Since G is unimodular, ζ∗(λ|U × ν˜) = c µ˜|X0 , for some c > 0, and θ preserves ν, it follows that θ˜ preserves m.

We claim that F2 y D admits a non-null measurable fundamental domain.

Since the actions of F2 and G on D commute, it will follow that D gives rise to a tychomorphism from F2 to G. To prove the claim, since K is compact, let n ˜ ˜ n ˜ g1, ..., gn ∈ G such that K ⊂ ∪j=1gjU. Then X = K · Y ⊂ ∪j=1gjX0 and therefore n 0 ˜ ˜ 0 D = ∪j=1gjD0, where D0 := {(x, x ) ∈ X0 × X0|Gx = Gx }.

Since D0 is F2-invariant, in order to prove the claim, it suﬃces to show that the action F2 y D0 admits a non-null measurable fundamental domain. To see ˜ this, using that the action F2 y (Y, ν˜) is ergodic, we choose a sequence {Ci}i≥1 ⊂ ˜ ˜ [R] such that [y]R˜ is the disjoint union of F2Ci(y), i ≥ 1, for almost every y ∈ Y 0 0 0 0 (see [IKT08, Remark 2.1]). Since D0 = {(hy, h y )|h, h ∈ U, (y, y ) ∈ R}, one 0 0 ˜ checks that F := {(hy, h Ci(y))|h, h ∈ U, y ∈ Y , i ≥ 1} is a non-null measurable fundamental domain for the action F2 y D0.

Note

Chapter 3 is, in part, material submitted for publication as it appears in [BHI15] Lewis Bowen, Daniel J. Hoﬀ, and Adrian Ioana, Von Neumann’s problem and extensions of non-amenable equivalence relations, preprint arXiv:1509.01723, 2015. of which the dissertation author was one of the primary investigators and authors. References

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