# Rigidity Theorems for Actions of Product Groups and Countable Borel Equivalence Relations

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*Greg Hjorth; Alexander S. Kechris*

This Memoir is both a contribution to the theory
of Borel equivalence relations, considered up to Borel reducibility,
and measure preserving group actions considered up to orbit
equivalence. Here \(E\) is said to be Borel reducible
to \(F\) if there is a Borel function \(f\) with \(x
E y\) if and only if \(f(x) F f(y)\). Moreover,
\(E\) is orbit equivalent to \(F\) if the
respective measure spaces equipped with the extra structure provided
by the equivalence relations are almost everywhere isomorphic.

We consider product groups acting ergodically and by
measure preserving transformations on standard Borel probability
spaces. In general terms, the basic parts of the monograph show that
if the groups involved have a suitable notion of
“boundary” (we make this precise with the definition of
near hyperbolic), then one orbit equivalence relation can
only be Borel reduced to another if there is some kind of algebraic
resemblance between the product groups and coupling of the
action. This also has consequence for orbit equivalence. In the case
that the original equivalence relations do not have non-trivial almost
invariant sets, the techniques lead to relative ergodicity
results. An equivalence relation \(E\) is said to be
relatively ergodic to \(F\) if any \(f\) with
\(xEy \Rightarrow f(x) F f(y)\) has \([f(x)]_F\)
constant almost everywhere.

This underlying collection of lemmas and structural
theorems is employed in a number of different ways.

One of the most pressing concerns was to give completely
self-contained proofs of results which had previously only been
obtained using Zimmer's superrigidity theory. We present
“elementary proofs” that there are incomparable countable
Borel equivalence relations (Adams-Kechris), inclusion does not imply
reducibility (Adams), and \((n+1)E\) is not necessarily
reducible to \(nE\) (Thomas).

In the later parts of the paper we give applications
of the theory to specific cases of product groups. In particular, we
catalog the actions of products of the free group and obtain
additional rigidity theorems and relative ergodicity results in this
context.

There is a rather long series of appendices, whose primary goal is
to give the reader a comprehensive account of the basic techniques.
But included here are also some new results. For instance, we show
that the Furstenberg-Zimmer lemma on cocycles from amenable groups
fails with respect to Baire category, and use this to answer a
question of Weiss. We also present a different proof that
\(F_2\) has the Haagerup approximation property.

#### Table of Contents

# Table of Contents

## Rigidity Theorems for Actions of Product Groups and Countable Borel Equivalence Relations

- Contents v6 free
- Introduction 110 free
- Chapter 0. Preliminaries 918 free
- Chapter 1. Actions of Free Groups and Treeable Equivalence Relations 1524
- Chapter 2. A Cocycle Reduction Result 2130
- Chapter 3. Some Applications 2534
- Chapter 4. Factoring Homomorphisms 4150
- Chapter 5. Further Applications 4554
- Chapter 6. Product Actions, I 4958
- Chapter 7. Product Actions, II 5564
- Chapter 8. A Final Application 6170
- Appendix A: Strong Notions of Ergodicity 6372
- A1. Homomorphisms and relative ergodicity 6372
- A2. E[sub(0)]–ergodicity and almost invariant sets 6372
- A3. Almost invariant vectors 6675
- A4. E[sub(0)]–ergodicity of the shift action 6978
- A5. Characterizations of amenable and Kazhdan groups 7079
- A6. Mixing 7180
- A7. Non–orbit equivalent relations 7483

- Appendix B: Cocycles and Cocycle–invariant Functions 7786
- Appendix C: Actions on Boundaries 8796
- Appendix D: K–structured Equivalence Relations 95104
- Appendix E: Proof of the General Case of Theorem 4.4 99108
- Bibliography 107116