The Borel complexity of von Neumann equivalence
Introduction
A fundamental problem of ergodic theory is the conjugacy problem: Given two measure preserving actions of a countable discrete group Γ on a standard probability space, how does one determine if they are conjugate actions? A solution to the conjugacy problem should ideally be a method that, when given two measure preserving actions, can be applied systematically, and will produce a yes-or-no answer to the conjugacy question.
In those cases where we have nice classification theorems for the probability measure preserving (p.m.p.) actions of Γ, then the classification also solves the conjugacy problem: For instance, when (or, more generally is amenable, [27], or even is a non-amenable free group, [1]) and we consider only Bernoulli actions of Γ, then the conjugacy problem can be solved by computing the entropy of the two actions, which by a celebrated theorem of Ornstein [25] is a complete invariant for conjugacy. However, the conjugacy problem may be viewed as distinct from the classification problem: Having a method for answering the yes-or-no question of conjugacy clearly does not provide a classification, and a classification may assign invariants for which it is difficult to determine if the assigned invariants are isomorphic or not.
The conjugacy problem arguably goes back to Halmos, who posed it in the context of as Problem 3 in [12, p. 96]. As stated above the conjugacy problem is vague since it is not clear what is meant by a “method”. One possible way of posing the conjugacy problem in a precise mathematical way is the following:
Problem 1.1 Kechris [22, 18.(IVb)] Is the conjugacy relation for p.m.p. (ergodic, a.e. free) actions of a countable discrete group Γ a Borel or analytic set? If it is analytic, is it complete analytic?
In §2 below we will give an explanation for why this question is closely related to the conjugacy problem. Roughly speaking, if the conjugacy relation were Borel, then the description of the Borel set (that is, how it is built up using countable unions and complements) would provide a method for determining if two actions are conjugate, and this method would use only countable resources. If, on the other hand, the conjugacy relation is analytic and not Borel, then no generally applicable method that relies on only countable resources could solve the conjugacy problem; and in case the conjugacy relation is complete analytic, then the worst possible general method (which we describe in §2) would also be the best possible.
When , the conjugacy problem was solved by Hjorth in [14], however the solution suffered from the obvious defect that it only showed that the conjugacy relation on non-ergodic measure preserving transformations is analytic and not Borel. Only recently was the conjugacy problem for -actions given a satisfactory solution: In [7], Foreman, Rudolph and Weiss showed that the conjugacy relation on ergodic actions of is a complete analytic set (see also [8]). In this paper we will prove the following:
Theorem 1.2 Let Γ be a countable discrete group containing a non-amenable free group , , as a normal subgroup. Then the conjugacy relation for weakly mixing a.e. free p.m.p. actions of Γ is complete analytic, and so it is not Borel. This in particular settles the conjugacy problem for for .
Since this paper was first circulated, Eusebio Gardella and Martino Lupini [10], in a technical tour-de-force, have vastly improved our result above, proving the theorem for all countable discrete non-amenable groups. The argument given in this paper, however, is somewhat less involved than the argument given by Gardella and Lupini, and some of the ideas of our proof inform the proof of Gardella and Lupini's proof. In particular, the meta-mathematical method and results we develop in §5 below to analyze so-called countable-to-1 Borel reductions and their relation to the conjugacy problem, also plays an important part in the work of Gardella and Lupini.
Our proof already allows that the hypothesis on Γ be weakened somewhat, and we will state our result in full in the next section. We note that Theorem 1.2 and the result of Foreman, Rudolph and Weiss stand in contrast to the recent result of Hjorth and Törnquist [17], where it is shown that conjugacy of unitary representations of any countably infinite discrete Γ is always Borel.
In addition to conjugacy, there are two other important equivalence relations for the p.m.p. actions that merit close consideration, namely orbit equivalence and von Neumann equivalence (also known as -equivalence). Let be a standard Borel probability space,1 and let be measure preserving actions. Denote by the orbit equivalence relation induced by , . Recall that and are orbit equivalent, written , if there is a measure preserving Borel bijection such that for almost all . Recall also that and are said to be von Neumann equivalent, written , if the associated group-measure space von Neumann algebras and are isomorphic (see e.g. [30] for a thorough discussion of von Neumann equivalence).
When Γ is amenable, it was shown in [26] and [2] that all ergodic p.m.p. Γ-actions are both orbit equivalent and von Neumann equivalent. However, it has recently been shown that when Γ is non-amenable then Γ admits uncountably many orbit inequivalent (see [5]) and von Neumann inequivalent (see [18]) ergodic, a.e. free p.m.p. actions. The natural question whether orbit equivalence of a.e. free p.m.p. Γ-actions is Borel or analytic was raised by Kechris in [22, 18.(IVb)] along with Problem 1.1 above. We will prove the following:
Theorem 1.3 Let Γ be a countable discrete group containing a non-amenable free group , , as a normal subgroup. Then orbit equivalence and von Neumann equivalence of weakly mixing a.e. free p.m.p. actions of Γ are analytic relations, but they are not Borel.
Again, Gardella and Lupini have improved this, proving the theorem for all Γ which are countable discrete and non-amenable. The assumptions on Γ can be weakened, and we state our results in full in §2. Furthermore, we note that the proofs of both Theorem 1.2, Theorem 1.3 use entirely different techniques than those used by Foreman, Rudolph and Weiss in [7]. Namely, our proofs use rigidity techniques, and rely in particular on Popa's cocycle superrigidity theorems [30], [31]. Our arguments are also closer to [38], where it was proven that conjugacy and orbit equivalence are complete analytic equivalence relations (on the weakly mixing a.e. free actions) when Γ is a countably infinite group with the relative property (T).
As a consequence of Theorem 1.3, we obtain the following:
Theorem 1.4 The isomorphism relation for separably acting factors of type and type , for each , is analytic but not Borel when the space of separably acting factors is given the Effros Borel structure.
It was shown in [33] that the isomorphism relation for separably acting factors of type is in fact complete analytic, however the argument there did not extend to factors of type and .
The proof of both Theorem 1.3, Theorem 1.4 relies crucially on establishing a technical set-theoretic result in the theory of Borel reducibility, Theorem 5.6, which shows that there is a sequence , , of Borel equivalence relations which is increasing and unbounded in the class of Borel equivalence relations, when this class is ordered under the relation of countable-to-1 Borel reductions. This uses a deep set-theoretic method due to Stern, known as Stern's absoluteness method from [35], which generalizes a similar result due to Harrington for the usual Borel reducibility hierarchy.
The paper is organized as follows: §2 is dedicated to preliminaries and background, as well as the statement of our results in full: We review Popa's cocycle superrigidity theorems and the related concepts, and we also briefly review the descriptive set theory (“Global theory”) of measure preserving actions, as well as the concept of Borel reducibility. In §3 we introduce a variant of the 1-cohomology group, called the relative 1-cohomology group, which will be our main tool to distinguish actions up to conjugacy. In §4 we compute the relative 1-cohomology group for certain families of actions. §5 is dedicated to establishing the technical result on countable-to-1 Borel reductions described above. Finally, in §6 we combine the results of §4 and §5 to prove Theorem 1.2, Theorem 1.3, Theorem 1.4.
Acknowledgments. We wish to thank Alexander Kechris for useful discussions through the early stages of this work. We also thank Benjamin Weiss for his remarks on a version of this paper that was circulated in the summer of 2011. Finally, we thank the anonymous referee for carefully reading of the manuscript and making many valuable suggestions for improving the paper.
Asger Törnquist is grateful for the kind hospitality and support he received during a visit to Caltech in February 2010 where part of the work for the paper was done. Thanks are also due to the Kurt Gödel Research Center in Vienna, where part of the paper was written while Asger Törnquist was employed there under FWF project P 19375-N18.
Asger Törnquist also received support from grant no. 7014-00145B from the Danish Council for Independent Research while finishing the manuscript for publication.
Section snippets
Global theory
The main reference for the global theory of measure preserving actions is Kechris' book [22].
Let be a standard Borel probability space. The group of measure preserving transformations of is denoted . We equip this group with the weak topology, i.e., the initial topology making all the maps continuous, where are Borel sets. This makes a Polish group. Let Γ be a countable group. The set
The relative 1-cohomology group of a measure preserving action
Let Λ be a countable group, and let be a probability measure preserving action. A 1-cocycle is a cocycle with target group , i.e., a measurable map such that for all and almost all we have The set of measurable 1-cocycles is denoted and forms a subgroup under pointwise multiplication. We give the topology it inherits from , which makes it a Polish group. A 1-coboundary is a 1-cocycle of the form
Families of actions with non-trivial and calculable
In this section, we will compute the Δ-relative 1-cohomology group of certain p.m.p. actions of the form under reasonably general conditions. We make our computations in a somewhat more general setting than what is narrowly needed for our applications in §6, where it turns out that we always have that is weakly mixing, in which case it follows from Lemma 3.5 that , and so only must be calculated. The extra effort this requires is mostly found in the proof of
A Stern absoluteness argument
The present section is dedicated to a technical result on many-to-one Borel reductions. The main aim is to prove Theorem 5.1 below, which is a consequence of results of Harrington and Hjorth. The argument uses a metamathematical technique developed by Stern in [35], known now as Stern's forcing absoluteness, which we briefly review below, and which requires some basic knowledge of forcing (see e.g. [24]) and effective descriptive set theory as found in e.g. [20, Ch. 2]. Since the
Conjugacy, orbit equivalence and von Neumann equivalence are not Borel
In this section we prove Theorem 2.7, Theorem 2.8, Theorem 2.10, Theorem 2.11. Recall that ABEL denotes the set of countably infinite Abelian groups with underlying set , the set of torsion-free abelian groups. For , let be the character group (which is then a compact group that we equip with the Haar measure). Fix an action and, as in §4, let be the generalized Bernoulli action and the quotient of σ by . We will need the following Lemma.
Lemma 6.1 Let and
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