On Orbit Equivalence of Measure Preserving Actions

Damien Gaboriau

Unité de Mathématiques Pures et Appliquées, Unité Mixte de Recherche CNRS 5669 46, allée d’Italie 69364 Lyon cedex 07, France

Abstract

We give a brief survey of some classification results on orbit equivalence of probability measure preserving countable group actions. The notion of ℓ^{2} Betti numbers for groups is gently introduced. An account of orbit equivalence invariance for ℓ^{2} Betti numbers is presented together with a description of the theory of equivalence relation actions on simplicial complexes. We relate orbit equivalence to a measure theoretic analogue of commensurability and quasi-isometry of groups : measure equivalence. Rather than a complete description of these subjects, a lot of examples are provided.

1 Equivalence Relations

1.1 Equivalence Relation defined by an action

Let (X, μ) be a standard Borel space, where μ is a probability measure without atoms. Remember that it is Borel isomorphic to the unit interval of the reals, with Lebesgue measure.

Let Γ be a countable group and α an action of Γ on (X, μ) by μ-preserving Borel automorphisms. Consider the orbit equivalence relation on X:

R_{α} = {(x, γ.x) : x ∈ X, γ ∈ Γ}.

As a subset of X × X, this equivalence relation is just the union of the graphs of the γ ∈ Γ. In this measured context, null sets are neglected. Thus the action is free if the only element of Γ with a fixed-point set of positive measure is the identity element.

Examples 1.1 The first examples to keep in mind are the following:

1. The action of ℤ^{n} on the circle 𝕊^{1} by rationally independent rotations (belongs to the next two families). This gives an idea of the wildness of the quotient spaces Γ\X.

2. Free volume-preserving group actions on finite volume manifolds.

3. A compact group K, its Haar measure μ and the action of a countable subgroup Γ by left multiplication on K.

4. The shift action of Γ on the space X = {0, 1}^{Γ} of sequences of 0, 1 indexed by Γ, with any invariant probability measure, for example the product of equiprobability on {0, 1}. The action is free when Γ is infinite. Thus, every countable group admits at least one free probability measure preserving (p.m.p.) action.

Question: Which properties of the group Γ are determined by the orbit equivalence relation R_{Γ}?

1.2 Standard Countable Borel Equivalence Relations

The orbit equivalence relation R = R_{α} enjoys the following properties:

1. the equivalence classes (or orbits) of R are countable

2. R is a Borel subset of X × X

3. every partial isomorphism φ : A → B whose graph is contained in R preserves the measure (R preserves μ).

A partial isomorphism φ : A → B is a Borel isomorphism between two Borel subsets A and B of X. Observe that if its graph {(x, φ(x)) : x ∈ A} is contained in R then A admits a partition A = ∐_{γ∈Γ} A_{γ} where x ∈ A_{γ} ⇒ φ(x) = α(γ)(x) (replaced soon by the notation γ.x). The third item is now obvious.

A countable standard measure preserving equivalence relation R on (X, μ) is an equivalence relation that satisfies items 1–3. This more general notion was introduced by J. Feldman and C. Moore who immediately showed [FM77a, th. 1] that every such equivalence relation is in fact the orbit relation of some group Γ acting by μ-preserving Borel automorphisms of X. The question of finding a freely acting Γ remained open until A. Furman’s work [Fur99b, th. D], exhibiting a lot of examples where it is impossible.

Examples 1.2 There are at least two kinds of examples – where an underlying Γ is not obvious – for introducing this generalization:

1. Let Y ⊂ X be a Borel subset which meets all orbits of R. The induced equivalence relation R_{Y} := R ∩ Y × Y on Y, whose classes are restrictions of classes to Y, preserves the normalized probability-measure μ_{Y} = μ/μ(Y).

2. When looking at a minimal measured lamination, choose a total transversal X of finite measure. The holonomy pseudogroup gives rise to such a general equivalence relation on X, with the normalized (to a probability measure) transverse measure. It is generated by the “return maps”. Two points of the transversal are in the same class iff they belong to the same leaf of the lamination.

1.3 Orbit Equivalence

Definition 1.3 Two such actions α_{i} of groups Γ_{i} on (X_{i}, μ_{i}), i = 1, 2 are said to be Orbit Equivalent (OE) if they define the same equivalence relation, i.e. if there exists a Borel isomorphism f : X_{1} → X_{2} such that f_{∗}(μ_{1}) = μ_{2} and for μ_{1} almost all x ∈ X_{1},

f(Γ_{1}.x) = Γ_{2}.f(x).

An orbit equivalence between free actions gives rise to a cocycle (see R. Feres’ contribution to the present volume [Fer01]) σ : Γ_{1} × X_{1} → Γ_{2}, where σ(γ, x) is the unique element λ ∈ Γ_{2} s.t. λ.x = γ.x. The cocycle identity is easily checked: σ(γ_{1}γ_{2}, x) = σ(γ_{1}, γ_{2}.x)σ(γ_{2}, x).

The coarseness of this notion may be really distressing for certain ergodic theorists: H. Dye showed that (for Γ_{1} = Γ_{2} = ℤ) any two ergodic probability measure preserving ℤ-actions are orbitally equivalent [Dye59, th. 1 p. 143 and th. 5 p. 154]. A. Vershik obtained the same result about at the same time.

Later on, in [Dye63, th. 1, p. 560], Dye showed the same result for any group Γ of polynomial growth or Γ infinite abelian: any free ergodic probability measure preserving Γ-action is orbitally equivalent to an ergodic ℤ-action. All these actions thus define THE same equivalence relation R_{a}. The sadness of those people I just mentioned is far from decreasing.

R. Zimmer [Zim78] introduced the notion for an equivalence relation to be amenable and showed, in particular, that if Γ has a free p.m.p. action which is orbitally equivalent to a ℤ-action then Γ is amenable. The natural conjecture was that any p.m.p. ergodic action of an amenable group is orbitally equivalent to THE relation R_{a}, and this was proved by Ornstein-Weiss [OW80], and by Connes-Feldman-Weiss [CFW81], in the context of general non-singular (rather than p.m.p.) amenable equivalence relations.

This p.m.p. equivalence relation R_{a} is characterized as ergodic and hyperfinite: it is an increasing union of equivalence relations with finite classes.

Recall that abelian, nilpotent, solvable groups are amenable groups. By contrast, the free groups F_{n}, n ≥ 2 are not, as well as any group containing F_{2}, and it’s not easy to produce a non amenable group not containing F_{2}.

1.4 Stable Orbit Equivalence

It’s important to realize that when the classes are infinite (in all non trivial situations), the equivalence relation doesn’t admit any Borel fundamental domain, i.e. a Borel subset D ⊂ X that meets once and only once the orbit of μ-almost all x ∈ X. In words it is not possible to measurably pick one point in each orbit, or the space of orbits is ugly. This is obvious in the free actions context, where the iterates γ.D of a hypothetical fundamental domain would furnish a partition of almost all of X (finite measure) with infinitely many equal-measure pieces. And it is almost that obvious in general.

Nevertheless, the following notion gives a sense to the idea of two equivalence relations R_{1} and R_{2} having the “same space of orbits”: First, if a Borel subset Y ⊂ X meets almost all orbits of R, the induced equivalence relation R_{Y} (example 1.2.(1)) would have the same space of orbits as R. Now comes the:

Definition 1.4 Two equivalence relations R_{1} and R_{2} on (X_{i}, μ_{i}), i = 1, 2 are said to be Stably Orbit Equivalent (SOE) if there exist Borel subsets Y_{i} ⊂ X_{i}, i = 1, 2 which meet almost all orbits of R_{i} on which the induced equivalence relations are the same: R_{1Y_{1}} ≃ R_{2Y_{2}}, by an isomorphism f : Y_{1} → Y_{2} which preserves the normalized measures f_{∗}(μ_{1}/μ_{1}(Y_{1})) = μ_{2}/μ_{2}(Y_{2}). The number i(R_{1}, R_{2}) = μ_{2}(Y_{2})/μ_{1}(Y_{1}) is called the index of R_{2} in R_{1} (given by this SOE).

In example 1.2.(2), choosing various transversal leads to SOE equivalence relations. Notice that for a given R, the set of possible i(R, R) forms a subgroup of ℝ_{+}^{∗}, called its fundamental group.

Remark 1.5 Why is it called stable? The countable stabilization of an equivalence relation (X, μ, R) is the equivalence relation R′ on the space (X × C, μ′), where C is any infinite discrete countable set and μ′ = μ×counting measure, defined by (x_{1}, c_{1}) ∼ (x_{2}, c_{2}) iff (x_{1}, x_{2}) ∈ R. It can be shown that SOE equivalence relations R_{1} and R_{2} have isomorphic countable stabilizations. More precisely, there exists a Borel isomorphism X_{1} × C_{1} →^{f} X_{2} × C_{2} sending μ′_{1} to λ · μ′_{2}, with λ^{−1} = i(R_{1}, R_{2}), and sending almost every R′_{1}-class onto an R′_{2}-class.

We refer to [FM77a], [Moo82], [Sch87] and [Zim84] for more material on this section.

2 Measure Equivalence

Stable orbit equivalence must be related to another notion introduced by M. Gromov and developed by A. Furman [Fur99a]: Measure Equivalence (ME) between countable groups which is a measure theoretical analog of quasi-isometry.

Criterion for quasi-isometry. [Gro93, 0.2.C′_{2}] Two finitely generated groups Γ_{1} and Γ_{2} are quasi-isometric (QI) iff there exist commuting, continuous actions of Γ_{1} and Γ_{2} on some locally compact space M, such that the action of each of the groups is properly discontinuous and has a compact fundamental domain.

Similarly:

Definition 2.1 [Gro93, 0.5.E] Two countable groups Γ_{1} and Γ_{2} are Measurably Equivalent (ME) iff there exist commuting, measure preserving, free actions of Γ_{1} and Γ_{2} on some Lebesgue measure space (Ω, m) such that the action of each of the groups admits a finite measure fundamental domain (D_{i} for Γ_{i}). In this case, we say that (Ω, m) (together with the actions) is a coupling between Γ_{1} and Γ_{2} and that Γ_{1} is ME to Γ_{2} with index i_{Ω}(Γ_{1}, Γ_{2}) = m(D_{2})/m(D_{1}).

Examples 2.2

(1) The basic example of QI groups are cocompact lattices in the same locally compact second countable group G (applying the criterion to M = G and Γ_{1} and Γ_{2} acting by left and right multiplication).

(2) The basic example of ME groups are general lattices Γ_{1} and Γ_{2} in the same locally compact second countable group G. Existence of a lattice (= a discrete, finite covolume subgroup) forces G to be unimodular (The modulus λ(g) of an element g ∈ G is the number by which the left Haar measure is multiplied when g acts from the right on G as well as on Γ_{1}\G): Haar measure is invariant under the commuting actions by left and right multiplication of Γ_{1} and Γ_{2}.

Thus, (G, Haar) gives a measure equivalence between Γ_{1} and Γ_{2} which are ME with index i_{G}(Γ_{1}, Γ_{2}) = Vol(G/Γ_{2})/Vol(Γ_{1}\G).

(3) Notice that the definition also makes sense when Γ_{1} ≃ Γ_{2} and that the values i_{Ω}(Γ, Γ) for various couplings may a priori assume any number in ℝ_{+}^{∗}, as shown by the example of Γ_{1} = ℤ, Γ_{2} = αℤ acting on ℝ by translations which (produces the value i_{ℝ}(ℤ, ℤ) = |α|) (see sect. 2.2).

2.1 Comparison SOE-ME

ME is an equivalence relation on countable groups and is strongly related to SOE (as was already noticed by A. Furman [Fur99b]).

Theorem 2.3 Groups Γ_{1} and Γ_{2} are measurably equivalent (with index i) iff they admit SOE free actions (with index i).

Proof. Suppose, that the free actions R_{Γ_{i}}, i = 1, 2 are in fact orbit equivalent (via a measure preserving isomorphism f : X_{1} → X_{2}, def. 1.3). Then one gets an identification of the equivalence relations R_{Γ_{1}} = R_{Γ_{2}}, as subsets of X_{1} × X_{2}. The product free action on X_{1} × X_{2} of the group Γ_{1} × Γ_{2} (by (γ_{1}, γ_{2}).(x, y) = (γ_{1}.x, γ_{2}.x)) restricts to an action on R_{Γ_{1}} = R_{Γ_{2}}, which preserves the natural measure ν (cf. ex. 5.1). Each group Γ_{i} admits as finite measure fundamental domain: the diagonal of the relation {(x_{1}, x_{2}) ∈ X_{1} ×X_{2}; f(x_{1}) = x_{2}}. We have constructed the desired coupling.

Given now SOE free actions R_{Γ_{i}}, i = 1, 2. The products X_{1} × ℕ ∼ X_{2} × ℕ are isomorphic by a scaling measure isomorphism f, which allows the identification of the countable stabilizations R′_{1} = R′_{2} := R′ as a subset of (X_{1} × ℕ) × (X_{2} × ℕ) (cf. remark 1.5). The two possible measures on it are proportional: choose, say, that given by R′_{2}, denoted by ν. The same as above may now be done on the intersection R′ ∩ (X_{1} × {0}) × (X_{2} × {0}) leading to a free action of Γ_{1} × Γ_{2}. Any Borel section of (X_{2} × {0}) is a fundamental domain for the Γ_{1}-action (resp. (X_{1} × {0}) for Γ_{2}). Their respective ν-measures are 1 and i(R_{Γ_{1}}, R_{Γ_{2}}).

Conversely, start from a coupling (Ω, m) and finite measure fundamental domains D_{i} ⊂ Ω for Γ_{i}, i = 1, 2. By commutativity, one gets (extending the left-right notations of example 2.2 to the commuting actions) an action of Γ_{1} on Ω/Γ_{2} ≃ D_{2} and an action of Γ_{2} on Γ_{1}\Ω ≃ D_{1}.

These actions are SOE: The corresponding equivalence relations R_{Γ_{1}} and R_{Γ_{2}} have countable stabilizations on Ω = Ω/Γ_{2} × Γ_{2} and on Ω = Γ_{1}\Ω × Γ_{1} that both identify to the orbit equivalence relation of the (Γ_{1} × Γ_{2})-action on Ω.

Recall that the measure must be normalized on Ω/Γ_{2} and Γ_{1}\Ω. The SOE map (f = id : Ω → Ω) sends \frac{m}{m(D_{2})} to λ · \frac{m}{m(D_{1})}, where λ = \frac{m(D_{1})}{m(D_{2})} = i_{Ω}^{−1}. The SOE index i(R_{Γ_{1}}, R_{Γ_{2}}) thus equals i_{Ω} (cf. remark 1.5).

If the actions are not free, just choose any free probability measure preserving action of Γ_{1} on a standard Borel space (X, μ), let Γ_{2} act trivially (γ_{2}x = x) and replace (Ω, m) by (Ω × X, m × μ) with the diagonal Γ_{1} × Γ_{2}-action, which turns out to be free. This new coupling leads to free actions. ■

2.2 Values of indices

Remark that two groups Γ_{1} and Γ_{2} are commensurable (i.e. there exists a group Λ which is isomorphic to a finite index subgroup in both of them) iff they are ME for a countable coupling Ω (with counting measure) and then i_{Ω}(Γ_{1}, Γ_{2}) = [Γ_{1} : Λ]/[Γ_{2} : Λ]. Given Γ, the set of values {i_{Ω}(Γ, Γ)} is denoted by I_{ME}(Γ) or I_{com}(Γ), according to whether one considers all possible couplings Ω, or only countable couplings. One easily checks the inclusions ℚ_{+}^{∗} ⊃ I_{com}(Γ_{1}, Γ_{2}) ⊂ I_{ME}(Γ_{1}, Γ_{2}) ⊂ ℝ_{+}^{∗}, and notice that I_{ME}(Γ) is a ME invariant.

The condition I_{ME}(Γ) = {1} means that for each free p.m.p. action of Γ, the induced equivalence relation (R_{Γ})_{Y} (ex. 1.2) cannot be generated by any free action of Γ, when Y is a proper (μ(Y) < μ(X)) Borel subset which meets all orbits. This implies, in particular, property:

(*) for each free p.m.p. action of Γ, the equivalence relation R_{Γ} has trivial fundamental group (i.e. for every self SOE, i(R_{Γ}, R_{Γ}) = 1).

It also implies that i_{Ω}(Γ, Γ_{1}) only depends on Γ_{1} in the ME class of Γ and not on the particular coupling Ω.

Amenable groups do not have (*) and neither do groups Γ which are ME to the direct product of an infinite amenable group by any group. In fact, I_{ME}(Γ) = ℝ_{+}^{∗}, as for ℤ. On the other hand, it follows from th. 4.1 below, that if one of its ℓ^{2}-Betti numbers is ≠ 0, ∞ (see sect. 4), then I_{ME}(Γ) = {1}. Also lattices in higher rank connected simple Lie groups satisfy I_{ME}(Γ) = {1} [GG88, Cor. B.3],[Fur99a, Fur99b].

Question: Does there exist groups Γ with an infinite discrete I_{ME}(Γ) ? For that question to make sense, one should restrict attention to ergodic situations.

2.3 Comparison QI-ME

- Amenability is both a QI and a ME invariant. In fact, Ornstein-Weiss’ result (cf. section 1.3) implies that all infinite (countable) amenable groups belong to the same ME class. This single class splits into many QI classes (distinguished for example by growth).

This could suggest that QI implies ME and that the latter in just a coarser equivalence relation. The QI classification of lattices in semi-simple Lie groups (where lattices in a given G split into several QI classes, but remain in the same ME class, ex. 2.2) also support such an idea. However it turns out to be wrong :

- A. Furman proved that Kazhdan’s property (T) is a ME invariant [Fur99a, th. 8.2]. But Kazhdan’s property (T) is not a QI invariant: If Γ has property (T) and admits a non-trivial 2-cocycle to ℤ which is bounded (many property (T) hyperbolic groups admit such cocycles, e. g. cocompact lattices in Sp(1, n)) then the direct product Γ_{1} = Γ × ℤ does not have property (T), but is QI to a non-trivial central extension of Γ by ℤ which has (T). I’m grateful to A. Furman and N. Monod for discussions on this example.

- For ME groups, the Euler characteristics are positively proportional (th. 4.1 below). This is not the case for QI: By forming the free product Γ∗F_{p} of a group Γ with various free groups (p ≥ 2), one gets QI groups (Gromov hyperbolic if Γ was). This is because the F_{p} are in fact Lipschitz equivalent (a theorem due to P. Papasoglu; see P. De la Harpe’s book [Har00, IV-B.46] for further results and references). Now, if χ(Γ) ≥ 2, the Euler characteristics χ(Γ ∗ F_{p}) = χ(Γ) − p do not all have the same sign. In fact, th. 4.1 states that ME groups have proportional ℓ^{2} Betti numbers β_{n} while P. Pansu showed that for QI groups Γ_{1} and Γ_{2}, one has β_{n}(Γ_{1}) = 0 iff β_{n}(Γ_{2}) = 0.

To complete the picture, let’s come back to lattices in Lie groups :

- A. Furman, improving R. Zimmer’s super-rigidity for cocycles, showed [Fur99a, th. 3.2] that for a higher rank simple Lie group G, the collection of all its lattices (up to finite groups) forms a single ME class. More precisely: Let Γ be a lattice in a simple, connected Lie group G with finite center and ℝ-rank ≥ 2. If a countable group Λ is measure equivalent to Γ, then Λ has a finite index subgroup which maps with finite kernel onto a lattice of Ad(G).

- The ME class of the free group F_{2} on two generators, in addition to all finitely generated (non cyclic) free groups and compact surface fundamental groups, contains all free products of a finite number of amenable groups (but ℤ/2ℤ ∗ ℤ/2ℤ).

3 Cost of an Equivalence Relation

3.1 Graphings

Given a group Γ, there is an associated number n(Γ), the minimal number of generators of Γ. The cost of an equivalence relation is a similar quantity. It was first introduced by G. Levitt [Lev95].

One way to produce an equivalence relation is by considering a graphing, i.e. a countable family Φ = (φ_{i} : A_{i} → B_{i})_{i∈I} of μ-preserving partial isomorphisms between Borel subsets A_{i}, B_{i} ∈ X. A graphing generates a μ-preserving standard countable Borel equivalence relation R_{Φ}: the smallest equivalence relation that satisfies

x ∼ φ_{i}(x) for all i ∈ I and all x ∈ A_{i}.

In other words, (x, y) ∈ R_{ϕ} iff there exists a Φ^{±1}-word (i.e. a word in the φ_{i}’s and their inverses) whose associated partial isomorphism is defined at x and sends x to y.

The cost of the generating system Φ is the number of elements in Φ, weighted by the measure of their domain: C(Φ) = ∑_{i∈I} μ(A_{i}).

The cost of the equivalence relation is the infimum of the costs over all the generating graphings: C(R) := inf{C(Φ) : R_{Φ} = R}. This is by definition an invariant of the equivalence relation (of OE), and the difficulty is now the computation. If all classes are infinite, then C(R) ≥ 1. The costs of SOE equivalence relations R_{1} and R_{2} are related as follows : μ_{2}(Y_{2})(C(R_{1}) − 1) = μ_{1}(Y_{1})(C(R_{2}) − 1), where μ_{1}, μ_{2} are probability measures on X_{1}, X_{2} ([Gab00a, Cor. II.12.]).

A generating graphing Φ associates in a Borel varying way a Cayley graph like structure to each orbit (this explains the terminology). Precisely, for each x in X, its orbit under R is the vertex set of a graph Φ_{x}, where two points are neighbors if one of the φ_{i} sends one to the other. The generating condition implies connectedness of these graphs. Graphings appeared in a paper of S. Adams [Ada90].

Examples 3.1

1. For a free action α of a group Γ, given a generating family (γ_{1}, γ_{2}, ⋯) of Γ, one has a particular graphing Φ where each φ_{i} is the automorphism α(γ_{i}), defined on all of X. Each orbit graph for that graphing is then isomorphic to the Cayley graph of Γ.

2. Ornstein-Weiss’ theorem (cf. section1.3) implies that almost all orbits of a free action of an infinite amenable group can be given a line structure and thus the corresponding orbit relation has cost 1.

A treeing is a graphing where the graphs associated to almost all x ∈ X are trees. If Φ is not a treeing, there is a Φ^{±1}-word with a non trivial fixed point set. One can then remove a subset from the domain of one of its letters (opening cycles in the graphs Φ_{x}), in order to define a new generating graphing with smaller cost (when finite). Conversely:

Theorem 3.2 [Gab98], [Gab00a, th. 2] If Φ is a treeing of the equivalence relation R, then it realizes the infimum of the costs: C(R) = C(Φ).

While the general result involving free products with amalgamation over amenable groups can be found in [Gab00a, th. 2, th. IV.15], the reader interested in the proof of that restricted statement is advised to look at the simplified proof in the announcement [Gab98].

As a corollary, any free action of the free group F_{n} has cost n (see ex. 3.1 with a basis of F_{n} as generating set), thus free groups of different ranks cannot have orbit equivalent actions. Now comes the natural: Question: How many

non orbitally equivalent p.m.p. free actions does the free group F_{2} admit?

Only finitely many of them are known (three following a personnal communication of S. Popa). Note that S. Gefter and V. Golodets [GG88, cor.A.9, p. 843] showed that non cocompact lattices in a simple, connected Lie group with finite center and ℝ-rank ≥ 2, have a continuum of non OE free p.m.p. ergodic actions.

Costs of actions are computed for a lot of groups Γ in [Gab00a]. For example, If Γ is an amalgamated product A ∗_{C} B of finite groups, then cost= 1 − (1/|A| + 1/|B| − 1/|C|) (it is always treeable); if Γ is the fundamental group π_{1}(S_{g}) of an orientable genus g compact surface then cost= 2g − 1.

Question: Does there exist a group with two actions of different costs?

3.2 Non treeability

But not every equivalence relation admits a treeing. Countable groups with Kazhdan’s property (T) are not tree-friendly. For example: Any simplicial action of such a group on a tree has to fix a point. Adams and Spatzier [AS90, th. 1.8, lem.2.1] showed: Probability measure preserving ergodic free actions of (infinite) Kazhdan property (T) groups don’t have any treeing. The analogy between these two result will be made more transparent in example 6.4, section 6.

The difficulty in showing that the cost is a non trivial invariant comes from the fact that a lot of groups have only cost 1 actions (like direct products of infinite groups, see [Gab00a, part. VI-D] for other examples). Thus, the following statement delivers a lot of groups with no treeable free action: The only groups which admit both a cost 1 and a treeable free action are amenable [Gab00a, cor.VI.22].

Question: Does the surface group π_{1}(S_{g}) admit a non treeable action?

3.3 An application

The theory of graphings is useful for a classification of equivalence relations. It can also be applied to group theory in order to show a result similar to Schreier’s theorem for normal subgroups of the free group:

Theorem 3.3 [Gab01] If the first ℓ^{2} Betti number of Γ is not zero, then any finitely generated normal subgroup N of Γ is either finite or has finite index.

Notice that finite generation may be replaced by β_{1}(N) finite and that W. Lück got this result, under an additional hypothesis on the quotient Γ/N, like containing an infinite order element or arbitrarily large finite groups [Lüc94] and [Lüc98b, th. 3.3]. Let’s show the following weaker statement:

Theorem 3.4 If Γ has no cost 1 free action, then any finitely generated normal subgroup of Γ is either finite or has finite index.

Proof. Consider an extension 1 → N → Γ →^{f} Q → 1. Suppose N and Q are infinite and N is finitely generated; let’s show that Γ has a cost 1 free action (by exhibiting generating graphings of small costs).

Consider a free probability measure preserving ergodic action of Γ (resp. Q) on X (resp. Y), then the diagonal Γ-action γ.(x, y) = (γ.x, f(γ).y) on Z = X×Y is free and preserves the product measure. It generates an equivalence relation R = R_{Γ}. The fibers of the projection π : Z → Y are preserved by N.

Fix ϵ > 0. The relation R_{Q}, as any equivalence relation whose classes are infinite, contains a hyperfinite (see section 1.3) subrelation S, whose classes are all infinite. The latter is generated by an automorphism t of Y and Y admits a partition Y = ∐_{q∈Q} Y_{q} where x ∈ Y_{q} iff q(x) = t(x). Put Z_{q} = π^{−1}(Y_{q}), choose a pullback γ_{q} ∈ f^{−1}(q) and denote by φ_{q} the (partial isomorphism of Z) restriction of γ_{q} to Z_{q}. As the Z_{q}’s form a partition of Z, the cost of Φ_{1} := (φ_{q}) equals 1.

Now let A ⊂ Z be a π-saturated Borel subset of measure ϵ and denote by φ′_{j} the restriction of n_{j} to A where n_{1}, ⋯, n_{r} is a generating set for N. The graphing Φ_{2} := (φ′_{j})_{j=1,···,r} generates the restriction of R_{N} to A and its cost equals (rϵ).

Claim: The relation generated by Φ_{1} ∪ Φ_{2} contains R_{N}. For each (x, y) ∈ Z there exists a Φ_{1}^{±1}-word m_{1} (defined at and) sending (x, y) into A. Let ω be the corresponding element of Γ. If n ∈ N, the point n.(x, y) lies in the same π-fiber, so that ω is also defined at it. By normality, ωnω^{−1} belongs to N, thus ω(x, y) and ωn(x, y) are connected by a Φ_{2}^{±1}-word m_{2}. The word m_{1}^{−1}m_{2}m_{1} connects (x, y) to n.(x, y).

Choose a series of positive numbers (η_{γ})_{γ∈Γ} with sum ≤ ϵ and Borel subsets A_{γ} ⊂ Z of measure ≤ η_{γ} which meet every N-orbit – since N is infinite, this is possible; one could also have supposed that the N-action is ergodic on X. Denote by φ_{γ} the restriction of γ to A_{γ}. The graphing Φ_{3} = (φ_{γ})_{γ∈Γ} has cost ≤ ϵ.

Claim: Φ_{1} ∪ Φ_{2} ∪ Φ_{3} generates R_{Γ}. For each (x, y) ∈ Z, there is an element of N sending it into A_{γ}, and also a (Φ_{1} ∪ Φ_{2})^{±1}-word m. By normality, γ.(x, y) and φ_{γ}m(x, y) are N-equivalent, thus they are connected by a (Φ_{1} ∪Φ_{2})^{±1}-word m′. Now m′φ_{γ}m connects (x, y) to γ.(x, y).

Since Φ_{1} ∪ Φ_{2} ∪ Φ_{3} has cost 1 + rϵ + ϵ, one concludes that R_{Γ} has cost 1. ■

4 ℓ^{2} Betti numbers for groups

To each countable group Γ is associated a sequence of numbers ∈ [0, ∞] called its ℓ^{2} Betti numbers (β_{n}(Γ))_{n∈ℕ} that are defined using the ℓ^{2} chains of CWcomplexes on which Γ acts. They were defined in this generality in [CG86].

Theorem 4.1 [Gab01] If Γ_{1} and Γ_{2} are measurably equivalent, then they have proportional ℓ^{2} Betti numbers.

More precisely if (Ω, m) is a coupling between them with index i_{Ω}(Γ_{1}, Γ_{2}), then for all n ∈ ℕ, one has β_{n}(Γ_{2}) = i_{Ω}(Γ_{1}, Γ_{2}) · β_{n}(Γ_{1}).

In particular, ME groups have positively proportional Euler characteristics (when defined) χ(Γ) = χ^{(2)}(Γ) = ∑_{n}(−1)^{n}β_{n}(Γ).

Corollary 4.2 Lattices in the same locally compact second countable group have ℓ^{2} Betti numbers that are proportional as the covolumes are.

4.1 ℓ^{2} homology

In this section, we give some indications about ℓ^{2} homology and ℓ^{2} Betti numbers for cocompact free actions on simplicial complexes. We refer to W. Lück’s and B. Eckmann’s surveys ([Lüc98], [Eck00]) for a general exposition of these ideas, or to Lück’s forthcoming book [Lück].

Let K be a simplicial complex on which Γ acts freely and simplicially. The space of n-chains is the free ℤ-module C_{n}(K, ℤ) with the family S_{n} of n-simplices as a basis. It is the space of (finite) formal integer linear combinations of elements of S_{n}. The boundary map ∂_{n} sends an n-simplex to the obvious n−1-chain (One has at some point to order simplices, let’s forget about it) and extends by linearity.

If now S_{n} is considered as a Hilbert orthonormal base, one gets the space of ℓ^{2} n-chains C_{n}^{(2)}(K), i.e. the space of (infinite) formal linear combinations of elements of S_{n} with square summable coefficients.

Once a representative is chosen in each orbit of S_{n}, there are natural Γequivariant identifications of S_{n} with α_{n} copies of Γ (where α_{n} is the number of Γ-orbits in S_{n}), inducing C_{n}(K, ℤ) ≃ ⊕_{i=1}^{α_{n}}ℤΓ and C_{n}^{(2)}(K) ≃ ⊕_{i=1}^{α_{n}}ℓ^{2}(Γ) where Γ acts by left translations, and the latter is a Hilbert sum. The spaces of chains thus get a Γ-module structure.

The ∂_{n} , defined on the Hilbert basis, requires some finiteness condition to extend to a well defined and bounded (=continuous) operator on ℓ^{2} chains.

For (counter)-example suppose a vertex v is the endpoint of infinitely many edges e_{1}, e_{2}, ⋯ of K. The chain ∑_{j∈ℕ} \frac{1}{j}e_{j} is ℓ^{2} (i.e. belongs to C_{1}^{(2)}(K)) since ∑_{j∈ℕ} \frac{1}{j^{2}} converges, but by the boundary operator ∂_{1}, the vertex v would like to receive the coefficient ∑_{j∈ℕ} \frac{1}{j} !! More precisely, the sequence ∂_{1}(∑_{j=1}^{p} \frac{1}{j}e_{j}) is not bounded when p → ∞ since, when expressed in the vertices Hilbert base, its v-coefficient is ∑_{j=1}^{p} \frac{1}{j}.

If K is cocompact, then the boundary maps extend to bounded operators still denoted ∂_{n}. They give a chain complex (∂ ∘ ∂ = 0)

(∗)  0 ←^{∂_{0}} C_{0}^{(2)}(K) ←^{∂_{1}} C_{1}^{(2)}(K) ←^{∂_{2}} C_{2}^{(2)}(K) ←^{∂_{3}} C_{3}^{(2)}(K) ←^{∂} ⋯

As usual, consider its homology H_{n}^{(2)}(K) := Ker ∂_{n}/Im ∂_{n+1}. If one wants to keep dealing with Hilbert spaces one has to divide out by a closed subspace, i.e. to consider the closure \overline{Im ∂_{n+1}} of Im ∂_{n+1}, thus considering the so called reduced ℓ^{2} homology \overline{H}_{n}^{(2)}(K) := Ker ∂_{n}/\overline{Im ∂_{n+1}}.

4.2 ℓ^{2} Betti numbers

In fact, all of this is Γ-equivariant, so that the reduced ℓ^{2} homology has the additional structure of a Hilbert Γ-module: It is a Hilbert space, with a Γ representation and it admits an isometric equivariant embedding into an orthogonal sum L_{r} = ⊕_{i=1}^{r}ℓ^{2}(Γ) (where Γ acts by λ_{r}, the sum of its left regular representations λ on each ℓ^{2}(Γ)). Namely, \overline{H}_{n}^{(2)}(K) embeds into C_{n}^{(2)}(K) ≃ ⊕_{i=1}^{α_{n}}ℓ^{2}(Γ) as the space H_{n}(K) of harmonic n-chains: the orthocomplement of \overline{Im ∂_{n+1}} in Ker ∂_{n}.

It thus gets a Γ-dimension (see section 4.4), in some sense a dimension modulo the Γ-action (cf. items 1–2, section 4.4).

The ℓ^{2} Betti numbers for the action of Γ on K are by definition:

β_{n}(K, Γ) := dim_{Γ} \overline{H}_{n}^{(2)}(K).

These numbers were first introduced by M. Atiyah in [Ati76], and Cheeger-Gromov have extended the notion to not necessarily cocompact simplicial actions [CG86], and even in a much more general context (singular ℓ^{2} cohomology). We mention a very nice alternative theory developed by W. Lück ([Lüc98a], [Lüc98b]).

The ℓ^{2} Betti numbers of the group Γ are those of any free Γ-action on a contractible simplicial complex EΓ:

β_{n}(Γ) = β_{n}(EΓ, Γ),

they are shown to depend only on Γ [CG86].

4.3 Example

Let’s consider the example of the free group Γ = F_{n} acting on the regular tree T which is the universal cover of a bouquet of n circles (built of one vertex and n oriented edges e_{1}, e_{2}, ⋯, e_{n}). The tree is the Cayley graph of F_{n} for a generating basis. Once a vertex \overline{x} and an edge \overline{e}_{i} in each of the n orbits of edges are chosen, C_{0}(T, ℤ) = ℤΓ.\overline{x} and C_{1}(T, ℤ) = ⊕_{i=1}^{n}ℤΓ.\overline{e}_{i} give a chain complex 0 ← ℤΓ ←^{∂_{1}} ⊕_{i=1}^{n}ℤΓ ← 0 where ∂_{1} is injective and not surjective (the image 0-chains are those for which the sum of the coefficients vanishes), leading to a 1-dimensional H_{0} and trivial H_{1} which is not very interesting (just related to the fact that T is contractible). To get something interesting one has to “divide out first” by Γ.

By contrast, at the ℓ^{2} level and for n ≥ 2, in the associated chain complex 0 ← ℓ^{2}(Γ) ←^{∂_{1}} ⊕_{i=1}^{n}ℓ^{2}(Γ) ← 0, the operator ∂_{1} becomes, surjective (as Kesten’s theorem shows, since Γ is non amenable) and far from injective since dim_{Γ} Ker ∂_{1} = dim_{Γ} ⊕_{i=1}^{n}ℓ^{2}(Γ)−dim_{Γ} \overline{Im ∂_{1}} = n−1 (item 7, section 4.4). Thus \overline{H}_{0}^{(2)}(T) = 0 while \overline{H}_{1}^{(2)}(T) ≃ ⊕_{i=1}^{n−1}ℓ^{2}(Γ). Taking the von Neumann dimension amounts in some sense to “divide out afterwards”.

An example of a harmonic 1-chain is obtained in the most obvious manner: put the coefficient +1 on some oriented edge e = [a, b]. Tail a and head b of e receives −1, resp. +1. Compensate this by putting the coefficient (2n − 1)^{−1} on each edge with head a, resp. with tail b, and so on, put (2n − 1)^{−d} on edges at distance d oriented towards a, resp. from b. Since n ≥ 2, this is an ℓ^{2} chain.

4.4 Von Neumann dimension

Hilbert Γ-modules M possess a well defined generalized dimension dim_{Γ} M. It enjoys the following properties:

1. dim_{Γ} ℓ^{2}(Γ) = 1

2. if Γ is finite, dim_{Γ} M = \frac{1}{|Γ|} dim M (usual vector space dimension)

3. dim_{Γ} M ≥ 0

4. dim_{Γ} M = 0 ⟺ M = 0

5. M ⊂ N ⇒ dim_{Γ} M ≤ dim_{Γ} N

6. dim_{Γ} M ⊕ N = dim_{Γ} M + dim_{Γ} N

7. dim_{Γ} Ker f +dim_{Γ} \overline{Im f} = dim_{Γ} M_{1}, when f : M_{1} → M_{2} is a Γ-equivariant bounded operator between Hilbert Γ-modules

The Γ-dimension is defined as follows: consider first a closed Γ-invariant subspace M of ℓ^{2}(Γ) = L_{1} and let p be the orthogonal projection onto M. By Γ-invariance, the operator p belongs to the von Neumann algebra of the group Γ, the algebra N(Γ) of those operators that commute with all the unitary operators λ(γ), γ ∈ Γ. The crucial feature is the existence of a finite trace τ on N(Γ): for a ∈ N(Γ), τ(a) = ⟨a(δ_{e})|δ_{e}⟩ (where δ_{e} is the characteristic function of the identity element of Γ). Now the von Neumann dimension of M is the trace of its projector dim_{Γ} M = τ(p).

If M is a closed Γ-invariant subspace of L_{r}, its projection belongs to the commuting algebra of λ_{r}(Γ) and admits a bloc decomposition as an r ×r matrix with coefficients p_{i,j} in N(Γ). By definition dim_{Γ} M = ∑_{i=1}^{r} τ(p_{i,i}).

The trace property (τ(ab) = τ(ba)) ensures that any two Γ-equivariant isometric embeddings in L_{r} of a Hilbert Γ-module have indeed the same dimension, giving rise to the well defined notion of Γ-dimension for such a module. This definition has a natural extension when r = ∞.

5 Simplicial Actions of an Equivalence Relation

5.1 Fibered spaces

5.1.1 Fundamental example.

Consider a free measure preserving action of a countable group Γ on (X, μ) and the orbit equivalence relation R_{Γ}. Consider moreover an honest free simplicial action of Γ on a countable simplicial complex K. The space Σ_{K} := X × K is equipped with the diagonal action of Γ: γ.(x, τ) = (γ.x, γ.τ). It is fibered over X.

Choose a fundamental domain D^{(0)} := {v_{1}, v_{2}, ⋯, } (= a set of Γ-orbit representatives) for the Γ-action on the 0-skeleton K^{(0)} of K. The set X × D^{(0)} is a fundamental domain for the diagonal action on X × K^{(0)} which permits us to identify the latter with countably many copies of R (one for each vertex v_{j} ∈ D^{(0)}) by Θ_{j} : (x, γ.v_{j}) ↦ (x, γ^{−1}.x). Notice that the Θ_{j} are equivariant when R is given the Γ-action on the left coordinates: Θ_{j}(γ′.(x, γ.v_{j})) = (γ′.x, γ^{−1}.x) = γ′.Θ_{j}(x, γ.v_{j}).

An analogous treatment can be made in each dimension, by choosing a fundamental domain D^{(n)} for the Γ-action on the set of n-simplices K^{(n)} of K. This permits us to identify the space X × K^{(n)} with countably many copies of R. Notice that an n-simplex in X × K^{(n)} is made of an (n + 1)-tuple of points in X × K^{(0)}, with the same projection on X.

5.1.2 Fibered spaces.

A standard Borel fibered space over X is a standard Borel space U together with a specified Borel map (projection map), with countable fibers p : U → X.

The natural measure ν_{U} on U is defined as μ (the measure on X) times counting measure in the fibers of p. Thus, the measure of a Borel subset V ⊂ U is obtained by integrating over (X, μ) the function x ↦ number of points in the intersection of V with the fiber of x – Alternatively, ν_{U} is built by considering any countable Borel partition U = ∐ U_{i} such that the restrictions of p to each U_{i} are injective and by putting on each U_{i} the pull back of μ.

When U and V are standard Borel fibered spaces over X, via projection maps p and q, their fibered product

U ∗ V = {(u, v) ∈ U × V/ p(u) = q(v)}

is a standard Borel fibered space over X.

Example 5.1 The equivalence relation R has two natural fiberings over X, given by the projection maps p_{l} : (x, y) ↦ x and p_{r} : (x, y) ↦ y. – the r in p_{r} stands for right while p_{l} is the range map and p_{r} is the source map for the groupoid R –. Due to the invariance of μ for R, the natural measures defined by these two fiberings coincide; just denote them by ν.

5.2 Groupoid actions

5.2.1 Space with Standard Left R-Action

A standard left R-space or space with standard left R-action consists of a (standard Borel) fibered space U over X and a map called the action map defined on the fibered product, where R fibers via p_{r},

R ∗ U → U, ((y, z), u) ↦ (y, z).u,

such that (x, y).[(y, z).u] = (x, z).u and (z, z).u = u. – In particular, z = p(u) and y = p((y, z).u) –. The space X itself is a standard left R-space. The projection map p of a left R-space is R-equivariant: p((y, z).u) = (y, z).p(u). The orbit of u is the set R.u := {(y, z).u : (y, z) ∈ R, y = p(u)} and the saturation R.B of a Borel subset B ⊂ U is the union of the orbits that meet B.

5.2.2 Discrete Actions

A standard left R-space U is discrete if the action admits a Borel fundamental domain D, i.e. a Borel subset D ⊂ U that meets once and only once the orbit of ν_{U}-almost all u ∈ U.

Example 5.2 An obvious instance of such a left R-space is the fibered space (U, p) = (R, p_{l}) itself, with the action map R ∗ U → U, ((x, y), (y, z)) ↦ (x, z) and the diagonal of R = U as a fundamental domain.

Given a discrete standard left R-space U, choose any countable Borel partition of a fundamental domain D = ∐_{i∈I} D_{i}, such that p restricts to bijections: D_{i} →^{p} p(D_{i}) ⊂ X, on each D_{i}. The natural identification of D_{i} with the diagonal subset Δ_{i} = {(z, z) : z ∈ p(D_{i})} ⊂ R extends by R-equivariance to an identification of the saturation R.D_{i} with the saturation R.Δ_{i} = p_{r}^{−1}(p(D_{i})) One thus gets an isomorphism of discrete standard left R-spaces between U and the disjoint union ∐_{i∈I} R.Δ_{i}.

If U is a discrete left R-space (with fundamental domain D), then also U ∗ U ∗ ⋯ ∗ U is a discrete left R-space (with f.d. D ∗ U ∗ ⋯ ∗ U).

6 Actions of the equivalence relation on a simplicial complex

Definition 6.1 A simplicial complex with standard left R-action or more briefly an R-simplicial complex Σ consists of the following data:

– a discrete left R-space Σ^{(0)} →^{p} X (space of vertices)

– for each n ∈ ℕ a Borel subset Σ^{(n)} ⊂ \underbrace{Σ^{(0)} ∗ ⋯ ∗ Σ^{(0)}}_{n+1 times} (space of ordered nsimplexes)(perhaps empty for large n’s) satisfying four conditions

1. (permutations) Σ^{(n)} is invariant under permutation of the coordinates

2. (non degeneracy) if (v_{0}, v_{1}, ⋯, v_{n}) ∈ Σ^{(n)}, then v_{0} ≠ v_{1}

3. (boundary condition) if (v_{0}, ⋯, v_{n}) ∈ Σ^{(n)}, then (v_{1}, ⋯, v_{n}) ∈ Σ^{(n−1)}

4. (invariance) R.Σ^{(n)} = Σ^{(n)}

The data in the fiber of each x ∈ X is just an ordinary (countable) simplicial complex, denoted by Σ_{x}. Notice that the first two conditions could be slightly modified according to your favorite definition of a simplicial complex.

The R-simplicial complex Σ is n-connected, resp. contractible, resp. ndimensional if for almost all x in X, the simplicial complex Σ_{x} has the corresponding property.

Examples 6.2

1. The basic and motivating example for this is the fundamental example in 5.1.1, where each Σ_{x} is identified with a copy of K.

2. A graphing Φ = (φ_{i})_{i∈I} of R (cf. 3.1), with no loops or double edges, gives a 1-dimensional connected R-simplicial complex: As left R-spaces (cf. ex. 5.2) Σ^{(0)} = R and Σ^{(1)} = {((x, y), (x, z)) ∈ R ∗ R : y = φ_{i}^{±1}(z) for some i ∈ I}.

3. The complete structure where each Σ^{(n)} = R ∗ ⋯ ∗ R with all diagonals removed (for non degeneracy) is a contractible infinite dimensional example.

Definition 6.3 The geometric dimension of an equivalence relation R is the smallest dimension of a contractible R-simplicial complex.

The ergodic dimension of a countable group Γ is the smallest dimension of the equivalence relations R_{Γ} produced by free actions of Γ.

Example 6.4 Adams and Spatzier’s result already mentioned (section 3.2) about non treeability for ergodic free actions of Kazhdan’s property (T) groups Γ can now be rephrased: R_{Γ} doesn’t admit any 1-dimensional contractible R_{Γ}- simplicial complex and also Kazhdan’s property (T) groups have ergodic dimension > 1.

One easily concludes that measure equivalent groups have the same ergodic dimension and that geometric (resp. ergodic) dimensions decrease when passing to subrelations (resp. subgroups).

7 ℓ^{2} Betti Numbers for Equivalence Relations and their Actions

7.1 ℓ^{2} homology of Σ

An R-simplicial complex Σ defines a Borel field of simplicial complexes x ↦ Σ_{x} and, for each n ∈ ℕ, a Borel field of Hilbert spaces x ↦ C_{n}^{(2)}(Σ_{x}).

What is a Borel vector field? Notice that a Borel section s of the fibering Σ^{(n)} → X leads to a C_{n}(Σ_{x})-valued vector field x ↦ s(x). A vector field x ↦ σ(x) ∈ C_{n}^{(2)}(Σ_{x}) is Borel if x ↦ ⟨σ(x)|s(x)⟩_{x} is a Borel function for every Borel section s.

The Hilbert integral of the ℓ^{2} n-chains of Σ_{x} is called the space of ndimensional ℓ^{2} chains of Σ and denoted by:

C_{n}^{(2)}(Σ) := ∫_{X}^{⊕} C_{n}^{(2)}(Σ_{x})dμ(x).

It is the Hilbert space of those Borel vector fields that are square integrable: x ↦ ||σ(x)||_{x} ∈ L^{2}(X, μ).

If the Σ_{x} are uniformly locally bounded, then the fields of the boundary operators lead by integration to bounded operators ∂_{n} : C_{n}^{(2)}(Σ) → C_{n−1}^{(2)}(Σ)

still satisfying ∂ ∘ ∂ = 0. The reduced homology \overline{H}_{n}^{(2)}(Σ) of the associated chain complex is called the reduced ℓ^{2} homology of Σ. Like in the group case (section 4), these spaces have additional structure: they are Hilbert modules – The von Neumann algebra M involved is that of the equivalence relation R (see section 7.2) – giving them a von Neumann dimension, “dimension modulo the groupoid action”, which is called ℓ^{2} Betti number of Σ and denoted:

β_{n}(Σ, R) := dim_{M} \overline{H}_{n}^{(2)}(Σ).

By using some ideas from [CG86], this definition is extended from uniformly locally bounded Σ’s to general Σ’s (see [Gab01]). A major difficulty arises here from the fact that operators appearing in homotopy equivalences don’t lead to bounded operators (see [Gab00b]). The next result must be compared to the notion of foliation Betti numbers of A. Connes [Co82, p. 549] and to the claim in [Gro93, 8.A_{4}, p. 233] where the hypothesis forces boundedness of certain operators.

Theorem 7.1 [Gab01] Any two contractible R-simplicial complexes have the same ℓ^{2} Betti numbers, called the ℓ^{2} Betti numbers of the equivalence relation R and denoted by β_{n}(R). If R is defined by a free measure preserving action of a countable group Γ, then for all n ∈ ℕ, β_{n}(R) = β_{n}(Γ).

If β_{p}(R) > 0 then the geometric dimension of R is ≥ p. One deduce the existence of equivalence relations of any geometric dimension: consider a free action of F_{2} × F_{2} × ⋯ × F_{2}, p times, to get a relation of dimension exactly p.

By the Morse inequalities, one gets a relation between ℓ^{2} Betti numbers and cost: cost(R) − 1 ≥ β_{1}(R) − β_{0}(R) with no known example of strict inequality.

Question: Is it always the case, that cost(R) − 1 = β_{1}(R) − β_{0}(R)?

7.2 Von Neumann algebra of an equivalence relation

Consider the Hilbert space L^{2}(R, ν) arising from the Borel space R with its measure ν (see ex. 5.1).

For each partial isomorphism φ : A → B whose graph is contained in R (see section 1.2) denote by L_{φ} the operator of L^{2}(R, ν) defined by L_{φ}(η(x, y)) = η(φ^{−1}(x), y) if x ∈ B and 0 otherwise. The von Neumann algebra M of R is the algebra of those bounded operators that commute with the family L of all the L_{φ}’s (see [FM77b] and [Moo82]). Notice that adjoining to L the L_{ψ}’s defined for ψ ∈ L^{∞}(X, μ) by L_{ψ}(η(x, y)) = ψ(x)η(x, y) wouldn’t change M.

If R = R_{Γ}, then by the observation in section 1.2, one could replace L by the L_{γ}’s, γ ∈ Γ, together with the L_{ψ}’s. In the free action case, M is described as a “ cross product” of L^{∞}(X, μ) with Γ (see [MvN36, part IV, p. 192-209]).

The algebra M possesses a trace τ(a) = ⟨a(χ_{Δ})|χ_{Δ}⟩, where χ_{Δ} is the characteristic function of the diagonal of R. This permits us to define, as in section 4.4, the von Neumann dimension of closed L-invariant subspaces of ⊕L^{2}(R, ν).

The embedding of C_{n}^{(2)}(Σ) into a sum of L^{2}(R, ν) is obtained from the identification at the end of section 5.2.2.

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