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L^2-Betti numbers of locally compact groups and their cross section equivalence relations

We prove that the L^2-Betti numbers of a unimodular locally compact group G coincide, up to a natural scaling constant, with the L^2-Betti numbers of the countable equivalence relation induced on a cross section of any essentially free ergodic probability measure preserving action of G. As a consequence, we obtain that the reduced and un-reduced L^2-Betti numbers of G agree and that the L^2-Betti numbers of a lattice Gamma in G equal those of G up to scaling by the covolume of Gamma in G. We also deduce several vanishing results, including the vanishing of the reduced L^2-cohomology for amenable locally compact groups.


Introduction
The theory of L 2 -Betti numbers, as well as related notions of L 2 -invariants, provides a set of powerful invariants in geometry, topology and group theory, which are computable in many interesting cases. In [At76], Atiyah introduced L 2 -Betti numbers for free cocompact group actions on manifolds. This was generalized by Connes [Co79] to a set of invariants of measured foliations. For arbitrary countable groups Γ, the L 2 -Betti numbers β n (2) (Γ), n ∈ N, were defined by Cheeger and Gromov in [CG85]. Gaboriau, in [Ga01], defined the L 2 -Betti numbers β n (2) (R) of an arbitrary countable probability measure preserving (pmp) equivalence relation. As a consequence, the L 2 -Betti numbers of a measured foliation with contractible leaves only depend on the associated equivalence relation. Furthermore, Gaboriau proves that β n (2) (Γ) = β n (2) (R Γ ) for every countable group Γ with an essentially free ergodic pmp action Γ (X, µ) and orbit equivalence relation R Γ . So L 2 -Betti numbers are invariant under orbit equivalence and, as also shown in [Ga01], scale under measure equivalence of groups by the compression constant of the measure equivalence.
L 2 -Betti numbers have been generalized further to a variety of different settings, see [Sa03,CS04,Ky06], and we refer to [Lü02] for an extensive monograph on the subject.
In all cases, L 2 -Betti numbers are defined as the Murray-von Neumann dimension of certain (co)homology modules with coefficients in the group von Neumann algebra LΓ, or the von Neumann algebra LR of a countable probability measure preserving (pmp) equivalence relation. By the work of Lück [Lü97], one can define the dimension of an arbitrary (purely algebraic) module over a tracial von Neumann algebra (M, τ ) and this provides a reinterpretation of the L 2 -Betti numbers of a countable group Γ by means of the formula β n (2) (Γ) = dim LΓ H n (Γ, ℓ 2 (Γ)) . Lück's dimension theory was extended to von Neumann algebras equipped with semifinite traces in [Pe11,Appendix B] (see also Appendix A in this article for details on dimension theory). Hence L 2 -Betti numbers of unimodular locally compact second countable (lcsc) groups could be defined in [Pe11,Section 3.1] by the formula β n (2) (G) = dim LG H n (G, L 2 (G)) .
This definition was motivated in part by the following two well-known facts for discrete groups. In [Pe11, Theorems 4.8 and 5.9], such a result was proved for cocompact lattices and also in the case when G is totally disconnected. One of our main results is to prove (1.1) in its full generality.
Our method is based on an observation, following [Fo74], that in a measurable sense every unimodular lcsc group G admits a cocompact lattice. More precisely, for every essentially free ergodic pmp action G (X, µ), there exists a cocompact cross section Y ⊂ X (see Sections 1.2 and 4.1 for terminology). This implies that the restriction of the orbit equivalence relation of G X to Y is a countable pmp equivalence relation R and that there exists a compact subset K ⊂ G such that K · Y is conegligible in X. Our main theorem says that the L 2 -Betti numbers β n (2) (G) of G are proportional to the L 2 -Betti numbers β n (2) (R) of the equivalence relation R, in the sense of Gaboriau [Ga01]. The proportion between the two is given by a natural constant that we call the covolume of Y .
We can then reduce several questions about L 2 -Betti numbers of G to known results for L 2 -Betti numbers of countable pmp equivalence relations. In this way, we prove that the reduced and unreduced L 2 -Betti numbers of G coincide and we establish several vanishing results. This includes the vanishing of all L 2 -Betti numbers and of the reduced cohomology groups H n (G, L 2 (G)) whenever G admits a noncompact amenable closed normal subgroup, in particular when G is noncompact and amenable. This extends a well-known result of Cheeger-Gromov [CG85] for countable groups. For connected amenable groups the vanishing of reduced L 2 -cohomology in degree one was proved essentially by Delorme in [De77] (see also [Ma04]). In contrast to the proof of Delorme, our more general result follows directly from the vanishing of L 2 -Betti numbers for the (unique) amenable ergodic II 1 equivalence relation.

Notation and standing assumptions
In what follows, all topological groups are implicitly assumed to be Hausdorff and we will use the abbreviation lcsc for 'locally compact second countable'. A nonsingular action of a lcsc group G on a standard measure space (X, µ) is an action of G on the set X such that the map G × X → X : (g, x) → g · x is Borel and such that µ(g · A) = 0 whenever A ⊂ X is a Borel set of measure zero. We say that the action is pmp (probability measure preserving) if µ is a probability measure and µ(g · A) = µ(A) for all g ∈ G and all Borel sets A ⊂ X.
When G (X, µ) is a nonsingular action, one can show that the set of points x ∈ X having a trivial stabilizer is a Borel set (see e.g. [MRV11, Lemma 10] for a proof of this well known result). If this Borel set is conegligible, we say that the action is essentially free. For later use, we recall the following.
Remark 1.1. Every lcsc group G admits an essentially free ergodic (even mixing) pmp action G (X, µ). Indeed, it suffices to denote by (X 0 , µ 0 ) the Gaussian probability space that corresponds to the real Hilbert space L 2 R (G). The Gaussian action G (X 0 , µ 0 ) is pmp and faithful. Since the Koopman representation on L 2 (X 0 , µ 0 ) ⊖ C1 is a multiple of the regular representation of G, the action G (X 0 , µ 0 ) is mixing. The diagonal action of G on the infinite direct product (X, µ) = (X 0 , µ 0 ) N is then essentially free, mixing and pmp (see [AEG93, Proposition 1.2] for details).

Statement of the main results
Let G be a lcsc group and G (X, µ) an essentially free pmp action. We call a Borel set Y ⊂ X a cross section of G (X, µ) if there exists a neighborhood of the identity U ⊂ G such that the map θ : U × Y → X : (g, y) → g · y is injective and such that µ(X − G · Y ) = 0.
We recall the following classical results and refer to Section 4.1 for a more detailed explanation and proofs. Every essentially free pmp action admits a cross section. Then R := {(y, y ′ ) ∈ Y × Y | y ∈ G · y ′ } is a countable Borel equivalence relation on Y , which is called the cross section equivalence relation. Assume that G is unimodular and fix a Haar measure λ on G. Then Y is equipped with a unique R-invariant probability measure ν satisfying θ * (λ × ν) = covol Y · µ |U ·Y , for some positive scaling factor covol Y .
Our main result relates the L 2 -Betti numbers of G to those of R by means of the following theorem. The precise definition for the L 2 -Betti numbers of G, resp. R, is given in Sections 2 and 3. Theorem A. Let G be a lcsc unimodular group and G (X, µ) an essentially free ergodic pmp action. For every cross section Y ⊂ X with corresponding cross section equivalence relation R and for every n ∈ N, we have where Φ(gH, h) = θ(gH)h and θ : G/H → G is any Borel cross section. We denote covol H := λ G/H (G/H).
Theorem B. Let G be a lcsc unimodular group and H < G a closed unimodular subgroup of finite covolume. Given fixed Haar measures on G and H, we have In particular, if Γ is a lattice in the lcsc group G, then β n (2) (G) = covol(Γ) −1 β (2) n (Γ) for all n 0.
For the following result, note that a closed normal subgroup of a lcsc unimodular group is again unimodular.
Theorem C. Let G be a lcsc unimodular group.
2. If G is noncompact and amenable, then β n (2) (G) = 0 for all n 0. 3. Let G be a lcsc unimodular group and H ⊳ G a closed normal subgroup. If d 0 and β n (2) (H) = 0 for all 0 n d, then β n (2) (G) = 0 for all 0 n d.
In Proposition 4.6 we will show that β 1 (2) (G) < ∞ whenever G is compactly generated, in particular when G is connected. We therefore obtain the following corollary.
Corollary D. Let G be a lcsc unimodular group. If G admits a closed normal subgroup H such that H is compactly generated and such that both H and G/H are noncompact, then β 1 (2) (G) = 0.
The following corollary, which follows directly from Theorem B and Theorem C gives an alternative approach to the vanishing of L 2 -Betti numbers in [BFS12, Remark 1.9].
Corollary E. Let G be a lcsc unimodular group. If G admits a noncompact, amenable, closed, normal subgroup then β n (2) (G) = 0 for all n 0, whence in particular the L 2 -Betti numbers vanish for any lattice in G.
Notice that by 2.10 below, the vanishing of the n-th L 2 -Betti number β n (2) (G) is equivalent with the vanishing of the n-th reduced cohomology group H n (G, L 2 (G)). So the vanishing results C-D-E can also be viewed as vanishing results for reduced cohomology groups.
Finally note that a combination of Corollary E with the structure theory of lcsc groups and the Künneth formula allows in principle to reduce all computations of L 2 -Betti numbers β n (2) (G) to computations where G is totally disconnected. We refer to [Pe11, Chapter 7] for details.

Cohomology and L 2 -Betti numbers of locally compact groups
In this section we fix the definitions of cohomology and L 2 -Betti numbers that we will use for locally compact groups.
2.1. L 2 -Betti numbers of locally compact unimodular groups Let G be a lcsc group, P an algebra and H a Fréchet space. Denote by S(H) the set of continuous seminorms on H.
• We call H a left Fréchet G-module if H is equipped with a left action of G by linear maps such that G × H → H : (g, ξ) → g · ξ is continuous.
• We call H a right Fréchet P -module if H is a right P -module and if for every a ∈ P , the map H → H : ξ → ξ · a is continuous.
• We call H a Fréchet G-P -bimodule if H is both a left Fréchet G-module and a right Fréchet P -module and if the two actions commute.
• If X is a lcsc space, then the vector space C(X, H) of continuous functions from X to H is again a Fréchet space, using the seminorms If H is a right Fréchet P -module, then C(X, H) naturally is a right Fréchet P -module.
By a complex of Fréchet spaces we mean a sequence C : H 0 −→ · · · such that each H n is a Fréchet space and the maps d n : H n → H n+1 are continuous and satisfy d n+1 • d n = 0 for all n 0.
For n 0 the n'th cohomology of C, respectively the n'th reduced cohomology of C, are defined as If the H n are Fréchet P -modules and the maps d n are P -linear, we call C a complex of Fréchet P -modules. Then, H n (C) and H n (C) are P -modules, with the latter being a Fréchet P -module.
When the H n are Fréchet G-P -bimodules and the maps d n are G-P -linear, we call C a complex of Fréchet G-P -bimodules.
Further, H n (G, H) is defined as the n-th reduced cohomology group of the complex (2.1) of Fréchet P -modules. Also H n (G, H) is a right P -module.
Following [Pe11], we now define the L 2 -Betti numbers, β n (2) (G), of a lcsc unimodular group G as the Murray-von Neumann dimension of the cohomology groups H n (G, L 2 (G)). Recall that the group von Neumann algebra LG of a lcsc group G is defined as the von Neumann algebra generated by the left regular representation of G on L 2 (G). If G is unimodular with a fixed Haar measure λ, the von Neumann algebra LG is equipped with a natural semifinite trace. It is the unique normal semifinite faithful trace Tr on LG satisfying for every continuous compactly supported function f : G → C. Note that L 2 (G) naturally is an LG-LG-bimodule, using the left and the right regular representations.
Whenever (N, Tr) is a von Neumann algebra equipped with a normal semifinite faithful trace, one can define the dimension dim N H of an arbitrary N -module H, see Definition A.14 in Appendix A on dimension theory.
Definition 2.2. Let G be a lcsc unimodular group. We define for all n 0, When G is discrete this definition agrees with the standard definitions, see e.g. [PT07, Section 2].
The main purpose of this article is to prove that, up to a natural rescaling, the L 2 -Betti numbers β n (2) (G) are equal to the L 2 -Betti numbers of the cross section equivalence relation associated with an arbitrary free ergodic pmp action of G (see Theorem A). As a byproduct, we get that β n (2) (G) = β n (2) (G) for all lcsc unimodular groups G. This equality was already shown in [Pe11, Theorem 5.6] whenever G is totally disconnected or admits a cocompact lattice.

Basic cohomology theory for locally compact groups
To identify H n (G, L 2 (G)) with a cohomology theory of the associated cross section equivalence relations, we need some basic tools from homological algebra. The cohomology theory for locally compact groups defined by continuous cochains was first considered by Mostow in [Mo61]. Standard references are the monographs [BW80,Gu80]. For the convenience of the reader, we list the needed properties in the rest of this section. Definition 2.3. Let G be a lcsc group and P an algebra.

A complex of right Fréchet
It is called strongly exact if there exist continuous P -linear maps S n : H n → H n−1 , for all n 1, such that S n+1 • d n + d n−1 • S n = id Hn for all n 1 .

A Fréchet
is strongly exact. Here H G denotes the Fréchet P -submodule of H that consists of the G-fixed points in H, and the coboundary maps d n , n 0, are the ones given in (2.2).
The following proposition is standard (cf. [Bl77, Proposition 2.9]) and we leave the proof as an exercise.
Proposition 2.4. Let G be a lcsc group, P an algebra and H a Fréchet G-P -bimodule. Assume that is a complex of Fréchet G-P -bimodules that is strongly exact as a complex of Fréchet P -modules. Assume that for all n 0, the Fréchet G-P -bimodule H n is strongly acyclic. Let C be the complex of Fréchet P -modules given by Then there are natural P -linear isomorphisms H n (G, H) ∼ = H n (C) and H n (G, H) ∼ = H n (C) for all n 0 .
We will apply Proposition 2.4 to G-P -bimodules H n of the form H n = L 2 loc (G, K n ). So we first recall some Fréchet-valued integration theory.
Let (Z, η) be a standard Borel space equipped with a σ-finite measure. Let H be a Fréchet space. Denote by S(H) the set of continuous seminorms on H. For every 1 p < +∞, we denote by L p (Z, H) the space of functions f : Z → H that are strongly Borel (in the sense that inverse images of open sets are Borel sets) and that have the property that q • f ∈ L p (Z) for every q ∈ S(H), where we implicitly identify functions that are equal a.e. The family of seminorms f → q • f p , q ∈ S(H), turns L p (Z, H) into a Fréchet space. All f ∈ L 1 (Z, H) have an integral Z f dη in H and the map f → Z f dη is a continuous linear map from L 1 (Z, H) to H.
In order to define locally square integrable functions, assume that the standard σ-finite measure space (Z, η) comes with an increasing sequence of Borel sets Z n ⊂ Z such that n Z n has complement of measure zero. We define L p loc (Z, H) as the space of strongly Borel functions f : Z → H such that f |Zn belongs to L p (Z n , H) for every n. Using the seminorms f → (q • f ) |Zn p , for all q ∈ S(H) and all n, we turn L p loc (Z, H) into a Fréchet space. Observe that if P is an algebra and H is a right Fréchet P -module, then also L p loc (Z, H) is a right Fréchet P -module.
Note that L p loc (Z, H) only depends on the choice of the sequence (Z n ) up to cofinality: if (Z ′ n ) is another increasing sequence of Borel sets whose union has complement of measure zero, and if for every n, there exists an m such that Z n ⊂ Z ′ m and Z ′ n ⊂ Z m , then the Fréchet spaces L p loc (Z, H) defined w.r.t. (Z n ) and (Z ′ n ) coincide. If G is a lcsc group, we always define L p loc (G, H) with respect to the Haar measure on G and an increasing sequence of compact subsets K n ⊂ G such that the interiors of K n cover G. Note that two such increasing sequences are always cofinal, so that L p loc (G, H) is unambiguously defined. Note that we have the continuous inclusion C(G, H) ⊂ L p loc (G, H). Also the following lemma is standard and we omit the proof.
Lemma 2.5. Let G be a lcsc group, P an algebra and K a Fréchet G-P -bimodule. Define the Fréchet G-P -bimodule H := L 2 loc (G, K) with the left G-action and right P -action given by Then the complexes of Fréchet P -modules given by with d n defined by (2.2), are strongly exact. In particular, H is a strongly acyclic G-P -bimodule.
As a consequence, we recover the fact proven in [Bl77] that cohomology for G may be computed by using locally square integrable functions.
Proposition 2.6 ( [Bl77]). Let G be a lcsc group, P an algebra and K a Fréchet G-P -bimodule.
Define a complex C of Fréchet P -modules given by with d n defined by (2.2). Then the inclusion maps C(G n , K) → L 2 loc (G n , K) induce P -linear isomorphisms H n (G, K) ∼ = H n (C) and H n (G, K) ∼ = H n (C) .
Proof. This is an immediate corollary of Proposition 2.4 and Lemma 2.5.
Note that Proposition 2.6 has the following slightly unexpected consequence: if a continuous ncocycle ω : G n → K is approximately inner in the L 2 loc -topology, then it must be approximately inner in the stronger topology of uniform convergence on compact sets. Assume now that Tr is a normal semifinite faithful trace on M and that N ⊂ M is a von Neumann subalgebra such that Tr |N is semifinite. Equip N with the trace given by restricting Tr to N . Denote by E : M → N the unique Tr-preserving conditional expectation. Whenever H is a right Hilbert N -module, the Connes tensor product H ⊗ N L 2 (M ) is defined as the separation-completion of H ⊗ alg M with respect to the scalar product

Change of coefficients : a dimension formula
The formula (ξ ⊗ a) · b = ξ ⊗ ab turns H ⊗ N L 2 (M ) into a right Hilbert M -module. If H was a Hilbert G-N -bimodule, the formula g · (ξ ⊗ a) = (g · ξ) ⊗ a turns H ⊗ N L 2 (M ) into a Hilbert G-M -bimodule. The following proposition is analogous to [Lü02, Theorem 6.29].
Proposition 2.7. Let G be a lcsc group and (M, Tr) a von Neumann algebra with separable predual equipped with a normal semifinite faithful trace. Let N ⊂ M be a von Neumann subalgebra such that Tr |N is semifinite. Equip N with the trace given by restricting Tr to N .
For every Hilbert G-N -module H, we have In order to prove Proposition 2.7, we need two elementary lemmas.
Proof. Denote by P : H → K the orthogonal projection of H onto K and note that P is Nlinear. Put Q = 1 − P . Then P ⊗ N 1 is the orthogonal projection of H ⊗ N L 2 (M ) onto K ⊗ N L 2 (M ). Assume that ξ ∈ H ⊗ N L 2 (M ) and that E(ξ · a) ∈ K for all a ∈ M . It follows that for all a ∈ M . Hence (Q ⊗ N 1)(ξ) = 0, so that ξ ∈ K ⊗ N L 2 (M ).
Lemma 2.9. Let (M, Tr) be a von Neumann algebra with separable predual equipped with a normal semifinite faithful trace. Let N ⊂ M be a von Neumann subalgebra such that Tr |N is semifinite. Denote by E : M → N the unique Tr-preserving conditional expectation.
Let H be an N -module and let K be an M -module. Assume that θ : H → K and E : K → H are N -linear maps satisfying Proof. We first prove the lemma when Tr is a tracial state. Assume that p ∈ M k (C) ⊗ N is a projection and ϕ : p(M k,1 (C) ⊗ N ) → H is an injective N -linear map. We construct an injective M -linear map p(M k,1 (C) ⊗ M ) → K. The inequality dim N H dim M K then follows directly from (A.1).
A direct computation yields that ξp = ξ and that ϕ(η) = ξη for all η ∈ p(M k,1 (C) ⊗ N ). Define By construction, ψ is M -linear. We claim that ψ is injective. So assume that η ∈ p(M k,1 (C) ⊗ M ) and that ψ(η) = 0. Then also Since E is faithful, we conclude that η = 0. So ψ is injective and the lemma is proven in the case where Tr is a finite trace.
In the general case, choose an increasing sequence of projections p n ∈ N with Tr(p n ) < ∞ for all n and with the central support of p n in N converging to 1 strongly. Applying the previous case to the p n N p n -module Hp n and the p n M p n -module Kp n , we conclude that for all n. Taking the limit n → ∞ and using Lemma A.16, the lemma follows.
We are now ready to prove Proposition 2.7.
Proof of Proposition 2.7. Denote K := H ⊗ N L 2 (M ) and denote by E : M → N the unique Tr-preserving conditional expectation. The map The maps θ and E naturally induce N -linear maps satisfying the assumptions of Lemma 2.9. So it follows from Lemma 2.9 that It remains to consider the reduced cohomologies. We first make the following observation : whenever L is a Hilbert N -module, we have is the canonical trace on B(ℓ 2 (N)).
Using Lemma 2.8, we get that K G = H G ⊗ N L 2 (M ). In combination with (2.4), we get that To prove the same formula for the reduced n-cohomology, n 1, consider the complexes where d n is defined by (2.2). Fix n 1 and define We similarly define Z n (K) and B n (K). By Proposition 2.6, we have .
Fix an increasing sequence of compact subsets K k ⊂ G whose interiors cover G. Denote by ϕ k : L 2 loc (G n , H) → L 2 (K n k , H) the restriction map. We define Z n k (H) as the closure of ϕ k (Z n (H)), and we define B n k (H) as the closure of ϕ k (B n (H)). By construction, we have N -linear maps Also by construction, Ker ϕ k is a decreasing sequence of N -submodules with trivial intersection. It then follows from Lemma A.17 that . (2.5) We similarly have that .
To conclude the proof of the proposition, we identify the Hilbert M -modules Once (2.6) is proven, the proposition follows by using (2.4).
To prove (2.6), note that L 2 (K n k , H) = L 2 (K n k ) ⊗ H, so that we can identify We therefore get inclusions and it remains to prove that these inclusions are actually equalities. To prove this, we use Lemma 2.8 and denote by E : L 2 (K n k , K) → L 2 (K n k , H) the orthogonal projection. Fix ξ ∈ Z n k (K) and fix a ∈ M . We must show that E(ξ · a) ∈ Z n k (H). The definition of Z n k (K) provides a sequence ω i ∈ Z n (G, K) such that ξ = lim i ϕ k (ω i ). We also have E : Z n (G, K) → Z n (G, H) and get that E(ξ · a) = lim i ϕ k (E(ω i · a)) .
Since E(ω i · a) is a sequence in Z n (G, H), we indeed get that E(ξ · a) ∈ Z n k (H). This proves that the first inclusion in (2.7) actually is an equality. We similarly get that the second inclusion in (2.7) is an equality. The required identification (2.6) follows and the proposition is proven.
As a consequence of the above proof we obtain the following result, which also appears in [Pe11, Proposition 3.8].
Porism 2.10. For any lcsc unimodular group G and any n 0 we have β n (2) (G) = 0 if and only if H n (G, L 2 (G)) vanishes.
Proof. Choosing H := L 2 (G) in Proposition 2.7 and its proof, we see that the modules appearing in (2.5) are actually Hilbert L(G)-modules. Since the dimension function is faithful on the class of Hilbert LG-modules, if we assume that β n (2) (G) = 0 this forces for each k 0. At the same time, the kernels of the maps ϕ k :

Cohomology of countable equivalence relations
Fix a countable Borel pmp equivalence relation R on the standard probability space (Y, ν).

Denote by [[R]
] the full pseudogroup of R, i.e. the set of all partial Borel bijections ψ with domain D(ψ) ⊂ Y and range R(ψ) ⊂ Y , such that (y, ψ(y)) ∈ R for all y ∈ D(ψ). We write R (0) := Y and R (n) := {(y 0 , . . . , y n ) | (y i , y j ) ∈ R for all i, j}. All R (n) are equipped with the natural σ-finite measure ν (n) , with ν (0) = ν and with ν (n) given by integrating w.r.t. ν the counting measure over the projection R (n) → Y onto any of the coordinates.
The von Neumann algebra LR of the equivalence relation R is defined as the von Neumann algebra acting on L 2 (R (1) , ν (1) ) generated by the partial isometries u ϕ , ϕ ∈ [[R]], given by The unit vector χ ∈ L 2 (R (1) , ν (1) ) given by χ(y, z) = 1 if y = z and χ(y, z) = 0 if y = z implements a faithful normal tracial state τ on LR satisfying . We refer to [FM75] for the details of the construction of LR.
We can identify L 2 (LR, τ ) with L 2 (R, ν (1) ) and under this identification, the right action of LR on L 2 (R, ν (1) ) is given by For later use, we write in this section a concrete complex of Fréchet LR-modules such that the LR-dimensions of the cohomology modules precisely are the L 2 -Betti numbers of R, as defined in [Ga01]. Enumerate Λ = k Λ k as an increasing sequence of finite subsets with id ∈ Λ 0 , and define Σ (n) := {(y 0 , . . . , y n , z) ∈ R (n+1) | y i = y j whenever i = j} . (3.1) We then consider the increasing sequence of subsets Σ and equip Σ (n) with the σ-finite measure given by restricting ν (n+1) . We consider the Fréchet spaces L 2 loc-Σ (Σ (n) ) where we use the notation L 2 loc-Σ to stress that we take functions that are square integrable on all the subsets Σ (n) k . Note that for n = 0, we just obtain L 2 (R), because In this way, L 2 loc-Σ (Σ (n) ) becomes a right Fréchet LR-module. We denote by β Proposition 3.1. Consider the complex C of right Fréchet LR-modules given by where d n is given by Then Proof. Using the same formulae as for d n , we also have, for every k ∈ N, the complexes of finitely generated Hilbert LR-modules given by We write C n := L 2 loc-Σ (Σ (n) ) and C n k := L 2 (Σ (n) k ). By definition of L 2 loc-Σ , the Fréchet LRmodule C n is the inverse limit of the Hilbert LR-modules C n k . Denote By construction, the Fréchet LR-module Z n is the inverse limit of the finitely generated Hilbert LR-modules Z n k . By Proposition A.13, we have For every bounded operator T between two Hilbert spaces, we denote by ImT the closure of the image of T . So, using Proposition A.12, we get that . (3.4) It remains to prove that the expression in (3.4) equals β (2) n (R). Whenever K is a closed subspace of a Hilbert space, denote by P K the orthogonal projection onto K. Denote by π α,β : C n β → C n α the LR-linear operator given by restricting functions on Σ For every LR-linear operator T between Hilbert LR-modules, we know that Therefore, the LR-dimension of (3.5) equals the LR-dimension of . (3.7) Denote by ∇ n (α, β) the LR-dimension of the Hilbert LR-module in (3.7). We have shown that Equipped with the projection π : Σ (n) → Y : π(y 0 , . . . , y n , z) = z and the action of R on the last variable of Σ (n) , we get that Σ is an R-simplicial complex in the sense of [Ga01, Définition 2.6]. Since Σ is n-connected for all n, it follows from [Ga01, Définition 3.14] that β Definition 4.1. Let G be a lcsc group, (X, µ) a standard probability space and G (X, µ) an essentially free nonsingular action.
Note that the injectivity of U × Y → X implies that the map G × Y → X : (g, y) → g · y is countable-to-one and hence maps Borel sets to Borel sets.
The following theorem was proven in [Fo74, Proposition 2.10], although the cocompactness was not studied there. Since it is crucial for us to have cocompact cross sections, we give a detailed proof.
Theorem 4.2 ([Fo74, Proposition 2.10]). Every essentially free nonsingular action of a lcsc group G on a standard probability space admits a cocompact cross section.
Proof. Fix a lcsc group G, a standard probability space (X, µ) and an essentially free nonsingular action G (X, µ). Fix a compact neighborhood K 0 of e in G and put K 1 := K −1 0 K 0 . We start by proving the following claim.
Step 1. If W ⊂ X is a nonnegligible Borel subset of X, there exists a Borel subset Y ⊂ W such that the map K 0 × Y → X : (k, y) → k · y is injective and has nonnegligible image.
Proof of step 1. By [Va62, Theorem 3.2], there exists a compact metric space (P, d) and a continuous action G P by homeomorphisms such that we can view X as a G-invariant Borel subset of P . We extend µ to a measure on P by putting µ(P − X) = 0. Put L := K 1 K 1 − interior(K 1 ). Then L is compact and e ∈ L. Since G acts continuously on P and since L is compact, we can define the continuous function Since G (X, µ) is essentially free, we see that δ(x) > 0 for a.e. x ∈ X. So we can take ε > 0 such that the set satisfies µ(W 1 ) > 0. Take a closed ball B ⊂ P with diameter smaller than ε such that µ(B ∩ W 1 ) > 0. Denote by X 0 the conegligible G-invariant Borel set of all x ∈ X that have trivial stabilizer. So also µ(B ∩ W 1 ∩ X 0 ) > 0. By regularity of µ (see e.g. [Ke95,Theorem 17.10]), take a compact subset P 1 ⊂ B ∩ W 1 ∩ X 0 with µ(P 1 ) > 0. Define the compact set S ⊂ P 1 × P 1 given by It is clear that (x, x) ∈ S for all x ∈ P 1 and that (y, x) ∈ S if and only if (x, y) ∈ S, because K 1 = K −1 1 . But S is also transitive: if (x, y) ∈ S and (y, z) ∈ S, then (x, z) ∈ S. Indeed, take r, s ∈ K 1 such that y = r · x and z = s · y. Then sr ∈ K 1 K 1 and (sr) · x = z. Since x and z both belong to the ball B with diameter ε, we get that d((sr) · x, x) ε. Since x ∈ W 1 , we know that δ(x) 2ε. So we must have that sr ∈ interior(K 1 ), and hence (x, z) ∈ S.
It follows that S is an equivalence relation on the compact metric space P 1 . The S-orbit of x ∈ P 1 is given by K 1 · x ∩ P 1 and hence, S has closed orbits. So by [Ta79,Theorem A.15], S admits a fundamental domain: we can choose a Borel subset Y ⊂ P 1 that meets every S-orbit exactly once. By construction, we have Y ⊂ P 1 ⊂ W ∩ X 0 . We can see as follows that Y satisfies all the conditions in the claim.
It follows that (y, z) ∈ S and hence y = z, because y and z belong to the fundamental domain Y of S. Since y has trivial stabilizer, also s = k.
Since K 0 has a nonempty interior, we can write G = n g n K 0 for a sequence of group elements g n ∈ G.
This proves step 1.
Step 2. There exists a Borel set Z ⊂ X such that the map K 0 × Z → X : (k, y) → k · y is injective and such that K 1 · Z has complement of measure zero.
Proof of step 2. Take a maximal family of disjoint nonnegligible Borel subsets W n ⊂ X that can be written as W n = K 0 · Z n for some Borel set Z n ⊂ X and with the map K 0 × Z n → X : (k, y) → k · y being injective. Since µ is a probability measure, this family (W n ) is countable. Put Z = n Z n . Then Z is a Borel set and since the sets W n are disjoint, the map K 0 × Z → X : (k, y) → k · y is injective. We claim that X − K 1 · Z has measure zero. If not, step 1 provides us with a Borel subset Y ⊂ X − K 1 · Z such that the map K 0 × Y → X is injective and has nonnegligible image. Since Y ∩ K 1 · Z = ∅, also K 0 · Y ∩ K 0 · Z = ∅. So we could add the nonnegligible set K 0 · Y to the family (W n ), contradicting its maximality. This ends the proof of step 2.
End of the proof of Theorem 4.2. Since G (X, µ) is essentially free, we start by discarding a G-invariant Borel set of measure zero so that G X becomes a free action. By step 2, take a Borel set Z ⊂ X such that the map K 0 × Z → X : (k, y) → k · y is injective and such that K 1 · Z has complement of measure zero. Put W := K 1 · Z. Since K 0 has a nonempty interior, we can choose a sequence g n ∈ G such that G = n K 0 g n . Put A = n g −1 n · W. Then A is a Borel set and µ(X − A) = 0. By [Zi84, Lemma B.8], we can choose a Borel set B ⊂ A such that µ(A − B) = 0 and such that X 0 := G · B is a Borel set. Since B ⊂ X 0 , we have µ(X − X 0 ) = 0. For every n ∈ N, we have Putting K := K 0 K 1 and taking the union over n, we get that X 0 ⊂ K · Z. Since X 0 is G-invariant, this means that X 0 = K · (Z ∩ X 0 ).
We define Y := Z ∩ X 0 . We have proven that the map K 0 × Y → X : (k, y) → k · y is injective and that K · Y = X 0 is a G-invariant Borel set with complement of measure zero. So Y is a cocompact cross section for G (X, µ).
The following proposition contains the basic properties of the cross section equivalence relation. The results are well known but not explicitly stated in the literature, so for the convenience of the reader we include a proof in Appendix B.
Proposition 4.3. Let G be a lcsc unimodular group and G (X, µ) an essentially free pmp action on a standard probability space. Let Y ⊂ X be a cross section and fix a Haar measure λ on G.

The formula
The projection on the first coordinate π ℓ : Z → X is countable-to-one. Define the measure η on Z by integrating w.r.t. µ the counting measure over the map π ℓ .
There exist a unique probability measure ν on Y and a unique 0 < covol Y < +∞ such that In particular, whenever U is a neighborhood of e in G such that θ : U × Y → X : (g, y) → g · y is injective, we have 3. The probability measure ν is R-invariant.
4. If Y ′ ⊂ X is a different cross section with corresponding equivalence relation R ′ , then there exist Borel subsets Y 0 ⊂ Y and Y ′ 0 ⊂ Y ′ and a Borel bijection α : Y 0 → Y ′ 0 satisfying the following properties.
• α is an isomorphism between the restricted equivalence relations R |Y 0 and R ′ |Y ′ 0 .
In particular, when G (X, µ) is ergodic, the equivalence relations R and R ′ are stably orbit equivalent with compression constant covol(Y )/ covol(Y ′ ). We end this section with a simple lemma which will be needed in the proof of Theorem A. ] such that for all y ∈ Y , we have Y ∩ (C · y) = F · y. Here we use the notation F · y := {ϕ(y) | ϕ ∈ F, y ∈ D(ϕ)}.
Take a neighborhood U of e in G such that θ : U × Y → X : (g, y) → g · y is injective. Take κ 1 and elements g 1 , . . . , g κ ∈ G such that For every fixed y ∈ Y , we have By the injectivity of θ, the sets in the union on the right hand side have at most one element. So, #(S ∩ π −1 ℓ ({y})) κ for all y ∈ Y and the lemma is proven.

Notation and conventions
Fix a lcsc unimodular group G and fix an essentially free ergodic pmp action G (X, µ). By Theorem 4.2 and after discarding a G-invariant Borel set of measure zero, we get that G X is a free action that admits a cocompact cross section Y ⊂ X. We fix a Haar measure λ on G and we define the cross section equivalence relation R and the probability measure ν on Y as in Proposition 4.3.
We fix a neighborhood U of e in G such that θ : U × Y → X : (g, y) → g · y is injective. We also fix a compact set K such that K · Y = X. Since K × Y → X : (g, y) → g · y is surjective and countable-to-one, we can choose a Borel right inverse x → (ρ(x), π(x)). We make this choice such that ρ(g · y) = g and π(g · y) = y for all g ∈ U and y ∈ Y . Note that by construction, ρ(x) · π(x) = x, so that π(x) ∈ G · x for all x ∈ X.
We put M := L ∞ (X)⋊ G and denote by (u g ) g∈G the canonical group of unitaries in the crossed product L ∞ (X) ⋊ G.
The von Neumann algebra M is equipped with a normal semifinite faithful trace Tr satisfying, for all continuous compactly supported functions f : G → C and a ∈ L ∞ (X), and note that the map X × G → R G : (x, g) → (x, g −1 · x) is a Borel bijection. We equip R G with the push forward of the measure µ×λ on X ×G. We can then identify the M -M -bimodule L 2 (M, Tr) with L 2 (R G ) with the left and right module action being given by

The crossed product L ∞ (X) ⋊ G is an infinite amplification of LR
We keep the notations introduced in Section 4.2. As in Proposition 4.3, define and equip Z with the σ-finite measure η given by integrating w.r.t. µ the counting measure over the map (x, y) → x. Then L 2 (Z) is a right LR-module with right action given, for ξ ∈ L 2 (Z) and ϕ ∈ [[R]], by We also define the left G-action on L 2 (Z) given by Note that L 2 (Z) becomes a Hilbert G-LR-bimodule.
Proof. Put W := U · Y and denote by p W ∈ L ∞ (X) the corresponding projection. Then L 2 (M )p W = L 2 (R G ∩ (X × W)). The map is Borel and bijective with inverse (x, y, g) → (x, g · y). Using Proposition 4.3, we get that this map is measure preserving. So we define the unitary operator Denote by ρ : p W M p W → B(L 2 (M )p W ) the * -antihomomorphism given by the right action of p W M p W . Similarly, denote by ρ : LR → B(L 2 (Z)) the * -antihomomorphism given by the right action of LR. A direct computation shows that So it follows that V ρ(p W M p W )V * ⊂ ρ(LR) ⊗ B(L 2 (U )). Denote by γ : M → B(L 2 (M )p W ) the * -homomorphism given by the left action of M . A direct computation also shows that V γ(au g )V * commutes with ρ(LR) ⊗ B(L 2 (U )) for all a ∈ L ∞ (X), g ∈ G. Since we find that V * (ρ(LR) ⊗ B(L 2 (U )))V ⊂ ρ(p W M p W ). So we have proven that The left hand side is equipped with the restriction of the trace Tr on M . The right hand side is equipped with the tensor product of the natural tracial state on LR and the semifinite trace Tr on B(L 2 (U )) that is normalized such that the trace of a minimal projection equals 1. Since M is a II ∞ factor both traces are a multiple of each other under the isomorphism Ad V . We must determine this multiple. To do so, choose a nonempty open subsets U 0 ⊂ U and choose a continuous compactly supported function f : G → C such that f (e) = 1 and gU 0 ⊂ U for all g ∈ supp f . Put W 0 := U 0 · Y . Define the element S ∈ p W M p W given by One computes that V ρ(S)V * = 1 ⊗ T , where T ∈ B(L 2 (U )) is given by and where (λ g ) g∈G denotes the left regular representation of G on L 2 (G). We have Tr(S) = µ(W 0 ) and (τ ⊗ Tr)(1 ⊗ T ) = λ(U 0 ). So, Ad V induces a * -isomorphism between ρ(p W M p W ) and ρ(LR) ⊗ B(L 2 (U )) that scales the trace with the factor covol Y .
Choose a minimal projection q ∈ B(L 2 (U )) and denote by p ∈ p W M p W the projection such that V ρ(p)V * = 1 ⊗ q. We get that Tr(p) = covol(Y ) −1 . We find the * -isomorphism φ : pM p → LR such that V ρ(a)V * = ρ(φ(a)) ⊗ q for all a ∈ pM p. The restriction of V to L 2 (M )p yields the required unitary U : L 2 (M )p → L 2 (Z).

Proof of Theorem A
It suffices to prove Theorem A for one particular choice of cross section. Indeed, if Y and Y ′ are two cross sections, with corresponding cross section equivalence relations R and R ′ , then by Proposition 4.3, R and R ′ are stably orbit equivalent with compression constant covol(Y )/ covol(Y ′ ). So, by [Ga01, Corollaire 5.6], we find that So Theorem A holds for the cross section Y if and only if it holds for the cross section Y ′ .
Therefore we can fix a cocompact cross section Y and use the notations introduced in Section 4.2. We also use the Hilbert G-LR-bimodule L 2 (Z) introduced in the beginning of Section 4.3 and we use cohomology of G with coefficients in L 2 (Z), in the sense of Definition 2.1.
Step 1. We have Using the unitary U : L 2 (M )p → L 2 (Z) and the isomorphism ϕ : pM p → LR, we get that dim pM p H n (G, L 2 (M )p) = dim LR H n (G, L 2 (Z)) and dim pM p H n (G, L 2 (M )p) = dim LR H n (G, L 2 (Z)) .
So, step 1 is proven.
Step 2. We have Proof. We fix an increasing sequence of compact subsets K k ⊂ G whose interiors cover G.
Since we have chosen a cocompact cross section, the increasing sequence of subsets of G × Σ (n) given by k was defined in (3.2). So we indeed find a G-LR-linear bijective homeomorphism D n ∼ = L 2 loc (G, C n ), with C n = L 2 loc-Σ (Σ (n) ). It then follows from Lemma 2.5 that the G-LR-bimodule D n is strongly acyclic.
Since moreover the complex in (4.4) is strongly exact, it follows from Proposition 2.4 that there are LR-linear isomorphisms H n (G, L 2 (Z)) ∼ = H n (C) and H n (G, L 2 (Z)) ∼ = H n (C) , (4.5) where C is the complex of Fréchet LR-modules given by (4.6) Using the isomorphism D n ∼ = L 2 loc (G, C n ) that we obtained in the previous paragraph, the complex in (4.6) is isomorphic with the complex that we considered in Proposition 3.1. So Proposition 3.1 gives us that dim LR H n (C) = dim LR H n (C) = β (2) n (R) . Using (4.5), step 2 is proven.
So the middle inequality must also be an equality and Theorem A is proven.

Proof of Theorem B
Fix Haar measures λ H on H and λ G on G. Fix a Borel cross section θ : G/H → G satisfying θ(eH) = e and denote by λ G/H the G-invariant measure on G/H given by (1.2). We define the 1-cocycle ω : G × G/H → H : g θ(kH) = θ(gkH) ω(g, kH) for all g, k ∈ G .
Note that ω(h, eH) = h for all h ∈ H.
Note that G (X ′ , µ ′ ) is an essentially free ergodic pmp action.
Fix a cross section Y ⊂ X for the action H (X, µ).
To prove our claim, it suffices to prove that the map So assume that g, k ∈ U ′ and y, z ∈ Y such that g · (eH, y) = k · (eH, z). Then gH = kH and by our choice of U ′ , we get that k = gh for some h ∈ U . But then y = h · z, so that y = z and h = e. Then also g = k and the required injectivity is proven.
The cross section equivalence relations on Y ′ and Y are identical. Only their canonical covolumes differ. As in Proposition 4.3, define Z ⊂ X × Y with its natural measure η and denote by ν the natural probability measure on Y . Define ν ′ on Y ′ such that ν = ν ′ under the obvious identification of Y and Y ′ . Finally define Z ′ ⊂ X ′ × Y ′ , again with its natural measure η ′ . The cross section θ induces a bijective Borel map G → G/H × H. We then have the obvious maps The first one is measure preserving, the second one scales the measure with a factor covol Y and the last one scales the measure with a factor covol(H). We conclude that Since the cross section equivalence relations on Y ′ and Y are identical, Theorem B follows from this formula. 4.6. Vanishing results : proof of Theorem C 1. If G is compact and λ is a Haar measure on G, we consider the action of G on itself, equipped with the probability measure λ(G) −1 · λ. Then {e} is a cross section. It has covolume λ(G) and the cross section equivalence relation is, obviously, the trivial equivalence relation on one point. So 1 follows.
2. Take G a lcsc unimodular amenable group that is noncompact. Take any essentially free ergodic pmp action G (X, µ) with cross section equivalence relation R. By 4.3, R is ergodic, amenable and has infinite orbits a.e. So by [CFW81], R is the orbit equivalence relation of an essentially free ergodic pmp action of Z. Then β (2) n (R) = 0 for all n 0 and 2 follows. 3 and 4. First make the following general observation. If G is a lcsc group with left Haar measure λ G , then the modular function ∆ G : G → R + * is defined such that λ G • Ad g = ∆ G (g) −1 · λ G . Whenever H ⊳ G is a closed normal subgroup, the Haar measure on the quotient group is G-invariant and therefore ∆ H = (∆ G ) |H ; cf. [BHV08, Corollary B.1.7].
In particular, if G is unimodular, also H is unimodular. Moreover, the uniqueness of the Haar measure on H allows to define the homomorphism α : G → R + * such that λ H • Ad g = α(g) −1 · λ H , and using (1.2), one deduces that (4.7) In the case where G is nonunimodular and H = Ker ∆ G , it follows that H is unimodular and that H is noncompact. Indeed if H would be compact, we have α(g) = 1 for all g ∈ G and also ∆ G/H = 1 because G/H is abelian. So (4.7) would then imply that G is unimodular.
We next prove 3 and 4 in the special case where also G/H is unimodular. So fix a unimodular lcsc group G and a closed normal subgroup H ⊳ G. Assume that G/H is unimodular and assume that H is noncompact. Note that both in 3 and 4, we assume that β 0 (2) (H) = 0, so that H is indeed noncompact because of 1. By Remark 1.1, we can choose a free mixing pmp action G (X, µ). Similarly choose a free ergodic pmp action G/H (X ′ , µ ′ ). Denote by π : G → G/H the quotient homomorphism and define the action G X × X ′ given by µ) is mixing and G/H (X ′ , µ ′ ) is ergodic, the action G (X ′′ , µ ′′ ) is ergodic as well. Since the action G (X, µ) is mixing and H is noncompact, the restricted action H (X, µ) is still ergodic.
Choose a cross section Y ⊂ X for the action H (X, µ) and denote by R the associated cross section equivalence relation. Choose a cross section Y ′ ⊂ X ′ for the action G/H (X ′ , µ ′ ) and denote by R ′ the associated cross section equivalence relation. We claim that Y ′′ := Y × Y ′ is a cross section for the action G (X ′′ , µ ′′ ). To prove this claim, choose a neighborhood U of e in H such that the action map U × Y → X is injective. Also choose a neighborhood U ′ of eH in G/H such that the action map U ′ × Y ′ → X ′ is injective. Take a neighborhood U ′′ of e in G such that π(U ) ⊂ U ′ and such that H ∩ (U ′′ ) −1 U ′′ ⊂ U . It follows that the action map U ′′ × Y ′′ → X ′′ is injective. Indeed, if g, k ∈ U ′′ and g · (x, x ′ ) = k · (y, y ′ ) for some (x, x ′ ), (y, y ′ ) in Y ′′ , we first conclude that π(g) · x ′ = π(k) · y ′ . Since π(g), π(k) ∈ U ′ , it follows that π(g) = π(k) and x ′ = y ′ . So k = gh with h ∈ H ∩ (U ′′ ) −1 U ′′ . So h ∈ U . But also x = h · y, so that x = y and h = e. This proves the injectivity of the action map U ′′ × Y ′′ → X ′′ .
To conclude the proof of the claim, we have to show that G · Y ′′ is conegligible in X ′′ . Define By the Fubini theorem, a.e. x ∈ X has the property that g −1 · x ∈ X 0 for a.e. g ∈ G. Since X 0 × Y ′ ⊂ G · Y ′′ , we conclude that a.e. x ∈ X has the property that (x, π(g) · y ′ ) = g · (g −1 · x, y ′ ) ∈ G · Y ′′ for all y ′ ∈ Y ′ and a.e. g ∈ G.
Using (4.1), it follows that a.e. x ∈ X has the property that (x, x ′ ) ∈ G · Y ′′ for a.e. x ′ ∈ X ′ . Again using the Fubini theorem, it follows that G · Y ′′ is conegligible. So we have proven the claim that Y ′′ is a cross section for the action G (X ′′ , µ ′′ ).
Then γ is a surjective homomorphism of equivalence relations and the kernel of γ is given by R × id. Moreover, since R is an ergodic equivalence relation, γ is strongly surjective in the sense of [ST07, Definition 3.7]. In [ST07, Theorems 1.3 and 1.5], it is shown that the equivalence relation version (even discrete measured groupoid version) of properties 3 and 4 holds for strongly normal subequivalence relations. So by the above construction, properties 3 and 4 hold whenever G/H is unimodular.
We finally deduce the general case. We still denote by π : G → G/H the quotient homomorphism. Denote G 0 := Ker(∆ G/H • π). Then H ⊳ G 0 is a closed normal subgroup and G 0 /H = Ker ∆ G/H is unimodular. Also G 0 ⊳ G is a closed normal subgroup and the quotient G/G 0 is abelian, hence unimodular. If β n (2) (H) = 0 for all 0 n d, we apply twice the already proven special case of property 3 and conclude that β n (2) (G) = 0 for all 0 n d. Finally assume that β n (2) (H) = 0 for all 0 n d and that β d+1 (2) (H) < ∞. Also assume that G/H is noncompact and nonunimodular. As we explained just after (4.7), it follows that G 0 /H is noncompact and unimodular. So by the already proven special case of 4, we get that β n (2) (G 0 ) = 0 for all 0 n d + 1. Then applying property 3 to the normal subgroup G 0 of G, we conclude that β n (2) (G) = 0 for all 0 n d + 1.

Proof of Corollary D
Proposition 4.6. Let G be a lcsc unimodular group and G (X, µ) an essentially free ergodic pmp action. Let Y ⊂ X be a cross section and denote by R the cross section equivalence relation. If G is compactly generated, then R has finite cost in the sense of [Ga99,Définition I.5]. In particular, β 1 (2) (G) < ∞.
Proof. Fix an essentially free ergodic pmp action G (X, µ). By [Ga99, Invariance II.2], the cost of an ergodic countable pmp equivalence relation is preserved under stable orbit equivalence. So using Proposition 4.3, it suffices to prove the proposition for a cocompact cross section Y ⊂ X. Discarding a G-invariant Borel set of measure zero, we may assume that there exists a compact subset K ⊂ G such that K · Y = X.
Take a compact subset C ⊂ G that generates G as a group. Take C such that C = C −1 and put L = K −1 CK. Since L is compact, Lemma 4.4 provides us with a finite subset F ⊂ [[R]] satisfying Y ∩ (L · y) ⊂ F · y for all y ∈ Y . We prove that F is a graphing for R, meaning that for all (y, z) ∈ R, there exist ϕ 1 , . . . , ϕ m ∈ F such that y = (ϕ m • · · · • ϕ 1 )(z). To prove this statement, fix (y, z) ∈ R. Since G is generated by C and C = C −1 , take g 1 , . . . , g m ∈ C such that y = (g m · · · g 1 ) · z. Since X = K · Y , we can take h 1 , . . . , h m−1 ∈ K such that z i := h −1 i · ((g i · · · g 1 ) · z) belongs to Y . Put h 0 = h m = e and put z 0 = z, z m = y. Finally put k i := h −1 i g i h i−1 , for all i = 1, . . . , m. By construction k i · z i−1 = z i for all i = 1, . . . , m. Since k i ∈ L and z i−1 , z i ∈ Y , we can take ϕ i ∈ F such that z i = ϕ i (z i−1 ). We have proven that y = (ϕ m • · · · • ϕ 1 )(z). So F is a graphing for R.
Since F is a graphing for R and since F is a finite set, it follows that R has finite cost.
It is now immediate to deduce Corollary D from Theorem C.
Appendix A. Some dimension theory for M -modules We need several other properties of the dimension function dim M . For the convenience of the reader, we provide detailed arguments.
Throughout this appendix, we fix a von Neumann algebra M with separable predual equipped with a faithful normal tracial state τ .

Generalities
Definition A.2. Let H be an M -module and H 0 ⊂ H an M -submodule. We say that H 0 is rank dense in H if for every x ∈ H and every ε > 0, there exists a projection p ∈ M with τ (p) > 1 − ε and xp ∈ H 0 .
We say that an M -module H is of rank zero if for every x ∈ H and every ε > 0, there exists a projection p ∈ M with τ (p) > 1 − ε and xp = 0.
Let K, H be M -modules and T : K → H an M -linear map. We call T an isomorphism in rank if Ker T is an M -module of rank zero and if Im T is rank dense in H.
Using Theorem A.1, one immediately gets the following result.
Proposition A.3. The dimension function satisfies the following properties.
Definition A.4. A Hilbert M -module H is said to be finitely generated if there exist finitely many ξ 1 , . . . , ξ n ∈ H such that ξ 1 M + · · · + ξ n M is dense in H. Equivalently, H is isomorphic with p(C n ⊗ L 2 (M )) for some projection p ∈ M n (C) ⊗ M .
Lemma A.5. Let H be a finitely generated Hilbert M -module and K ⊂ H an M -submodule. If K is dense in H, then K is rank dense in H.

Proof.
Replacing M by M n (C) ⊗ M , we may assume that H = pL 2 (M ) for some projection p ∈ M . Define P := {a ∈ pM + p | aM ⊂ K} .
Whenever ξ ∈ K and p k equals the spectral projection χ (1/k,k) (ξξ * ), we have p k ξ ∈ M and hence ξξ * p k M ⊂ K. Since ξξ * p k M = p k M , we get that p k ∈ P. If k → ∞, then p k increases to the left support projection of ξ. Further, P is closed under sums and under taking spectral projections χ (1/k,k) (a).
Using this, we first show that p can be approximated in the strong operator topology with projections from P. Since K is dense in H, we can find a sequence ξ n ∈ K such that ξ n − p 2 → 0. Denoting by q n the left support of ξ n we have p = ∨ n q n and, by what was just proven, each q n is a countable union of projections from P. It therefore suffices to show that r 1 ∨ · · · ∨ r k ∈ Proj(P) SOT whenever r 1 , . . . , r k ∈ Proj(P). But r 1 ∨ · · · ∨ r k = left support of r 1 + · · · + r k = lim m→∞ χ (1/m,m) (r 1 + · · · + r k ) ∈P , so r 1 ∨ · · · ∨ r k ∈ Proj(P) SOT as desired. We may therefore choose a sequence of projections p k ∈ P converging strongly to p.
We now prove several elementary lemmas in preparation for Proposition A.13 Lemma A.6. Let H be an M -module and K n ⊂ H n ⊂ H sequences of M -submodules. If K n is rank dense in H n for all n, then n K n is rank dense in n H n .
Proof. Take x ∈ n H n and choose ε > 0. For every n ∈ N, take a projection p n ∈ M with τ (p n ) > 1 − ε2 −n−1 such that xp n ∈ K n . Put p = n p n and note that τ (p) > 1 − ε. Then xp ∈ K n for all n, so that xp ∈ n K n .
The following lemma is a special case of [Lü02, Theorem 6.18], which is stated without proof in [Lü02]. Therefore we provide the details here. Proof. Put K := n K n . Note that dim M K n is a decreasing sequence. Denote its limit by α.
Since dim M K dim M K n for all n, we have dim M K α. We need to prove the converse inequality.
We first prove the converse inequality when H is the finitely generated Hilbert M -module p(C k ⊗ L 2 (M )). Then cl(K n ) = q n (C k ⊗ L 2 (M )), where q n ∈ p(M k (C) ⊗ M )p is a decreasing sequence of projections. By Lemma A.5, we have that dim M K n = (Tr ⊗τ )(q n ). Denote by q the strong limit of the decreasing sequence of projections q n . Then α = (Tr ⊗τ )(q). By Lemma A.5, every K n is rank dense in q n (C k ⊗ L 2 (M )). By Lemma A.6, K is rank dense in q(C k ⊗ L 2 (M )). Hence dim M K = (Tr ⊗τ )(q) = α.
We now prove the converse inequality in general. Fix ε > 0. Choose an injective M -linear map ϕ : p(C k ⊗ M ) → H such that (Tr ⊗τ )(p) > dim M H − ε. Denote H 0 = Im ϕ. By the rank theorem, we have that dim M (H/H 0 ) < ε. Again by the rank theorem, it follows that . So by the case proven in the previous paragraph, we know that The left hand side equals dim M (K ∩ H 0 ) and the right hand side equals dim M (K n ∩ H 0 ). Since we conclude that α − ε dim M K. Since this holds for every ε > 0, we are done.
Lemma A.8. Let K, H be finitely generated Hilbert M -modules and K n ⊂ K a decreasing sequence of closed M -submodules. Let T : K → H be a bounded M -linear operator. Then T ( n K n ) is dense in n cl(T (K n )).
Proof. Replacing M by matrices over M , we may assume that K = pL 2 (M ), H = qL 2 (M ) and T ∈ qM p. We have the decreasing sequence of projections p n p such that K n = p n L 2 (M ). Denote by p ∞ the strong limit of p n and note that n K n = p ∞ L 2 (M ). Define q n as the left support projection of T p n . Since the sequence p n is decreasing, also the sequence q n is decreasing and we denote its limit by q ∞ . By construction, cl(T (K n )) = q n L 2 (M ) and n cl(T (K n )) = q ∞ L 2 (M ). We must prove that the left support projection of T p ∞ equals q ∞ . Denote this left support projection by e. Clearly e q ∞ . Put f = q ∞ − e. Since the left support of T p n equals q n and since f q n , we have that the left support of f T p n equals f . On the other hand, f T p ∞ = 0, implying that f T (p n − p ∞ ) = f T p n . We conclude that the left support of f T (p n − p ∞ ) equals f for all n. Hence, τ (f ) τ (p n − p ∞ ) → 0. It follows that f = 0, so that e = q ∞ . Conversely assume that p ∈ M k (C) ⊗ M is a projection and ϕ : p(C k ⊗ M ) → H is an injective M -linear map. It remains to prove that (Tr ⊗τ )(p) r. The M -submodules ϕ −1 (H n ) form a decreasing sequence whose intersection equals {0} by the injectivity of ϕ. By Lemma A.7, we get that dim M (ϕ −1 (H n )) → 0. Denote by π n : H → H/H n the quotient map. By the rank theorem,

Inverse limits in a weak sense
Hence, dim M (Im(π n • φ)) → (Tr ⊗τ )(p) as n → ∞. Since dim M (Im(π n • ϕ)) dim M (H/H n ) r for all n, we conclude that (Tr ⊗τ )(p) r. Proof. Write H = p(ℓ 2 (N) ⊗ L 2 (M )) for some projection p ∈ B(ℓ 2 (N)) ⊗ M . Choose an increasing sequence of projections p n ∈ B(ℓ 2 (N)) ⊗ M such that p n p, p n → p strongly and such that for every n, the center valued trace of p n is bounded. This means that H n := p n (ℓ 2 (N) ⊗ L 2 (M )) is a finitely generated Hilbert M -module for every n. Write ϕ n : H → H n : ϕ n (ξ) = p n ξ. Then K ∩ Ker ϕ n is a decreasing sequence of M -submodules of K with trivial intersection. By Lemma A.9, we get that dim M K = lim n dim M ϕ n (K). Since K is dense in H, we get that ϕ n (K) is dense in H n . By Lemma A.5, we get that ϕ n (K) ⊂ H n is rank dense. Hence, Proof. The lemma is an immediate consequence of Lemmas A.9 and Lemma A.10.

Inverse limits in a strong sense
An inverse system of M -modules consists of a sequence of M -modules H n and, for every k n, an M -linear map π n,k : H k → H n such that π n,k • π k,m = π n,m . The inverse limit lim ← − H n of the inverse system is the M -module H consisting of all sequences (x n ) with x n ∈ H n for all n and π n,k (x k ) = x n for all k n. We denote by π n : H → H n : (x n ) → x n the natural M -linear map from H to H n . So we conclude that cl(π n (T (K))) = cl(π n (H)) for all n ∈ N .
We let n tend to infinity. By Lemma A.9, the left hand side converges to dim M (T (K)), while the right hand side converges to dim M (H) and remains bounded by 1. So we get that dim M (T (K)) = dim M (H) 1 .
It follows from Theorem A.1 that the quotient H/T (K) has dimension zero, meaning that T (K) is rank dense in H. So the first claim is proven.
Claim 2. If T n (K n ) is dense in H n for all n, then T (K) is rank dense in H.
Fix x ∈ H and ε > 0. Define H ′ n := cl(π n (x)M ) and H ′ = lim ← − H ′ n . We view H ′ as an Msubmodule of H. Put K ′ n := T −1 n (H ′ n ) and K ′ = lim ← − K ′ n . We also view K ′ as an M -submodule of K. By the first claim, T (K ′ ) is rank dense in H ′ . Since x ∈ H ′ , we find a projection p ∈ M with τ (p) > 1 − ε and xp ∈ T (K ′ ). So certainly xp ∈ T (K) and the second claim is proven.
Proof of the proposition. Define L n := cl(T n (K n )) and view the inverse limit L := lim ← − L n as a closed M -submodule of H. By construction, T n (K n ) is dense in L n . So by claim 2, we get that T (K) is rank dense in L. It follows that −→ H L are rank isomorphisms. From claim 2, it also follows that the natural map H → lim ← − H n /L n has a rank dense image. Its kernel is by construction equal to L, so that the natural map is a rank isomorphism.

Dimension theory for semifinite von Neumann algebras
The Murray-von Neumann dimension of arbitrary (purely algebraic) modules over a tracial von Neumann algebra (M, τ ) was defined in [Lü97]. This was extended to the case of semifinite von Neumann algebras (N, Tr) in [Pe11,Appendix B]. We give here a more direct approach to the results of [Pe11].
We define the N -dimension of an arbitrary N -module over a von Neumann algebra N equipped with a normal semifinite faithful trace Tr. In doing so, we systematically make use of the dimension of pN p-modules, where p ∈ N is a projection with Tr(p) < ∞. We always implicitly equip pN p with the faithful normal tracial state τ (x) = Tr(p) −1 Tr(x). Proof. Denote by τ p the faithful normal tracial state on pM p given by τ p (x) = τ (p) −1 τ (x). As explained in the beginning of this section, we always consider dim pM p with respect to this tracial state.
First assume that q ∈ M k (C) ⊗ M is a projection and that ϕ : q(C k ⊗ M ) → Hz is an injective M -linear map. Note that the injectivity of ϕ forces q 1⊗z. The restriction of ϕ to q(C k ⊗M p) is an injective pM p-linear map into Hp. Hence, dim pM p (Hp) dim pM p (q(C k ⊗ M p)) .
Since q 1 ⊗ z, the right hand side equals τ (p) −1 (Tr ⊗τ )(q). Since this holds for all injective M -linear maps ϕ, we get that Conversely assume that q ∈ M k (C)⊗ pM p is a projection and that ϕ : q(M k,1 (C)⊗ pM p) → Hp is an injective pM p-linear map. Define ξ ∈ M 1,k (C) ⊗ Hp given by e 1i ⊗ ϕ(q(e i1 ⊗ p)) .
Note that ξq = ξ and ϕ(η) = ξη for all η ∈ q(M k,1 (C) ⊗ pM p). Define Observe that ψ takes values in Hz and that ψ is an injective M -linear map. Hence, The left hand side equals τ (p) (Tr ⊗τ p )(q). Since this holds for all injective pM p-linear maps ϕ, we get that τ (p) dim pM p (Hp) dim M (Hz) .
Lemma A.16. Let (N, Tr) be a von Neumann algebra with separable predual equipped with a normal semifinite faithful trace. Let H be an N -module. If p n is an increasing sequence of projections in N with Tr(p n ) < ∞ and such that the central supports z n of p n converge strongly to 1, then the sequence Tr(p n ) dim pnN pn (Hp n ) is increasing and converges to dim N H .
In particular, if p ∈ N is a projection of finite trace and central support equal to 1, we have dim N H = Tr(p) dim pN p (Hp).
Proof. If p p ′ are projections in N with Tr(p) Tr(p ′ ) < ∞, we apply Lemma A.15 to the von Neumann algebra p ′ N p ′ . Denoting by z the central support of p in N , we conclude that So the sequence in the formulation of the lemma is indeed increasing. Denote its limit by α ∈ [0, +∞].
By construction, α dim N H. To prove the converse inequality, choose an arbitrary projection q ∈ N with Tr(q) < ∞. We must prove that Tr(q) dim qN q (Hq) α. Denote by z the central support of q in N . Put e n := q ∨ p n . Note that Tr(e n ) < ∞. Inside e n N e n , the central supports of p n and q are respectively equal to e n z n and e n z. Applying twice Lemma A.15, it follows that Tr(p n ) dim pnN pn (Hzp n ) = Tr(e n ) dim enN en (Hzz n e n ) = Tr(q) dim qN q (Hz n q) .
Since Hzp n ⊂ Hp n , the left hand side is smaller or equal than α. It remains to show that the right hand side converges to Tr(q) dim qN q (Hq).
Put H 0 = {ξ ∈ Hq | ξz n = 0 for all n ∈ N}. Since z n is an increasing sequence of projections that strongly converges to 1, we have by construction that H 0 is a qN q-module of rank zero. Hence, dim qN q (Hq) = dim qN q (Hq/H 0 ). Using the surjective qN q-linear maps it follows from Lemma A.9 that dim qN q (Hz n q) → dim qN q (Hq/H 0 ).
It is now easy to prove the semifinite version of Lemma A.11. Proof. Choose an increasing sequence of projections p k ∈ N such that Tr(p k ) < ∞ for all k and such that the central supports z k of p k converge strongly to 1. Consider the double sequence α n,k := Tr(p k ) dim p k N p k (cl(ϕ n (H))p k ). For fixed n and increasing k, by Lemma A.16, the sequence α n,k is increasing and converges to dim N cl(ϕ n (H)).
For fixed k and increasing n, we apply Lemma A.11 to p k N p k and the restriction of ϕ n to Hp k , and conclude that α n,k is increasing to the limit Tr(p k ) dim p k N p k Hp k . When k → ∞, this last sequence increases to dim N H by Lemma A.16. In combination with the previous paragraph, the lemma is proven.

Appendix B. Properties of cross section equivalence relations
In this section we prove the "folklore" Proposition 4.3. We do not claim any originality, but in order to keep our article as clear and self-contained as possible, we give a detailed argument. The construction of the invariant probability measure ν on the cross section equivalence relation R is a very special case of Connes's transverse measure theory, see [Co79] and [ADR00, Appendix A.1]. In particular, point 7 of Proposition 4.3 is a very special case of [ADR00, Theorem 3.2.16]. Nevertheless we think that the following explicit and direct approach is useful.
We need to introduce a bit of terminology from the theory of countable equivalence relations.
Following [CFW81, Definition 6], we say that R is amenable if there exists a (typically nonnormal) conditional expectation P : Here we used the natural left actions of R on Y and on R. We call P a left invariant mean on R.
Lemma B.1. Let R be an amenable countable pmp equivalence relation on the standard probability space (Y, ν). Assume that we are given a nonsingular action of R on the standard measure space (Z, η). Denote by L ∞ (Z) R the von Neumann subalgebra of L ∞ (Z) consisting of the R-invariant bounded measurable functions. Then there exists a (typically non-normal) conditional expectation Q : L ∞ (Z) → L ∞ (Z) R satisfying δ * • Q = Q • δ * for every nonsingular automorphism δ of (Z, η) that commutes with the action of R.
Proof. Define Z as in (B.1) and equip Z with the σ-finite measure given by integrating w.r.t. η the counting measure over the map Z → Z : (z, y) → z. Every normal conditional expectation E : L ∞ (Z) → L ∞ (Y ) uniquely extends to a normal conditional expectation E : L ∞ (Z) → L ∞ (R). Fix a left invariant mean P : L ∞ (R) → L ∞ (Y ). We claim that there is a unique conditional expectation for every normal conditional expectation E : L ∞ (Z) → L ∞ (Y ). To prove this claim, first fix a faithful normal conditional expectation E 0 : L ∞ (Z) → L ∞ (Y ). Denote by τ the state on L ∞ (Y ) given by integration w.r.t. ν. Denoteτ = τ • E 0 . Using the Cauchy-Schwarz inequality, we find a unique conditional expectation P : By construction, E 0 • P = P • E 0 . If E is another normal conditional expectation and E 2E 0 , there is a positive a ∈ L ∞ (Z) with a 2 and E(f ) = E 0 (af ) for all f ∈ L ∞ (Z). It follows that E(f ) = E 0 (af ) for all f ∈ L ∞ (Z) and hence E • P = P • E. If E is an arbitrary normal conditional expectation, put E 1 = (E 0 + E)/2. Then E 1 is a faithful normal conditional expectation and we find a unique P 1 satisfying E 1 • P 1 = P • E 1 . Since E 0 2E 1 , we also have E 0 • P 1 = P • E 0 . Hence P 1 = P. Since E 2E 1 , we have E • P 1 = P • E. Since P 1 = P, we have proven that P satisfies (B.2) for every normal conditional expectation E.
We similarly define P ψ by the formula P ψ (a) = P(a · ψ −1 ) · ψ. Since the unique normal conditional expectation of L ∞ (Z) onto L ∞ (R) extending E ψ is given by the formula So we get that E • P ψ = P • E for all normal conditional expectations E : L ∞ (Z) → L ∞ (Y ). Hence P ψ = P and the claim is proven.
We define the conditional expectation Q : L ∞ (Z) → L ∞ (Z) R given by Q(a) = P(b) where b(z, y) = a(α(z, y)) . (B.3) Finally take a nonsingular automorphism δ of (Z, η) that commutes with the action of R. The "functoriality" of the above construction of Q ensures that δ * • Q = Q • δ * .

Proof of Proposition 4.3
Proof of 1. Fix a cross section Y ⊂ X and a neighborhood U ⊂ G of e such that θ : U × Y → X : (g, y) → g · y is injective. Write W := U · Y . Define R as in 1. Then R is an equivalence relation on Y . Since θ is injective, the map G×Y → X : (g, y) → g·y is countable-to-one. Define the Borel maps Ψ : G×Y → X ×Y and π ℓ : X ×Y → X as in 2. Since π ℓ •Ψ is countable-to-one, also Ψ is countable-to-one. So Z = Im Ψ is a Borel set. Then also R = Z ∩ (Y × Y ) is a Borel set, meaning that R is a Borel equivalence relation. We get as well that R → Y : (y, y ′ ) → y is countable-to-one. So R is a countable Borel equivalence relation on Y . This proves 1, as well as the facts that Z is a Borel set and that π ℓ : Z → X is countable-to-one.
Proof of 2. To prove the other statements of the proposition, we may discard a conegligible G-invariant Borel subset of X and assume that G acts freely on X and that G · Y = X. Define the σ-finite measure η on Z as in 2. Since µ is invariant under G X, the measure η is invariant under the action G Z given by g · (x, y) = (g · x, y). Since the action G X is free, Ψ : G × X → Z is a bijection. So, (Ψ −1 ) * (η) is a G-invariant measure on G × Y . By the uniqueness of the Haar measure, there exists a unique σ-finite measure ν 1 on Y such that Ψ * (λ × ν 1 ) = η. Since π ℓ • Ψ is injective on U × Y , we get that λ(U ) ν 1 (Y ) = µ(U · Y ). In particular, λ(U ) and ν 1 (Y ) are finite. Putting covol Y := λ(U )/µ(U · Y ) and ν := covol Y · ν 1 , we have proven 2.
Proof of 3 and 4. Take another cross section Y ′ ⊂ X (and for the proof of 3, we will later take Y ′ = Y ). Define the equivalence relation R ′ , the probability measure ν ′ and the Borel set Z ′ as in 2. Define S := {(y, y ′ ) ∈ Y × Y ′ | y ∈ G · y ′ } .
Denote by π ℓ : Z → X the projection on the first coordinate. Denote by ρ the σ-finite measure on Z given by integrating w.r.t. µ the counting measure over π ℓ . Using the intermediate projection Z → Z → X on the first two coordinates and using the fact that Ψ * (λ × ν) = covol Y · η, we get that (Φ 1 ) * (λ × γ ℓ ) = covol Y · ρ . Since G acts freely on X, we can uniquely define the Borel map Ω : S → G satisfying Ω(y, y ′ ) · y ′ = y for all (y, y ′ ) ∈ S .
End of the proof of 3. In the particular case where Y = Y ′ , equation (B.4) precisely says that R preserves ν. So 3 is proven.
Proof of 5. Since the map G × Y → X : (g, y) → g · y is countable-to-one and surjective, it admits a Borel right inverse x → (ϕ(x), π(x)). Then note that a Borel map F : X → C is G-invariant if and only if it is of the form F 0 • π for some R-invariant Borel map F 0 : Y → C. From this, 5 follows immediately.
Proof of 6. If G is compact, one checks that R has finite orbits. Conversely assume that Y 0 ⊂ Y is a nonnegligible subset such that every y ∈ Y 0 has a finite orbit under R. Choose a fundamental domain Y 1 for the equivalence relation R ∩ (Y 0 × Y 0 ). So Y 1 ⊂ Y is nonnegligible and has the following property: if y, y ′ ∈ Y 1 and (y, y ′ ) ∈ R, then y = y ′ . Choose a compact neighborhood K of e such that K ⊂ U . Define W 1 := K · Y 1 . From (4.2), we know that W 1 is a nonnegligible subset of X. We prove the following claim: if g ∈ G and if g · W 1 ∩ W 1 = ∅, then g ∈ KK −1 . To prove this claim, assume that g ∈ G, x ∈ W 1 and g · x ∈ W 1 . Take h, h ′ ∈ K and y, y ′ ∈ Y 1 such that x = h · y and g · x = h ′ · y ′ . It follows that (y, y ′ ) ∈ R and hence y = y ′ . Since G acts freely on X, it then follows that gh = h ′ , so that indeed g ∈ KK −1 .
Since µ is a probability measure and since W 1 ⊂ X is nonnegligible, we can take a finite sequence of elements g 1 , . . . , g n ∈ G that is maximal with respect to the property that the sets (g k · W 1 ) k=1,...,n are disjoint. Using the claim in the previous paragraph, it follows that G = n k=1 g k KK −1 so that G is compact. This ends the proof of 6.
Proof of 7. Consider as above the measure space (Z, η), together with the left action of G given by g · (x, y) = (g · x, y) and the right action of R given by α(x, y, y ′ ) = (x, y ′ ) for all (x, y, y ′ ) ∈ Z where Z = {(x, y, y ′ ) ∈ X × Y × Y | G · x = G · y = G · y ′ } .
Note that these actions commute. First assume that G is an amenable lcsc group. Integrating over an invariant mean on G, we obtain a (non-normal) conditional expectation Q : L ∞ (Z) → L ∞ (Z) G satisfying Q(a·ψ) = Q(a)·ψ for all a ∈ L ∞ (Z) and all ψ ∈ [[R]]. Using (4.1), it follows that L ∞ (Z) G = L ∞ (Y ), where we view L ∞ (Y ) ⊂ L ∞ (Z) as functions that only depend on the second variable. To deduce that R is amenable, choose a Borel right inverse x → (ϕ(x), π(x)) for the countable-to-one and surjective Borel map G × Y → X : (g, y) → g · y. Make this choice such that ϕ(g · y) = g and π(g · y) = y for all g ∈ U and y ∈ Y . Note that π : X → Y is a factor map, so that we can define the factor map Z → R : (x, y) → (π(x), y). This factor map induces an inclusion L ∞ (R) → L ∞ (Z). The composition with Q yields a right invariant mean on R, i.e. a conditional expectation P : L ∞ (R) → L ∞ (Y ) satisfying P (a · ψ) = P (a) · ψ for all a ∈ L ∞ (R) and ψ ∈ [[R]]. So R is amenable.
Conversely assume that R is amenable. From Lemma B.1, we get a conditional expectation Q : L ∞ (Z) → L ∞ (Z) R satisfying Q(g · a) = g · Q(a) for all a ∈ L ∞ (Z) and g ∈ G. Note that L ∞ (Z) R = L ∞ (X), where we view L ∞ (X) ⊂ L ∞ (Z) as functions that only depend on the first variable. Composing Q with integration w.r.t. µ, we find a G-invariant mean on L ∞ (Z). Using the isomorphism Ψ given by (4.1), we find a G-invariant mean on L ∞ (G × Y ). The restriction to L ∞ (G) ⊗ 1 yields a left invariant mean on L ∞ (G) so that G is amenable.
This concludes the proof of Proposition 4.3.