Examples of Groups that are Measure Equivalent to the Free Group

D. Gaboriau^{∗}

Abstract

Measure Equivalence (ME) is the measure theoretic counterpart of quasi-isometry. This field grew considerably during the last years, developing tools to distinguish between different ME classes of countable groups. On the other hand, contructions of ME equivalent groups are very rare. We present a new method, based on a notion of measurable free-factor, and we apply it to exhibit a new family of groups that are measure equivalent to the free group. We also present a quite extensive survey on results about Measure Equivalence for countable groups.

1 Introduction

During the last years, much progress has been made in the classification of countable discrete groups up to Measure Equivalence (ME). All of them describe criteria ensuring that certain groups don’t belong to the same equivalence class (see for instance the work of R. Zimmer [Zim84, Zim91], S. Adams [Ada90, Ada94, Ada95], Adams-Spatzier [AS90], A. Furman [Fur99a, Fur99b], D. Gaboriau [Gab00, Gab02], Monod-Shalom [MS02],... We also would like to draw the attention of the reader to the very nice paper [HK05] of Hjorth-Kechris where similar results are developed.

This notion (ME), introduced by M. Gromov [Gro93] is a measure theoretic analogue of Quasi-Isometry (QI) (see sect. 2). But, it takes its roots in the pioneering work of H. Dye [Dye59, Dye63], and even, somehow, back to Murray-von Neumann [MvN36].

The simplest instances of ME groups are commensurable groups or more generally commensurable up to finite kernel^{1}, and groups that are lattices (=discrete, finite covolume subgroup) in the same locally compact second countable group. Recall that cocompact lattices are QI.

The first non-elementary iso-ME-class result is due to Dye [Dye59, Dye63] who puts together many amenable groups in the same ME-class, for example, all the infinite groups with polynomial growth. A series of improvements led to Ornstein-Weiss’ theorem [OW80] asserting that, in fact, all infinite amenable groups are ME to each other. On the other hand, since amenability is a ME invariant, the ME class of ℤ consists precisely in all the infinite amenable groups [Fur99a].

Next, to build further ME groups, one can elaborate on these constructions via elementary procedures: direct products and free products (see Section 2 for precise statements). And these are essentially the only known methods.

The family of groups ME to a free group^{2} contains for instance the finite and amenable groups, as well as the fundamental groups π_{1}(Σ_{g}) of the compact orientable surfaces of genus ≥ 2 (lattices in SL(2, ℝ)); and it is stable under taking free products (see Section 2.2, Property P_{ME}7) and subgroups (see Section 2.2, Property P_{ME}9). For instance, the following groups are ME to F_{2}: F_{2} ∗ π_{1}(Σ_{g}), ℤ^{2} ∗ ℤ^{2}, ℚ ∗ ℤ/3ℤ, the triangular groups T_{a,b,c} of isometries of the hyperbolic plane (with fundamental domain a triangle of angles \frac{π}{a} , \frac{π}{b} , \frac{π}{c}, 2 ≤ a ≤ b ≤ c and \frac{1}{a} + \frac{1}{b} + \frac{1}{c} < 1), (SL(2, ℤ) × ℤ/5ℤ) ∗ ℚ^{2} ∗ A, where A is any amenable group.

∗C.N.R.S.

1in the sense of [Har00, IV.B.27]: commensurability up to finite kernel is the equivalence relation on the set of countable groups generated by the equivalence of two terms of an exact sequence 1 → A → B → C → 1 as soon as the third term is a finite group.

2For consistency, we are led to include the free groups F_{∞} on countably many elements and F_{0} on 0 element (i.e. the trivial group {1}).

Following a theorem of G. Hjorth (see [Hjo02] or also [KM04, Th. 28.2, p. 98]), being ME to a free group is equivalent with the following elementarily equivalent conditions (sect. 2.2, P_{ME}8):

– being treeable in the sense of Peres-Pemantle [PP00]

– admitting a treeable p.m.p.^{3} free^{4} action in the sense of [Ada90, Gab00]

– having ergodic dimension 0 (for finite groups) or 1 in the sense of [Gab02] On the other hand, the free groups (and thus the family of groups ME to a free group) split into four different

ME-classes: the class of F_{0} (finite groups), that of F_{1} = ℤ (amenable groups), that of F_{p}, 2 ≤ p < ∞, (all the F_{p} are commensurable) and that of F_{∞}. The last two classes are distinguished by their ℓ^{2}-Betti numbers – β_{1} ∈ (0, ∞), resp. β_{1} = ∞ – [Gab02] (or also by their cost – C ∈ (1, ∞), resp. C = ∞ – [Gab00]) (see Section 2).

However, the classification of groups ME to a free group seems, nowadays, completely out of reach^{5}. In fact, the only groups whose ME-class is classified are finite groups, amenable groups and lattices in simple connected Lie groups with finite center and real rank ≥ 2 [Fur99a] (see Property P_{ME}19).

The contribution of this paper consists in a new construction of ME groups, leading to the exhibition of new groups that are ME to free groups F_{2} and F_{∞}.

Let Γ be a countable group and Λ a subgroup. We denote by ∗_{Λ}^{n} Γ the iterated amalgamated free product Γ ∗_{Λ} Γ ∗_{Λ} ⋯ ∗_{Λ} Γ of n copies of Γ above the corresponding n copies of Λ, where the injection morphisms are just the identity.

Consider the free group F_{2p} = ⟨a_{1}, ⋯, a_{p}, b_{1}, ⋯, b_{p}⟩ and its cyclic subgroup C generated by the product

of commutators: κ := ∏_{i=1}^{i=p}[a_{i}, b_{i}].

Theorem 1.1 (Cor. 3.7) For each n ∈ ℕ, the iterated amalgamated free product ∗_{C}^{n} F_{2p} is measure equivalent to the free group F_{2}.

This iterated amalgamated free product ∗_{C}^{n} F_{2p} is the fundamental group π_{1}(Σ) of a “branched surface” (figure 1).

Our examples for various n are not obviously lattices in the same l.c.s.c. group (i.e. the question asked to several

Figure 1: Branched surface

specialists remains open). However, they are certainly not cocompact lattices in the same l.c.s.c. group nor commensurable up to finite kernel since they are not quasi-isometric:

Proposition 1.2 The number n of factors F_{2p} in the iterated amalgamated free product ∗_{C}^{n} F_{2p} is a quasiisometry invariant.

3p.m.p.: probability measure preserving, on a standard Borel probability space.

4In this measure theoretic context, free means “essentially free”, i.e. up to removing a set of measure zero.

5To compare with, recall that a group quasi-isometric to a free group is virtually a free group [Sta68].

Proof: The boundary of the group is disconnected by pairs of points into either 1, 2 or n connected components. ■

Theorem 1.1 extends to an infinite amalgamated free product: ∗_{C}^{∞} F_{2p} is measure equivalent to the free group

F_{∞} (Cor. 3.7).

Corollary 1.3 If r is any non trivial element of ℤ, then the amalgamated free product Γ_{r} = ℤ ∗_{r=κ} F_{2p} is measure equivalent to the free group F_{2}.

Proof: Consider the natural homomorphism ℤ ∗_{r=κ} F_{2p} → ℤ/rℤ that kills F_{2p}. Its kernel has index r in Γ_{r} and is isomorphic with ∗_{C}^{r} F_{2p}. ■

More generally, one gets:

Theorem 1.4 (Cor. 3.18) Let G be any countable group, H an infinite cyclic subgroup and C the cyclic subgroup generated by κ in F_{2p}. Then G ∗_{H=C} F_{2p} is ME to G ∗ F_{2p−1}. In particular, if G is ME to F_{2}, then G ∗_{H=C} F_{2p} is ME to F_{2}.

By contrast, an amalgamated free product ℤ ∗_{ℤ} ℤ is measure equivalent to a free group if and only if the common

ℤ coincides with one the components or injects as a subgroup of index 2 in both (see P_{ME}15**). Also, if instead of C one considers 2C, generated by κ^{2}, and if H is contained as a subgroup of index at least three in a greater abelian group H′ of G, then G ∗_{H=2C} F_{2p} cannot be ME to a free group (by P_{ME}9 and P_{ME}15**): it contains

the nonamenable subgroup H′ ∗_{H=2C} C with β_{1} = 0.

To obtain these results we show that the subgroup C happens to appear as a free factor of the group F_{2p} in a measure theoretic sense (see Def. 3.1). Our proof makes use of standard percolation techniques developed in [LP05].

Acknowledgment: I would like to thank Russell Lyons for worthwhile discussions. I am deeply grateful to Yuval Peres for crucial indications in proving Theorem 3.2. Contrarily to his opinion, his contribution is really substantial. I’m also grateful to Roman Sauer for drawing my attention to the co-induced action construction, and together with Yves de Cornulier, Nicolas Monod and Yehuda Shalom for useful comments on a preliminary version.

2 Generalities about Measure Equivalence

Definition 2.1 ([Gro93, 0.5.E]) Two countable groups Γ_{1} and Γ_{2} are Measure Equivalent (ME) if 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 Γ_{1} is ME to Γ_{2} with index ι = [Γ_{1} : Γ_{2}]_{Ω} := m(D_{2})/m(D_{1}), and we denote the existence of such a coupling Ω by

Γ_{1} ∼^{ME}_{ι} Γ_{2} or simply Γ_{1} ∼^{ME} Γ_{2}

when the particular value of the index is irrelevant. Measure Equivalence is an equivalence relation on the set of countable groups. The grounds are established by A. Furman in [Fur99a, Sect. 2].

The notion of Measure Equivalence has been introduced by M. Gromov as the measure theoretic analogue of the notion of Quasi-Isometry, which is more topological, as testified by the following criterion:

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.

We anthologize basic and less basic properties of ME, with references, or proofs when necessary.

2.1 Basic Properties of ME

P_{ME}1 The ME class of the trivial group {1} consists in all finite groups.

P_{ME}2 Groups that are commensurable up to finite kernel are ME. Commensurability up to finite kernel (footnote 1) corresponds exactly to ME with a countable atomic space Ω.

P_{ME}3 Lattices in the same locally compact second countable group are ME. Actions of the two lattices by left multiplication and right multiplication by the inverse on the group deliver the desired ME coupling.

P_{ME}4 (Direct Product) Γ_{1} ∼^{ME} Λ_{1} and Γ_{2} ∼^{ME} Λ_{2} ⇒ Γ_{1} × Γ_{2} ∼^{ME} Λ_{1} × Λ_{2}

This is quite obvious, by taking the product actions on the product space. The converse fails in general. For instance by Ornstein-Weiss’ theorem recalled in the introduction (see also P_{ME}10), all infinite amenable groups

are ME so that^{6} F_{2} × ℤ ∼^{ME} F_{2} × (ℤ × ℤ) = (F_{2} × ℤ) × ℤ. but F_{2} ≁^{ME} F_{2} × ℤ (see [Ada94, Th. 6.1] or P_{ME}15**). However, it becomes true when restricted to a quite large class of groups (see P_{ME}20).

Recall that two p.m.p. actions α_{i} of Γ_{i} on (X_{i}, μ_{i}) (i = 1, 2) are stably orbit equivalent (SOE) if there are complete sections A_{i} ⊂ X_{i} such that the orbit equivalence relations (R_{α_{i}(Γ_{i})}, μ_{i}) restricted to A_{i} are isomorphic via a measure scaling isomorphism f : A_{1} → A_{2} (f_{∗}μ_{1|A_{1}} = λ μ_{2|A_{2}}). The compression constant is defined as ι := μ_{2}(A_{2})/μ_{1}(A_{1}) = 1/λ. The connection with ME is very useful and is proved in [Fur99b, Lem. 3.2, Th. 3.3], where it is credited to Zimmer and Gromov. An easy additional argument gives the freeness condition (see [Gab00b, Th. 2.3]). Orbit Equivalence (OE) corresponds to the case where A_{i} = X_{i} (and thus ι = 1) and where the fundamental domains D_{1}, D_{2} of the measure equivalence coupling may be taken to coincide.

P_{ME}5 (ME ⇔ SOE) – Two groups Γ_{1} and Γ_{2} are ME (with index ι) if and only if they admit stably orbit equivalent (SOE) free p.m.p. actions (with compression constant ι).

– They are ME with a common fundamental domain if and only if they admit orbit equivalent (OE) free p.m.p. actions. We shall denote this last situation by

Γ_{1} ∼^{OE} Γ_{2}

– If the only possible value of the index ι = [Γ_{1} : Γ_{2}] is 1, then Γ_{1} ∼^{OE} Γ_{2}.

For its conciseness, we present a proof of this result. Proof: ME ⇒ SOE. Consider an action of Γ_{1}×Γ_{2} on (Ω, m) witnessing Γ_{1} ∼^{ME}_{ι} Γ_{2}. The existence of fundamental domains of finite measures D_{i} ensures the existence of a Γ_{1}-equivariant Borel map onto a finite measure Borel space π_{1} : Γ_{1} × Γ_{2} ↷ (Ω, m) → Γ_{1} ↷ X_{1} := Ω/Γ_{2} ≃ D_{2} and similarly for π_{2} by exchanging the subscripts 1 and 2. Any two points y, y′ of Ω are Γ_{1} × Γ_{2}-equivalent iff their images under π_{i} are Γ_{i}-equivalent (i = 1, 2). Since π_{1}, π_{2} have countable fibers, one can find a non-null Y ⊂ Ω on which both π_{1} and π_{2} are one-to-one. The natural isomorphism f between π_{1}(Y) and π_{2}(Y) preserves the restriction of the equivalence relations. Normalizing the induced measure on X_{i} to a probability measure leads to identify index and compression constant. The freeness is obtained by just considering any free p.m.p. action of Γ_{1} on some space (Z, ν) (with trivial action of Γ_{2}) and replacing the action on Ω by the diagonal action on the product measure space Ω × Z. In case of coincidence, Y := D_{1} = D_{2} ≃ X_{1} ≃ X_{2} delivers OE actions. Proof: SOE ⇒ ME. Let Γ_{i} act on (X_{i}, μ_{i}) and assume X_{1} ⊃ Y_{1} ≃ Y_{2} ⊂ X_{2} witnesses the SOE. Rescale the measures μ_{1}, μ_{2} so that Y_{1} and Y_{2} get the same measure and so that the glueing X := (X_{1} ∐ X_{2})/(Y_{1} = Y_{2}) of X_{1}, X_{2} along Y_{1}, Y_{2} becomes a probability space. Define on X the equivalence relation R extending R_{Γ_{i}} on X_{i}. Consider the coordinate projections π_{1} and π_{2} from R to X. The Borel subset Ω ⊂ R ⊂ X × X defined as the preimage by π_{1} × π_{2} of X_{1} × X_{2} gives the desired ME coupling: Γ_{i} acts on the i-th coordinate (and trivially on the other one). Any Borel section of π_{1} gives a Borel fundamental domain for Γ_{2} and any Borel section of π_{2} gives a Borel fundamental domain for Γ_{1}. Their measures, for the natural measure on R (see [FM77]), are compared via the rescaling to the compression constant. In case of OE, Ω = R and the diagonal Δ ⊂ R is a fundamental domain for both groups. Observe that ergodic SOE actions on X_{i} are in fact OE if and only if ι = 1. The condition, when ι = 1 is the only possible index, follows by considering an ergodic component. ■

6More generally (Γ × A) ∼^{ME} (Γ × A) × B, for any group Γ, and any infinite amenable groups A and B.

2.2 Free Products and Amalgamations

Stability under taking free products requires the additional assumption that not only the groups are ME but also that they admit a coupling with a common fundamental domain. That such a result may be true is suggested in [MS02, Rem. 2.27]. We prove it below (after Lemma 2.6) as well as some generalizations.

P_{ME}6 (Free Product) Γ_{1} ∼^{OE} Λ_{1} and Γ_{2} ∼^{OE} Λ_{2} ⇒ Γ_{1} ∗ Γ_{2} ∼^{OE} Λ_{1} ∗ Λ_{2}

And more generally:

P_{ME}6^{∗} (Infinite Free Product) Γ_{j} ∼^{OE} Λ_{j} for all j ∈ ℕ ⇒ ∗_{j∈ℕ}Γ_{j} ∼^{OE} ∗_{j∈ℕ}Λ_{j}

The hypothesis ∼^{OE} cannot be replaced by ∼^{ME} since the following three groups belong to three different MEclasses^{7}: {1} ∗ ℤ/2ℤ (finite) (see P_{ME}1), ℤ/2ℤ ∗ ℤ/2ℤ (amenable) (see P_{ME}10) and ℤ/3ℤ ∗ ℤ/2ℤ (nonamenable). Similarly, free groups of different ranks are not OE (for a proof, we draw the attention of the reader to the simplified preliminary version [Gab98] of [Gab00]), and the groups F_{p} ∗ (F_{2} × F_{2}) all belong to different MEclasses for different p’s: their ℓ^{2} Betti numbers^{8} do not agree with P_{ME}15. However, this hypothesis may be relaxed when working with groups ME to free groups:

P_{ME}7 If each Γ_{j} is ME to some free group, then ∗_{j∈ℕ}Γ_{j} is ME to a free group.

In particular,

P_{ME}7^{∗} Γ_{1} ∼^{ME} F_{p} and Γ_{2} ∼^{ME} F_{q}, for p, q ∈ {1, 2, ∞} ⇒ Γ_{1} ∗ Γ_{2} ∼^{ME} F_{p+q}

For instance, non trivial free products of finitely many infinite amenable groups are ME to F_{2} (see [Gab00, p. 145]). Observe that free products of finitely many finite groups are virtually free. ^{9}

We make an intensive use of countable measured equivalence relations R on standard probability measure spaces (see [FM77] and the reminders of [Gab00, Gab02]) : they all are assumed to preserve the measure.

A countable group Γ is treeable in the sense of Peres-Pemantle [PP00] if the set of trees with vertex set Γ supports a Γ-invariant probability measure. The group Γ is not anti-treeable in the sense of [Gab00] if it admits a treeable free action α, i.e. an action whose associated equivalence relation R_{α} is generated by a treeing. Recall that a graphing of R is a countable family Φ = (φ_{j})_{j∈J} of partially defined isomorphisms φ_{j} : A_{j} → B_{j} between Borel subsets of X such that R is the smallest equivalence relation satisfying for all j ∈ J for all x ∈ A_{j}, x ∼ φ_{j}(x). It equips each orbit with a graph structure. When these graphs are trees, the graphing is called a treeing [Ada90] and the equivalence relation treeable.

The two following notions taken from [Gab02], which we mention just for completeness (see also P_{ME}16), won’t be seriously used in the remainder of the paper. Equivalence relations, as groupoids, admit discrete actions on fields of simplicial complexes. The ergodic dimension is the smallest possible dimension of such a field of contractible simplicial complexes. For the approximate ergodic dimension, one considers increasing exhaustions of R by measurable subrelations and the smallest possible ergodic dimensions of these approximations.

P_{ME}8 The following conditions on a countable group Γ are equivalent: -i- being treeable in the sense of Peres-Pemantle -ii- admitting a treeable p.m.p. free action in the sense of [Gab00] -iii- having ergodic dimension 0 (for finite groups) or 1 -iv- being ME to a free group The equivalence of the first three conditions is elementary, while the connection with the fourth one is a result of G. Hjorth: see [Hjo02] or also [KM04, Th. 28.2, Th. 28.5]. In order to apply [KM04, Th. 28.2, Th. 28.5], observe that after considering an ergodic component, the cost can be arranged, by SOE, to be an integer.

7 The same phenomenon occurs in the analogous setting of geometric group theory and for similar reasons: Bi-Lipschitz equivalence of groups passes to free products, but not just QI of them. However, the (counter-)example of {1} ∗ ℤ/2ℤ, ℤ/2ℤ ∗ ℤ/2ℤ and ℤ/3ℤ ∗ ℤ/2ℤ is essentially the only one and is satisfactorily worked around by a result of Papasoglu-Whyte: If two families of non trivial (≠ {1}) finitely generated groups (Γ_{i})_{i=1,··· ,s} and (Λ_{j})_{j=1,··· ,t} s, t ≥ 2 define the same sets of quasi-isometry types (without multiplicity), then the free products Γ = Γ_{1} ∗ Γ_{2} ∗ ⋯ Γ_{s} and Λ = Λ_{1} ∗ Λ_{2} ∗ ⋯ ∗ Λ_{t} are quasi-isometric unless Γ or Λ is ℤ/2ℤ ∗ ℤ/2ℤ [PW02, Th. 02].

8The sequence of ℓ^{2} Betti numbers of F_{p} ∗ (F_{2} × F_{2}) is (β_{i}(F_{p} ∗ (F_{2} × F_{2})))_{i∈ℕ} = (0, p, 1, 0, 0, ⋯, 0, ⋯). Incidentally, no one of QI or ME classification is finer than the other one, as illustrated by this family of groups (that they are QI is due to K. Whyte, see footnote 7).

9It follows from P_{ME}7 that all the groups Γ_{S} that appear in [HK05, Th. 1] are ME when S is finite (resp. infinite).

We introduce some terminology. Let R and S be two countable measured equivalence relations on the standard probability measure spaces (X, μ) and (Y, ν) respectively. A measurable map p : X → Y is a morphism from R to S if almost every R-equivalent points x, x′ of X have S-equivalent images p(x), p(x′) and if the pushforward measure p_{∗}μ is equivalent to ν. Say that such a morphism is locally one-to-one if, for almost every x ∈ X, p induces a one-to-one map from the R-class of x to the S-class of p(x).

Example 2.2 If R_{α} and S_{β} are the orbit equivalence relations of two actions α, β of a countable group Γ on X (resp. Y), then any Γ-equivariant measurable map p : X → Y is a morphism from R_{α} to S_{β}. If the action β is moreover free, then p is locally one-to-one.

Lemma 2.3 If p : X → Y is a locally one-to-one morphism from R to S and if S is the orbit equivalence relation of an action β of a countable group Γ, then there exists a unique action α of Γ on X inducing the equivalence relation R and for which p is Γ-equivariant. If β is moreover free, then α is free. ■

And more generally, for graphings (see [Gab00]):

Lemma 2.4 If p : X → Y is a locally one-to-one morphism from R to S and if S is generated by a graphing Φ = (φ_{1}, φ_{2}, ⋯, φ_{j}, ⋯), then there exists a unique graphing Φ̃ = (φ̃_{1}, φ̃_{2}, ⋯, φ̃_{j}, ⋯) generating R and for which p is equivariant (i.e. ∀φ_{j}, x ∈ Dom(φ̃_{j}) iff p(x) ∈ Dom(φ_{j}) and in this case pφ̃_{j}(x) = φ_{j}(p(x))). In particular, if S is treeable, then R is treeable. ■

We have introduced in [Gab00, Sect. IV.B] the notions of an equivalence relation R that splits as a free product of two (or more) subrelations R = R_{1} ∗ R_{2} and more generally of an equivalence relation R that splits as a free product with amalgamation of two subrelations over a third one R = R_{1} ∗_{R_{3}} R_{2}:

– R is generated by R_{1} and R_{2} (i.e. R is the smallest equivalence relation containing R_{1} and R_{2} as subrelations)

– for (almost) every 2p-tuple (x_{j})_{j∈ℤ/2pℤ} that are successively R_{1} and R_{2}-equivalent (i.e. x_{2i−1} ∼^{R_{1}} x_{2i} ∼^{R_{2}} x_{2i+1}

∀i), there exists two consecutive ones that are R_{3}-equivalent (i.e. there exists a j such that x_{j} ∼^{R_{3}} x_{j+1}).

Free products correspond to triviality of R_{3}, i.e. the second condition becomes: there exists a j such that x_{j} = x_{j+1}.

Standard examples are of course given by free actions of groups that split in a similar way.

Lemma 2.5 Assume that R splits as a free product R = R_{1} ∗ R_{2} and that R_{i} is produced by a free action of Γ_{i}. Then the induced action of the free product Γ_{1} ∗ Γ_{2} produces R and is free. ■

Lemma 2.6 If p : X → Y is a locally one-to-one morphism from R to S and if S splits as a free product with amalgamation S = S_{1} ∗_{S_{3}} S_{2}, then R admits a corresponding splitting R = R_{1} ∗_{R_{3}} R_{2} for which p is a locally

one-to-one morphism from R_{i} to S_{i}, for i = 1, 2, 3. ■

Proof of P_{ME}6^{∗} (the proof of P_{ME}6 follows by letting Γ_{j} = Λ_{j} = {1} for j ≥ 3): Let S_{j}, (j ∈ ℕ), be measured equivalence relations on (Y_{j}, ν_{j}), given by orbit equivalent free actions of Γ_{j} and Λ_{j}. Take any free p.m.p. action of ∗_{j∈ℕ}Γ_{j} on a standard probability measure space (Z, ν) and consider the diagonal action on the product measure space (X, μ) := (Z × ∏_{j∈ℕ} Y_{j}, ν × ∏_{j∈ℕ} ν_{j}) (i.e. γ.x = γ.(z, y_{1}, y_{2}, ⋯, y_{j}, ⋯) = (γ.z, γ.y_{1}, γ.y_{2}, ⋯, γ.y_{j}, ⋯), where ∗_{j∈ℕ}Γ_{j} acts on Y_{i} via the natural homomorphism ∗_{j∈ℕ}Γ_{j} → Γ_{i}). This action is free and p.m.p. Denote by R the equivalence relation it defines on X and by R_{j} the equivalence sub-relations defined by the restriction of this action to Γ_{j}. The measurable map p_{i} : X = Z × ∏_{j∈ℕ} Y_{j} → Y_{i} is Γ_{i}-equivariant and thus is a locally one-to-one morphism from R_{i} to S_{i} (ex. 2.2). Lemma 2.3 provides a free action of Λ_{i} on X producing R_{i}; and thus an action of the free product ∗_{j∈ℕ}Λ_{j} giving R. The free product structure R = R_{1} ∗R_{2} ∗⋯∗R_{j} ∗⋯ of the equivalence relation, given by the free product structure of the group ∗_{j∈ℕ}Γ_{j}, (see [Gab00, Déf. IV.9]) ensures that the action of the free product ∗_{j∈ℕ}Λ_{j} is free (see Lemma 2.5). We have constructed orbit equivalent free actions of ∗_{j∈ℕ}Γ_{j} and ∗_{j∈ℕ}Λ_{j}. ■

Proof of P_{ME}7: While following the lines of the preceding proof, forget about the groups Λ_{j} = F_{p_{j}} and just retain that S_{j}, generated by a free action of Γ_{j}, is treeable. The treeings of R_{j} (j ∈ ℕ), delivered by lemma 2.4, together define a graphing of R = R_{1} ∗ R_{2} ∗ ⋯ ∗ R_{j} ∗ ⋯ that happen to be a treeing due to the free product structure of R. Then apply P_{ME}8. ■

P_{ME}9 The family of groups ME to some free group is stable under taking subgroups.

This follows from [Gab00, Th. 5] and P_{ME}8.

2.3 Some Invariants

P_{ME}10 (Amenability) The ME class of ℤ consists in all infinite amenable groups. Moreover, any ergodic action of any two infinite amenable groups are OE and also SOE for any compression constant ι [OW80].

This property is sometimes interpreted as a kind of elasticity. This in wide contrast with the rigidity phenomenon described by Zimmer and Furman [Zim84, Fur99a, Fur99b] for lattice in higher rank semi-simple Lie groups see also P_{ME}19.

Recall that Kazhdan property T is often considered as opposite to amenability. More recently, the general opinion switched, after M. Gromov to oppose Kazhdan property T with the greater class of groups with Haagerup property (= a-T-menability, see [CCJJV01]).

P_{ME}11 (Kazhdan Property T) Kazhdan property T is a ME invariant [Fur99a]. Lattices of Sp(n, 1) are not ME to lattices of SU(p, 1) or SO(p, 1). We already knew from [AS90] that groups with property T are not ME to groups that split non trivially as free product with amalgamation or HNN-extension.

P_{ME}12 (Cowling-Haagerup constant) Cowling-Haagerup constant Λ(Γ) is a ME invariant [CZ89,

Jol01].

The Cowling-Haagerup constant Λ(Γ) of the group Γ is the infimum of the constants C such that the Fourier algebra A(Γ) admits an approximate unit bounded by C in the multipliers norm. Jolissaint in [Jol01] extends the above result of Cowling-Zimmer [CZ89] from OE to ME. Lattices in various Sp(n, 1) are not ME (Λ(Γ) = 2n−1). Recall that Λ(Γ) = 1 implies that Γ has Haagerup property, but that it is unknown whether the converse holds (Cowling conjecture).

P_{ME}13 (a-T-menability) Haagerup property is a ME invariant This result was obtained independently by several people, including Jolissaint[Jol01], Popa [Pop01] and Shalom (personal communication).

Two kinds of numerical invariants of countable groups are studied in [Gab00, Gab02] in connection with ME: Cost C(Γ) ∈ [0, ∞] and ℓ^{2} Betti numbers β_{0}(Γ), β_{1}(Γ), ⋯, β_{i}(Γ), · · ·.

P_{ME}14 (Cost) Having cost = 1 (resp. = ∞) is a ME invariant [Gab00]. For all the groups for which the computation has been carried out until now, C(Γ) = β_{1}(Γ) − β_{0}(Γ) + 1.

P_{ME}15 (ℓ^{2} Betti Numbers) ME groups Γ and Λ have proportional ℓ^{2} Betti numbers: there is

λ > 0 such that for every n ∈ ℕ, β_{n}(Γ) = λβ_{n}(Λ) [Gab02].

This gives^{10} the splitting of groups ME to free groups into four ME-classes according to whether β_{0} ≠ 0, β_{1} = 0 (finite groups) β_{0} = 0, β_{1} = 0 (amenable ones), β_{1} ∈ (0, ∞) (class of F_{p}, 1 < p < ∞) or β_{1} = ∞ (class of F_{∞}). Moreover (see P_{ME}15* below), if Γ is ME to a free group and β_{1}(Γ) = p − 1 is an integer, then Γ ∼^{OE} F_{p}. Also, F_{p_{1}} × F_{p_{2}} ×⋯ × F_{p_{j}} ∼^{ME} F_{q_{1}} × F_{q_{2}} × ⋯ × F_{q_{k}} iff j = k (with ∞ > p_{i}, q_{i} ≥ 2)^{11}, and in this case they are commensurable. In the following three claims, the only non zero ℓ^{2} Betti number is in the middle dimension of the associated symmetric space: If two lattices of Sp(n, 1) and Sp(p, 1) are ME, then p = n (see also P_{ME}12). If two lattices of SU(n, 1) and SU(p, 1) are ME, then p = n. If two lattices of SO(2n, 1) and SO(2p, 1) are ME, then p = n. If it would happen that lattices in SU(n, 1) and SO(p, 1) are ME, then p = 2n, but we suspect that lattices of SU(n, 1) and SO(2n, 1) are never ME.

10For p ≠ 0, β_{1}(F_{p}) = p − 1 and all the other β_{i} are zero

11The only non zero ℓ^{2} Betti number is the j-th, resp. the k-th.

Thanks to the great flexibility in constructing groups with prescribed ℓ^{2} Betti numbers (see [CG86]), it follows from P_{ME}15 that there are uncountably many different ME classes.

P_{ME}15* If one of the β_{n}(Γ) is different from 0, ∞, and if Γ ∼^{ME} Λ, with index ι = [Γ : Λ], then ι = \frac{1}{λ} is imposed. In this case, Γ ∼^{OE} Λ iff λ = 1 (see P_{ME}5).

P_{ME}15** An infinite group with β_{1} = 0 is ME to a free group if and only if it is amenable. An amalgamated free product ℤ ∗_{ℤ} ℤ is ME to a free group if and only if it is amenable, i.e. the common ℤ coincides with one the components or injects as a subgroup of index 2 in both.

Remark 2.7 If Γ = Γ_{1} ∗_{D} Γ_{2} is ME to a free group and nonamenable, with D infinite, then β_{1}(Γ_{1}) + β_{1}(Γ_{2}) − β_{1}(D) > 0. To see it, first observe that β_{1}(Γ) > 0 and β_{2}(Γ) = 0. The Mayer-Vietoris’ exact sequence then gives the estimate. For instance, amalgamated free products F_{2} ∗_{F_{3}} F_{2} are not ME to F_{2}.

P_{ME}16 Ergodic dimension and approximate ergodic dimension are ME invariants (see [Gab02,

Sect. 5.3 and Prop. 6.5]).

Both invariants are useful when all the ℓ^{2} Betti numbers are 0. For instance, very little is known about the ME classification of lattices in SO(m, 1) for m odd. However, a reasoning on ergodic dimension shows that: If two lattices in SO(2n + 1, 1) and SO(2p + 1, 1) are ME, with p ≤ n, then n ≤ 2p [Gab02, Cor. 6.9]. Similarly, such a group as the following Γ has approximate dimension j, and thus Γ = F_{p_{1}}× F_{p_{2}}×⋯ × F_{p_{j}} × ℤ ∼^{ME} F_{q_{1}}× F_{q_{2}}× ⋯ × F_{q_{k}} × ℤ iff j = k (all the p_{i}, q_{i} ≥ 2). In this case they are commensurable.

P_{ME}17 The set I_{ME}(Γ) of all indices of ME couplings of Γ with itself [Gab00b, Sect. 2.2] is a

ME invariant.

Since non-ergodic ME couplings are allowed, I_{ME}(Γ) is a convex subset of ℝ_{+}^{∗}. If one of the ℓ^{2} Betti numbers β_{n}(Γ) ≠ 0, ∞, then I_{ME}(Γ) = {1}. When Γ is the direct product of an infinite amenable group with any group then I_{ME}(Γ) = ℝ_{+}^{∗}.

Question 2.8 Are there groups Γ such that the set of all indices of ergodic ME couplings of Γ with itself is discrete ≠ {1} ?

Recall that the fundamental group of a p.m.p. ergodic Γ-action α on (X, μ) is the subgroup of ℝ_{+}^{∗} generated by the set of measures μ(A) for those A ⊂ X such that the restriction of R_{α} to A is isomorphic with R_{α} (equivalently the set of compression constants of α with itself). To compare with, I_{ME} is related to the bigger set of measures μ(A) for those A ⊂ X such that the restriction of R_{α} to A may be produced by some free action of Γ.

In [MS02], Monod-Shalom introduce the class C_{reg} of all groups Γ such that H_{b}^{2}(Γ, ℓ^{2}(Γ)) is non-zero and the larger class C of all groups Γ admitting a mixing unitary representation π such that H_{b}^{2}(Γ, π) is non-zero. Non amenable free products and non-elementary subgroups of hyperbolic groups all belong to the class C_{reg} ⊂ C [MS02, MMS04]. A group in the class C has finite center and is not a direct product of two infinite groups. Also, being in the class C passes to normal subgroups.

P_{ME}18 Belonging to the class C_{reg} (resp. C ) is a ME invariant [MS02]. For instance, a normal subgroup Λ of a group Γ ME to a non elementary hyperbolic group has finite center and is not a direct product of two infinite groups. This statement for Λ = Γ were obtained by S. Adams [Ada94, Ada95]. Lattices in SO(2n, 1) are not ME to lattices in SL(2, ℝ) × SL(2, ℝ) × ⋯ × SL(2, ℝ) (follows from [Ada94]). Observe that hyperbolicity itself is not preserved since ℤ^{2} ∗ ℤ^{2} ∼^{OE} F_{2}.

2.4 Some “Rigidity” Results

The following result is a spectacular achievement of the rigidity phenomena attached to higher rank lattices, after Margulis’ super-rigidity and Zimmer’s cocycle super-rigidity [Zim84]. In fact, Zimmer obtained the similar result with the additional assumption that the mysterious group admits a linear representation with infinite image [Zim84, Zim91].

P_{ME}19 (ME Rigidity – Higher Rank Lattices) Any countable group which is ME to a lattice in a connected simple Lie group G with finite center and real rank ≥ 2, is commensurable up to finite kernel with a lattice in G [Fur99a].

The prototype of such Lie groups are G = SL(n, ℝ) or SO(p, q), for n − 1, p, q ≥ 2.

Direct products sometimes appear as analogue to higher rank lattices. Here, bounded cohomology and belonging to the class C (see P_{ME}18) rigidifies the situation:

P_{ME}20 (ME Rigidity – Products) If Γ_{1} × ⋯ × Γ_{n} ∼^{ME} Λ_{1} × ⋯ × Λ_{p} are ME products of (non-

trivial) torsion-free countable groups with n ≤ p, where all the Γ_{i}’s are in the class C, then n = p and after permutation of the indices Γ_{i} ∼^{ME} Λ_{i} for all i [MS02, Th. 1.14].

For direct products of free groups, with p_{i}, q_{i} ≥ 2 the value p_{i}, q_{i} = ∞ being now allowed, F_{p_{1}} × F_{p_{2}} ×⋯ × F_{p_{j}} ∼^{ME} F_{q_{1}} × F_{q_{2}} × ⋯ × F_{q_{k}} iff j = k and the number of times ∞ occurs is the same; and in this case they are commensurable (compare with P_{ME}15 which gives only j = k).

P_{ME}21 (ME Rigidity – Quotients by Radicals) Let M ◁ Γ, N ◁ Λ be amenable normal subgroups of Γ, Λ such that Γ/M and Λ/N are in the class C and torsion-free. If Γ ∼^{ME} Λ, then Γ/M ∼^{ME} Λ/N [MS02, Th. 1.15].

2.5 Application

Consider Lie groups of the form ∏_{i∈I} Sp(m_{i}, 1) × ∏_{j∈J} SU(n_{j}, 1) × ∏_{k∈K} SO(p_{k}, 1), where I, J, K are finite sets and m_{i}, n_{j}, p_{k} ≥ 2. Let Γ and Γ′ be lattices of such Lie groups G and G′. Assume they are ME. Each of them is ME to a product of torsion free cocompact (thus hyperbolic) lattices of the factors ∏_{t∈I,J,K} Γ_{t} ∼^{ME} Γ ∼^{ME} Γ′ ∼^{ME} ∏_{t′∈I′,J′,K′} Γ_{t′}. By P_{ME}20, the pieces correspond under ME, after reordering. It follows that the number of factors coincide. Thanks to property T (P_{ME}11), the pieces in the Sp(·, 1)’s are ME. Moreover, P_{ME}12 or the examination of the ℓ^{2} Betti numbers (P_{ME}15) ensures that the sets {m_{i} : i ∈ I} and {m_{i′} : i′ ∈ I′} are the same. Similarly, the number of odd p_{k}, p_{k′} are the same and the set of n_{j} and even p_{k} on the one hand, coincide with the set n_{j′} and even p_{k′} on the other hand. For instance if K, K′ are empty, then Γ ∼^{ME} Γ′ implies that G, G′ are isomorphic. And similarly if J, J′ are empty and all the p_{k} are even.

3 A Construction of Measure Equivalent Groups

3.1 Measure Free-Factor

We first introduce a measure theoretic notion analogue to free factors in group theory.

Definition 3.1 A subgroup Λ < Γ is called a measure free-factor of Γ if Γ admits a free p.m.p. action α such that R_{α(Λ)} is freely supplemented in R_{α(Γ)}; i.e. there exists a subrelation S < R such that R = R_{α(Λ)} ∗ S.

This notion is clearly invariant under the automorphism group of Γ. Example: If Λ is a free-factor of Γ, i.e. if Γ decomposes as a free product Γ = Λ ∗ Λ′, then Λ is a measure free-factor. This definition, motivated by Theorem 3.6, of course challenges to exhibit non-trivial examples. This is done in the following:

Theorem 3.2 The cyclic subgroup C generated by the product of commutators κ := ∏_{i=1}^{i=p}[a_{i}, b_{i}] is a measure free-factor in the free group F_{2p} = ⟨a_{1}, ⋯, a_{p}, b_{1}, ⋯, b_{p}⟩.

Since it vanishes in the abelianization, C is not a free factor in the usual sense.

Proof of Th. 3.2: This result is obtained by exhibiting a free p.m.p. action σ of F_{2p} on an (X, μ) whose equivalence relation is also generated by a treeing Φ = (φ_{γ})_{γ∈{κ,a_{1},··· ,a_{p},b_{1},··· ,b_{p}}} made of an automorphism φ_{κ} = σ(κ), defined on the whole of X, and of partially defined isomorphims φ_{γ} that are restrictions of the generators γ.

It follows from the classification of surfaces that the free group F_{2p} is isomorphic with the fundamental group of the oriented surface of genus p with one boundary component (supporting the base point), via an isomorphism sending the product of commutators κ to the boundary curve and the generators to simple closed curves that are disjoint up to the base point.

The Cayley graph G and the universal cover of the Cayley complex associated to the (non-free) presentation

⟨κ, a_{1}, ⋯, a_{p}, b_{1}, ⋯, b_{p} | κ := ∏_{i=1}^{i=p}[a_{i}, b_{i}]⟩ are thus planar, with one orbit of bounded 2-cells, and boundary

edges labelled κ. Its dual graph G^{∗} is a regular tree of valency 2p which does not cross the edges labelled κ. It is equipped with a natural vertex-transitive and free action of F_{2p} which thus may be seen as a “standard” Cayley tree for the free group F_{2p}. Denote by E^{∗} its edge set and by F^{∗} the subset of {0, 1}^{E^{∗}}, corresponding to the (characteristic functions of the) forests, whose connected components are infinite trees with one end. The subset F^{∗} is equipped with a natural F_{2p}-action.

Proposition 3.3 The subset F^{∗} supports an F_{2p}-invariant probability measure ν.

This is an immediate consequence of [Häg98, Cor 2.6 or Th. 4.2]. For completeness, we will give an elementary proof below; now we continue with the proof of Th. 3.2.

Denote the edge set of G by E. A forest f^{∗} ∈ F^{∗} defines a subgraph H(f^{∗}) ∈ {0, 1}^{E} by removing the edges of E that f^{∗} crosses. The end-points of an edge removed (thanks to an edge e^{∗} ∈ f^{∗}) are connected in H(f^{∗}) by the finite planar path surrounding the finite bush of f^{∗} \ {e^{∗}}. It follows that H(f^{∗}) is a connected subgraph of G containing all the edge labelled κ (since the dual G^{∗} does not cross the edges of G labelled κ). By planarity and since all the connected components of f^{∗} are infinite, H(f^{∗}) has no cycle: it is a tree.

The equivariant map H : F^{∗} → {0, 1}^{E} thus pushes the measure ν to an F_{2p}-invariant probability measure μ, supported on the set of (connected) subtrees of G, containing all the edges labelled κ.

Making the action free if necessary (by considering the diagonal action on (X, μ) = (X′, μ′) × ({0, 1}^{E}, μ) for some free p.m.p. F_{2p}-space (X′, μ′), see [Gab04, 1.3.c]), and according, for example, to [Gab04, 1.3.f] this is equivalent with the existence of the required graphing on (X, μ). More precisely, the natural F_{2p}- equivariant map π : X → {0, 1}^{E} forgetting the first coordinate sends x ∈ X to the subset π(x) ⊂ E of the edge set of G. For γ ∈ {κ, a_{1}, ⋯, a_{p}, b_{1}, ⋯, b_{p}} define the Borel subset A_{γ} of those x ∈ X for which the edge [id, γ] of G belongs to π(x) and consider the partially defined isomorphism φ_{γ} := γ_{|A_{γ}}^{−1} . The graphing Φ = (φ_{κ}, φ_{a_{1}}, ⋯, φ_{a_{p}}, φ_{b_{1}}, ⋯, φ_{b_{p}}) matches the conditions: The graph associated with the orbit of x is isomorphic with the subgraph defined by π(x), and thus it is a tree for μ-almost all x ∈ X (Φ is a treeing), Φ generates the same equivalence relation as the F_{2}-action since the graph π(x) is connected, and A_{κ} = X (up to a μ-null set), i.e. φ_{κ} is defined almost everywhere. ■

Proof of Prop. 3.3: Let (Y, m) be a Borel standard probability space with a partition Y = ∐_{i∈ℕ}(U_{i} ∐ V_{i}), where m(U_{i}) = m(V_{i}) = \frac{1}{2^{i+1}} . Let g_{1}, ⋯, g_{2p} be a family of p.m.p. automorphisms of (Y, m) such that g_{1}(U_{i}) = U_{i+1} ∪ V_{i+1} and g_{2}(V_{i}) = U_{i+1} ∪ V_{i+1}. They define a p.m.p. (non-free) action α of F_{2p} = ⟨γ_{1}, ⋯, γ_{2p}⟩ on Y. Denote A_{1} = ∐_{i∈ℕ} U_{i} and A_{2} = ∐_{i∈ℕ} V_{i}. The restrictions of the automorphisms φ_{1} = g_{1|A_{1}} and φ_{2} = g_{2|A_{2}} define a graphing, which is a treeing whose associated graphs are all one-ended trees, and which generates a subrelation of R_{α}. If e_{i}^{∗} denotes the edge [id, γ_{i}^{−1}id] in the Cayley graph of F_{2p} associated with the generating family (γ_{i}), then the conditions π(y)(e_{1}^{∗}) = 1 iff y ∈ A_{1}, π(y)(e_{2}^{∗}) = 1 iff y ∈ A_{2}, and π(y)(e_{i}^{∗}) = 0 for the other i’s, extend by F_{2p}-equivariance to a map Y → F^{∗}. Pushing forward the invariant measure m delivers an instance of the required measure ν. ■

Proposition 3.4 If Λ < Γ is a measure free-factor of Γ and Γ′ is any countable group, then Λ as well as Λ ∗ Γ′ is a measure free-factor of Γ ∗ Γ′.

Proof: Consider an action of Γ, that witnesses the measure free-factor condition. Look at it as an action of Γ ∗ Γ′ via the obvious map Γ ∗ Γ′ → Γ, and consider the direct product action α with any free p.m.p. action of Γ ∗ Γ′. The lifting properties of locally one-to-one morphisms (see Lemma 2.6) applied to the restriction of α to Γ show that α matches the required condition. ■

Corollary 3.5 Let Σ be an oriented surface with r > 0 boundary components. Let π_{1}(Σ, ∗) ≃ F_{q} be its fundamental group and Λ ≃ F_{r} the subgroup generated by the boundary components. Then Λ is a measure free-factor of F_{q}.

Proof: π_{1}(Σ, ∗) admits a free generating set a_{i}, b_{i}, κ_{1}, κ_{2}, ⋯ κ_{r−1} for which ∏_{i=1}^{p}[a_{i}, b_{i}], κ_{1}, κ_{2}, ⋯ κ_{r−1} freely generates Λ. In fact, the product of the boundary components κ_{1}, κ_{2}, ⋯ κ_{r} equals ∏_{i=1}^{p}[a_{i}, b_{i}]. ■

3.2 Application

Theorem 3.6 If Γ is ME to the free group F_{p} (p = 2 or p = ∞) and admits a measure free-factor subgroup Λ, then the iterated amalgamated free product ∗_{Λ}^{n} Γ = Γ ∗_{Λ} Γ ∗_{Λ} ⋯ ∗_{Λ} Γ is ME to F_{p}, and the infinite iterated

amalgamated free product ∗_{Λ}^{∞} Γ = Γ ∗_{Λ} Γ ∗_{Λ} ⋯ ∗_{Λ} Γ ∗_{Λ} ⋯ is ME to F_{∞}. Moreover, Λ remains a measure free-factor in the resulting group.

Applied with Th. 3.2, this gives Th. 1.1 of the introduction:

Corollary 3.7 For each n ∈ ℕ, the iterated amalgamated free product ∗_{C}^{n} F_{2p} is measure equivalent to the free

group F_{2}. The infinite amalgamated free product: ∗_{C}^{∞} F_{2p} is measure equivalent to the free group F_{∞}. Moreover, C is a measure free-factor in the resulting group.

Proof of Th. 3.6: The diagonal action β on the product measure space X of the two actions witnessing each condition on Γ, besides being free and p.m.p.,both is treeable (Lemma 2.4) and admits a splitting as S = S_{β(Λ)} ∗ S′ (the splitting condition lifts, see Lemma 2.6). It thus admits a treeing that splits into two pieces Φ = Φ_{Λ} ∨ Φ′ where Φ_{Λ} generates S_{β(Λ)} and Φ′ generates S′ (S′ is treeable by [Gab00, Th. 5]). Consider

(1) the natural surjective homomorphism π : ∗_{Λ}^{n} Γ → Γ, defined by the identity on each copy of Γ,

(2) the (non free) p.m.p. action of ∗_{Λ}^{n} Γ on X via β ∘ π,

(3) any free p.m.p. action of ∗_{Λ}^{n} Γ on a standard Borel space Y,

(4) the diagonal ∗_{Λ}^{n} Γ-action α^{n} on the product measure space Z := X × Y (it is p.m.p. and free),

(5) the ∗_{Λ}^{n} Γ-equivariant projection to the first coordinate Π : X×Y →X.

For each i = 1, ⋯, n, denote by R_{i} := R_{α^{n}(Γ_{i})} the equivalence relation generated by the restriction of the action α^{n} to the i-th copy Γ_{i} of Γ (resp. R_{Λ} := R_{α^{n}(Λ)} for the restriction to Λ). The morphism Π is locally one-to-one when restricted to R_{i} (Ex. 2.2). Denote by Φ̃_{i} = Φ̃_{i Λ} ∨ Φ̃′_{i} the graphing of R_{i} given from Φ by the lifting lemma 2.4. More concretely: Up to subdividing the domains, each generator φ : A_{φ} → B_{φ} of the treeing Φ is the restriction of the β-action of one element γ_{φ} of Γ. Each σ_{j} of the n homomorphic sections σ_{j} : Γ → ∗_{Λ}^{n} Γ

of π delivers a graphing on Z of the following form: Φ̃_{j} = Φ̃_{jΛ} ∨ Φ̃′_{j}, where each element φ̃ : Ã_{φ} → B̃_{φ} is defined from the element φA_{φ} → B_{φ} of Φ as the restriction of the α^{n}-action of σ_{j}(γ_{φ}) to Ã_{φ} := Π^{−1}(A_{φ}).

By the coincidence of the copies of Λ, it follows that all the Φ̃_{i Λ} coincide, so that Φ̃ = Φ̃_{1 Λ} ∨Φ̃′_{1} ∨Φ̃′_{2} ⋯∨Φ̃′_{n} still generates R_{α^{n}}. Since it lifts a treeing, each Φ̃′_{i} (as well as Φ̃_{1 Λ}) is itself a treeing. The structure of free product with amalgamation of R = R_{1} ∗_{R_{Λ}} R_{2} ∗_{R_{Λ}} ⋯ ∗_{R_{Λ}} R_{n} ensures that Φ̃ is globally a treeing: ∗_{Λ}^{n} Γ admits a

treeable free action, and R_{Λ} is a free-factor. A similar reasoning give the similar result for ∗_{Λ}^{∞} Γ. These groups

are consequently ME to a free group by P_{ME}8. The precise ME-class is determined by the first ℓ^{2} Betti number (by P_{ME}15). ■

Remark 3.8 The treeing Φ produced in the proof of theorem 3.2 cannot be a quasi-isometry, i.e. there is some generator γ ∈ {a_{1}, ⋯, a_{p}, b_{1}, ⋯, b_{p}} for which the length of the Φ-words sending x to γ(x) is not bounded a.s. For otherwise, the graphing produced in the proof of theorem 3.6 for n = 2 would equip each equivalence class with a graph structure (tree) quasi-isometric to the group itself F_{2p} ∗_{C} F_{2p} (the fundamental group of a closed compact surface, without boundary). It follows from the particular form of Φ that the surface group ⟨a_{1}, ⋯, a_{p}, b_{1}, ⋯, b_{p}⟩ ∗_{C=C′} ⟨a′_{1}, ⋯, a′_{p}, b′_{1}, ⋯, b′_{p}⟩ has a treeing made of C and restrictions of the “natural” generators a_{i}, b_{i}, a′_{i}, b′_{i}.

Question 3.9 What are all the measure free-factors of the free group F_{2} ?

Observe that in a group Γ ME to a free group, a cyclic subgroup D strictly contained in a greater cyclic subgroup E cannot be a measure free-factor by Th. 3.6, since iterated amalgamated free product Γ ∗_{D} Γ ∗_{D} Γ is not ME to

a free group: it contains the nonamenable subgroup E ∗_{D} E ∗_{D} E for which β_{1} = 0, then use P_{ME}15** and P_{ME}9.

Question 3.10 It is known that if an amalgamated free product F_{p} ∗_{ℤ} F_{q} happens to be a free group, then ℤ is a free factor in one of F_{p} or F_{q} (see [BF94, Ex. 4.2]). Is it true that similarly if F_{p} ∗_{ℤ} F_{q} ∼^{ME} F_{2} then ℤ is a measure free-factor in F_{p} or F_{q} ?

3.3 Orbit Equivalence of Pairs

We will now make use of the following refined notion of Orbit Equivalence.

Definition 3.11 Consider for i = 1, 2 a group Γ_{i} and a subgroup Γ_{i}^{0}. An orbit equivalence of the pairs (Γ_{1}^{0} < Γ_{1}) and (Γ_{2}^{0} < Γ_{2}) is the data of two p.m.p. actions α_{1} and α_{2} of Γ_{1} and Γ_{2} on the probability measure standard Borel space (X, μ), that generate the same equivalence relation (R_{α_{1}(Γ_{1})} = R_{α_{2}(Γ_{2})}) and such that the restrictions of α_{1} to Γ_{1}^{0} and of α_{2} to Γ_{2}^{0} also define a common sub-relation: (R_{α_{1}(Γ_{1}^{0})} = R_{α_{2}(Γ_{2}^{0})}). The existence of such an equivalence of pairs is denoted by:

(Γ_{1}^{0} < Γ_{1}) ∼^{OE} (Γ_{2}^{0} < Γ_{2})

Definition 3.12 It is a strong orbit equivalence of the pairs if moreover the sub-groups Γ_{1}^{0} and Γ_{2}^{0} are isomorphic, via an isomorphism ϕ : Γ_{1}^{0} → Γ_{2}^{0} that turns the actions conjugate: ∀γ ∈ Γ_{1}^{0}, α_{1}(γ) = α_{2}(ϕ(γ)) in Aut(X, μ). The existence of such an equivalence of pairs is denoted by:

(Γ_{1}^{0} < Γ_{1}) ∼^{OE}_{st} (Γ_{2}^{0} < Γ_{2})

Measure free-factors naturally lead to orbit equivalence of pairs:

Theorem 3.13 Let Γ be a group ME to the free group F_{2} and let Γ_{0} be a subgroup such that β_{1}(Γ)−β_{1}(Γ_{0}) = q is an integer. If Γ_{0} is a measure free-factor of Γ, then there exists a strong orbit equivalence of pairs (Γ_{0} <

Γ) ∼^{OE}_{st} (Γ_{0} < Γ_{0} ∗ F_{q}).

Proof of Th. 3.13: Like in the proof of Th. 3.6, consider (by taking a diagonal action of two actions witnessing both properties of Γ) a free p.m.p. action β of Γ that both is treeable and admits a splitting S_{β} = S_{β(Γ_{0})} ∗ S′. Up to considering the ergodic decomposition, β may be assumed to be ergodic. The subrelation S_{β(Γ_{0})} admits a treeing Φ and S′ admits a treeing Ψ of cost q [Gab02, Cor. 3.16, Cor. 3.23]). A recursive use of Theorem 28.3 of [KM04] (due to Hjorth) gives that S′ may be replaced in the above decomposition by S′′ produced by a free action of the free group F_{q}. The free product decomposition of S_{β} = S_{β(Γ_{0})} ∗ S′′ asserts that the action of Γ_{0} fits well to produce the pair S_{β(Γ_{0})} < S_{β} by a free action of Γ_{0} ∗ F_{q}. The resulting orbit equivalence of pairs is strong by construction. ■

Example 3.14 If C is the cyclic subgroup generated by the product of commutators κ in F_{2p} then (C <

F_{2p}) ∼^{OE}_{st} (ℤ < ℤ ∗ F_{2p−1}).

Example 3.15 If Λ is the subgroup of F_{q} as in Corollary 3.5, then (F_{r} ≃ Λ < F_{q}) ∼^{OE}_{st} (F_{r} < F_{r} ∗ F_{q−r}).

Remark 3.16 Observe that in a free product situation (Γ^{0} < Γ_{1}) ∼^{OE} (Γ^{0} < Γ^{0} ∗ Γ′_{2}), the orbit equivalence of pairs may always be assumed strong, by changing the Γ^{0}-action in the free product.

Theorem 3.17 If (Γ_{1}^{0} < Γ_{1}) ∼^{OE}_{st} (Γ_{2}^{0} < Γ_{2}) are two pairs of groups admitting a strong orbit equivalence of pairs and G is a countable group with a subgroup Γ^{0} isomorphic with Γ_{1}^{0} and Γ_{2}^{0}, then the following pairs admit a strong orbit equivalence of pairs:

(G < G ∗_{Γ^{0}=Γ_{1}^{0}} Γ_{1}) ∼^{OE}_{st} (G < G ∗_{Γ^{0}=Γ_{2}^{0}} Γ_{2}).

In particular, if (Γ^{0} < Γ_{1}) ∼^{OE} (Γ^{0} < Γ^{0} ∗ Γ′_{2}) and G contains a subgroup isomorphic with Γ^{0}, then

(G < G ∗_{Γ^{0}} Γ_{1}) ∼^{OE}_{st} (G < G ∗ Γ′_{2}).

Corollary 3.18 Let G be any countable group and H an infinite cyclic subgroup. Let C be the cyclic subgroup generated by the product of commutators κ in F_{2p}. Then G ∗_{H=C} F_{2p} ∼^{OE} G ∗ F_{2p−1}. In particular, if G is ME to F_{2}, then G ∗_{H=C} F_{2p} ∼^{ME} F_{2}.

Proof of Th. 3.17: Let β_{1} and β_{2} be p.m.p. actions on (Y, ν) defining a strong OE of the pairs given by the assumption. Consider a free p.m.p. action α̃_{1} of G ∗_{Γ^{0}=Γ_{1}^{0}} Γ_{1} on some (X, μ), with a Γ_{1}-equivariant map

p : Y → X (the co-induced action from Γ_{1} to G ∗_{Γ^{0}=Γ_{1}^{0}} Γ_{1} satisfies this property, see section 3.4 below). Call α_{1}

the restriction of α̃_{1} to Γ_{1}. The locally one-to-one morphism p from R_{α_{1}(Γ_{1})} to S_{β_{1}(Γ_{1})} = S_{β_{2}(Γ_{2})} allows to lift (cf. Lemma 2.3) the action β_{2} to an action α_{2} of Γ_{2} on Y, which is orbit equivalent with α_{1}. By uniqueness in Lemma 2.3, strongness also lifts, i.e. α_{1} and α_{2} coincide on Γ_{1}^{0} and Γ_{2}^{0}, via the given isomorphism between these groups^{12}. Now, using α̃_{1}, one extends the Γ_{2}^{0}-restriction of α_{2} to a free action of G, so as to produce an action α̃_{2} of G ∗_{Γ^{0}=Γ_{2}^{0}} Γ_{2}. Given the amalgamated free product structure of Rα̃_{2} = R_{α̃_{2}(G)} ∗_{R_{α̃_{2}(Γ^{0}=Γ^{0}_{1})}} R_{α̃_{2}(Γ_{1})}, and since

α̃_{1} and α̃_{2} produce the same equivalence relation when restricted, on the one hand to G, on the second hand to Γ_{1}^{0} and Γ_{2}^{0}, and on the third hand (!) to Γ_{1} and Γ_{2}, it follows that the action α̃_{2} is free, and forms with α̃_{1} the required strong orbit equivalence of pairs. For the “in particular” part, just observe with remark 3.16 that

(G < G ∗_{Γ^{0}} Γ_{1}) ∼^{OE}_{st} (G < G ∗_{Γ^{0}} Γ^{0} ∗ Γ′_{2}) = (G < G ∗ Γ′_{2}). ■

Question 3.19 A limit group is a finitely generated group Γ that is ω-residually free, i.e. for every finite subset K ⊂ Γ there exists a homomorphism Γ → F to a free group, that is injective on K. It is a natural question to ask whether limit groups are ME to a free group.

3.4 Co-induced action

Let A be a countable group and B a subgroup. The co-induction is a canonical way, from an action β of the “small” group B on Y, to produce an action α of the overgroup A > B on a space X together with a surjective B-equivariant map X → Y. Let λ and ρ be the left A-actions by left multiplication and right multiplication by the inverse on A: λ(g) : h ↦ gh and ρ(g) : h ↦ hg^{−1}. The space X := coInd_{B}^{A}Y is the set of (B, ρ, β)-equivariant maps from A to Y:

coInd_{B}^{A}Y := {Ψ : A → Y | Ψ(ρ(b)a) = β(b)(Ψ(a)); ∀b ∈ B, a ∈ A}.

12Thus, α_{1} and α_{2} also form a strong orbit equivalence for the pairs (Γ_{1}^{0} < Γ_{1}) and (Γ_{2}^{0} < Γ_{2}); with an additional potentiality.

The co-induced action α := coInd_{B}^{A}β is the action of A on coInd_{B}^{A}Y defined from the A-action λ on the source:

coInd_{B}^{A}β(a) : Ψ(.) ↦ Ψ(λ(a)(.)), ∀a ∈ A

The map coInd_{B}^{A}Y → Y; Ψ ↦ Ψ(1), where 1 is the identity element of A, is B-equivariant.

A section s : A/B → A of the canonical map being chosen, the A-space coInd_{B}^{A}Y naturally identifies with Y^{A/B} = ∏_{h∈A/B} Y_{h}, equipped with an action σ of A which permutes the coordinates Y_{h} and then permutes the points in each coordinate Y_{h}, via the β action of an element of B. More precisely, for all a ∈ A, f ∈ Y^{A/B}, h ∈ A/B, the action gives [σ(a)(f)]_{h} = β(b^{−1})(f_{h′}), where b ∈ B and h′ ∈ A/B are defined by the equation s(h′)b = a^{−1}s(h).

It follows that if β preserves a probability measure ν on Y, then σ preserves the product probability measure on Y^{A/B}, and in view of the description of σ, the corresponding (A, α)-invariant measure μ on X is independent of the choice of the section s. If β is essentially free, then α is essentially free.

Remark: It is interesting to compare co-induction with the usual notion of induction (the A-action induced by λ on the second factor, on the quotient of Y × A by the diagonal B-action (β, ρ)). They produce respectively right-adjoint and left-adjoint functors for the restriction functor res : A → B between the categories A of spaces with an A-action and B of spaces with a B-action, with equivariant maps as morphisms. In our context above, induction delivers an action of A on (Y × A)/(B, β, ρ) ≃ Y × A/B with a natural invariant measure which is infinite when B has infinite index in A.

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D. G.: UMPA, UMR CNRS 5669, ENS-Lyon, 69364 Lyon Cedex 7, FRANCE

gaboriau@umpa.ens-lyon.fr