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On the fundamental group of II_1 factors and equivalence relations arising from group actions

Given a countable group G, we consider the sets S_factor(G), S_eqrel(G), of subgroups F of the positive real line for which there exists a free ergodic probability measure preserving action G on X such that the fundamental group of the associated II_1 factor, respectively orbit equivalence relation, equals F. We prove that if G is the free product of Z and infinitely many copies of a non-trivial group \Gamma, then S_factor(G) and S_eqrel(G) contain R_+ itself, all of its countable subgroups, as well as uncountable subgroups whose log can have any Hausdorff dimension in the interval (0,1). We then prove that if G=\Gamma*\Lambda, with \Gamma, \Lambda finitely generated ICC groups, one of which has property (T), then S_factor(G)=S_eqrel(G)={1}. We also show that there exist II_1 factors M such that the fundamental group of M is R_+, but the associated II_\infty factor M tensor B(l^2) admits no continuous trace scaling action of R_+.


Introduction
Some of the most intriguing phenomena concerning group measure space II 1 factors M = L ∞ (X)⋊G and orbit equivalence relations R = R(G X), arising from free ergodic probability measure preserving (p.m.p.) actions G X of countable groups G on probability spaces (X, µ), pertain to their fundamental group F(M ), F(R) ( [18]). Although much progress has been made in understanding and calculating these invariants, many natural questions on how the group G may affect the behavior of F(M ), F(R), remain open.
A first indication that certain properties of G can impact the invariants independently of the way it acts, appeared in Connes' ground breaking work on the classification and the structure of von Neumann factors, from the 1970's. Thus, a side effect of the uniqueness of the amenable II 1 factor [5] and of the amenable II 1 equivalence relation [7], is that F(M ) = F(R) = R + whenever the group G is amenable. On the other hand, arguments from Connes' rigidity paper [4] were used to show that if G is infinite conjugacy class (ICC) and has the property (T) of Kazhdan, then F(M ), F(R) are countable for any free ergodic p.m.p. action of G ( [23], [15]).
Then in the late 1990's, Gaboriau discovered that certain groups G, such as the free groups with finitely many generators, F n , 2 ≤ n < ∞, give rise to orbit equivalence relations R = R(G X) with F(R) = {1}, for any free ergodic p.m.p. action F X [13]. Moreover, many factors of the form L ∞ (X) ⋊ F n were shown to have trivial fundamental group as well (cf. [27], [22]) and it is strongly believed that, in fact, this holds true for all F n X.
In turn, a completely new type of phenomena emerged in the case G = F ∞ , where it was shown that there exist free ergodic p.m.p. actions F ∞ X with the fundamental group of the associated II 1 factors and equivalence relations, F(M ), F(R), ranging over a "large" family of subgroups F ⊂ R + , containing R + itself, all its countable subgroups, as well as "many" uncountable subgroups = R + [29]. In fact, it was conjectured in [29] that any group F that can be realized as a fundamental group of a II 1 factor or equivalence relation, can also be realised as F(L ∞ (X) ⋊ F ∞ ), F(R(F ∞ X)), for some free ergodic p.m.p. action F ∞ X.
Related to all these phenomena, we introduced in [29] the sets S factor (G), S eqrel (G), of subgroups F ⊂ R + for which there exists a free ergodic m.p. action G X such that F(L ∞ (X) ⋊ G) = F, respectively F(R(G X)) = F. Using this notation, the result in [29] shows, more precisely, that S factor (F ∞ ) ∩ S eqrel (F ∞ ) contains the set S centr of all subgroups F ⊂ R + for which there exists a free ergodic action of an amenable group Λ on an infinite measure space (Y, ν), such that the set of scalars t > 0 that can appear as scaling constants of non-singular automorphisms θ of (Y, ν) commuting with Λ Y , equals F. In turn, S centr is shown to contain R + , all its countable subgroups and uncountable subgroups F ⊂ R + with the Hausdorff dimension of log(F) ⊂ R ranging over all the interval (0, 1) ( [29]). While an abstract characterization of S factor (F ∞ ), S eqrel (F ∞ ) remains elusive, it was noticed in [29] that subgroups in either set, as in fact subgroups in S factor (G), S eqrel (G) for any G, must be Borel sets and Polishable.
Our purpose in this paper is to estimate (or even completely calculate) the invariants S factor (G), S eqrel (G) for other classes of groups G. We target two types of results: on the one hand, detecting classes of groups G for which S factor (G), S eqrel (G) are "large", containing for instance the set S centr defined above (like in the case case G = F ∞ ); on the other hand, detecting classes of groups G for which S factor (G), S eqrel (G) contain only "small" subgroups of R + (e.g. countable, or just {1}).
Thus, our first result enlarges considerably the class of groups G for which we can show that the set S centr is contained in both S factor (G) and S eqrel (G). Theorem 1.1. Let Γ be a non-trivial group, Σ an infinite amenable group and denote G = Γ * ∞ * Σ. Then, S centr ⊂ S factor (G) and S centr ⊂ S eqrel (G) .
Moreover, there exist free ergodic p.m.p. actions G (X, µ) such that the II 1 factor M = L ∞ (X)⋊ G has fundamental group F(M ) = R + , but the II ∞ factor M ⊗ B(ℓ 2 (N)) admits no trace scaling action of R + .
In Section 6, we will show that if the full group of an equivalence relation R on a probability space (X, µ) contains a property (T) group acting ergodically on X, then F(R) follows countable. Thus, if a group Γ appearing in Theorem 1.1 contains an infinite subgroup Λ with the property (T) and if G = Γ * ∞ * Σ X is a free ergodic p.m.p. action such that R G has fundamental group equal to an uncountable group in S centr , then the restriction of G X to Λ cannot be ergodic.
Note that the last part of Theorem 1.1 provides group measure space II 1 factors M = L ∞ (X) ⋊ G which do have fundamental group equal to R + yet cannot appear in the continuous decomposition of a type III 1 factor. The problem of whether such II 1 factors exist was posed over the years by several people, including Connes, Takesaki, and more recently Shlyakhtenko. The fact that there are even factors of the form L ∞ (X) ⋊ F ∞ satisfying this property (by simply taking Γ = Σ = Z in 1.1) should be contrasted with the fact that the II ∞ factor associated with L(F ∞ ) does admit a trace scaling action of R + , by [32].
Note that all groups of the form G = Γ * ∞ * Σ, covered by the above theorem, have infinite first ℓ 2 -Betti number, β 1 (G) = ∞, and in fact β (2) n (G) = ∞, 0, ∀n ≥ 2. On the other hand, by Gaboriau's scaling formula for ℓ 2 -Betti numbers [13], any free ergodic p.m.p. action of a group G with β (2) n (G) = 0, ∞, for some n, gives rise to an equivalence relation R G with trivial fundamental group, F(R G ) = {1}. In other words, S eqrel (G) = {{1}}. While it is still an open question whether the corresponding II 1 factors M = L ∞ (X)⋊ G satisfy F(M ) = {1} as well (i.e. S factor (G) = {{1}}), our next result provides a large class of groups G for which this is indeed the case. Theorem 1.2. Let Γ and Λ be infinite, finitely generated groups. Assume that Γ is ICC and that one of the following conditions holds. a) Γ = Γ 1 × Γ 2 , with Γ 1 non-trivial and Γ 2 non-amenable, b) Γ admits a non virtually abelian, normal subgroup Γ 1 with the relative property (T).
When viewed from the perspective of Connes' discrete decomposition of type III λ factors with 0 < λ < 1 ( [6]) and respectively Connes-Takesaki continuous decomposition of type III 1 factors ([10]), the above result provides a large class of groups G with the property that no II 1 factor M arising from an arbitrary free ergodic p.m.p. action of G can appear in the decomposition of a type III factor (i.e., as Connes puts it, no such M can appear as the "shadow" of a type III factor).
While II 1 factors M = L ∞ (X) ⋊ Γ arising from free ergodic p.m.p. actions Γ X of ICC property (T) groups always have countable fundamental group (cf. [4], [23], [15]), it was not known whether there exist cases when F(M ) = {1}. Our next result gives the first such examples. It also provides the first "concrete" examples of free ergodic p.m.p. actions Γ X with the associated II 1 factors M having fundamental group = {1}, R + . Indeed, the actions in Theorem 1.1 above and in [29] are shown to exist by using a Baire-category argument, at some point, while in 1.3 below they are specific G-actions, obtained as diagonal products of Bernoulli and profinite actions. Theorem 1.3. Let F ⊂ Q + be a subgroup generated by a subset of the prime numbers. Let G = Z n ⋊ SL(n, Z) with n ≥ 3. Then, G admits a free ergodic p.m.p. action G (X, µ) such that the fundamental group of L ∞ (X) ⋊ G and of R(G X) equals F.
We in fact believe that any subgroup of Q + can be realized as the fundamental group of a factor or equivalence relation arising from a free ergodic p.m.p. action of Z n ⋊ SL(n, Z), n ≥ 3. The question of whether there exist free ergodic p.m.p. actions of an ICC property (T) group G X such that F(R G ) or F(L ∞ (X) ⋊ G) contain irrational numbers, remains open. In fact, it is not even known whether the union of all the fundamental groups of II 1 factors and equivalence relations arising from free ergodic p.m.p. actions of a fixed ICC property (T) group G is necessarily countable or not.
Finally, noticing that for a large number of groups G it is known that {1} ∈ S factor (G) (see e.g. [25], [27], [28]), we conjecture that this is in fact the case for all non-amenable groups G. If true, this would also show that the only possibilities for S factor (G), S eqrel (G) to be single point sets are S factor (G) = S eqrel (G) = {R + }, S factor (G) = S eqrel (G) = {{1}}, the first situation corresponding to G being amenable. This would provide a new, interesting facet of the dichotomy amenable/nonamenable for groups.

Preliminaries
The fundamental group F(M ) of a II 1 factor M , introduced in [18], is defined as the following subgroup of R + .
We call II 1 equivalence relation on a standard probability space (X, µ) every ergodic probability measure preserving (p.m.p.) measurable equivalence relation with countable equivalence classes. The fundamental group F(R) of a II 1 equivalence relation R is defined as Whenever Γ (X, µ) is a free ergodic p.m.p. action, we denote by R(Γ X) the associated II 1 orbit equivalence (OE) relation and by L ∞ (X) ⋊ Γ the associated group measure space II 1 factor [18]. such that F(R(Γ X)) = F} .
Following [1, Section 4], we call ergodic measure on R any σ-finite measure ν on the Borel sets of R having the following properties, where we denote λ x (y) = x + y.
• There exists a countable subgroup Q ⊂ R such that ν • λ x = ν for all x ∈ Q and such that every Q-invariant Borel function on R is ν-almost everywhere constant.
For every ergodic measure ν on R, one defines As shown in [1], the groups H ν can have arbitrary Hausdorff dimension and all exp(H ν ) belong to S centr . We refer to [29, Section 2 and proof of Theorem 5.3] for a detailed exposition.
Intertwining by bimodules and the notation A ≺ M B In Sections 3 and 4, we use the method of intertwining by bimodules, introduced by the first author in [27]. Let (M, τ ) be a von Neumann algebra with faithful normal tracial state τ . We use the notation 3 Groups G for which S factor (G) contains uncountable groups The following theorem, whose proof is given at the end of the section, provides a large family of groups G such that S factor (G) and S eqrel (G) is large, in the sense that both contain S centr . Moreover, we prove that G admits free ergodic p.m.p. actions G (X, µ) such that the II 1 factor M := L ∞ (X)⋊ G has fundamental group R + , but nevertheless, the II ∞ factor M ⊗ B(ℓ 2 (N)) admits no strongly continuous trace scaling action of R + .
The groups G involved are infinite free product groups and should be opposed to the groups G treated in Theorem 4.1, for which S factor (G) is trivial (cf. Remark 4.2).
Moreover, there exist free ergodic p.m.p. actions G (X, µ) such that the II 1 factor M = L ∞ (X)⋊ G has fundamental group F(M ) = R + , but the II ∞ factor M ⊗ B(ℓ 2 (N)) admits no trace scaling action of R + .
In the course of the proof of Theorem 3.1, we will also obtain the following result. Let G be a countable group with subgroup Γ. Suppose that G (X, µ) is a free p.m.p. action such that the restriction to Γ is ergodic. Slightly changing notations compared to [29,Section 2], denote by Emb(Γ, G) the set of non-singular partial automorphisms φ of (X, µ) satisfying φ(g ·x) ∈ G·φ(x) for all g ∈ Γ and almost all x ∈ X with x, g · x ∈ D(φ). Denote by [[G]] the full pseudo-group of the OE relation R(G X), i.e. the set of a partial automorphisms φ of (X, µ) satisfying φ(x) ∈ G · x for almost all x ∈ D(φ).
• The restriction of α to Γ * ∞ is ergodic and rigid (in the sense of Definition 2.1).
• The restriction of α to any of the copies of Γ, resp. to Λ, is conjugate with the originally given action.
Extend the action α 0 inductively to free actions Γ n * Λ αn (X, µ) following the procedure in [29, Section 3] and such that the restriction of α n to G k ⊂ Γ n is conjugate with β for all k ≤ n. We end up with the free action Γ * ∞ * Λ α∞ (X, µ). For every infinite subset E ⊂ N, we denote by α E the restriction of α ∞ to Γ E * Λ. Following the proof of [29, Theorem 4.1], there exists an infinite Since Γ E ∼ = Γ * ∞ , the lemma is proved.
Remark 3.4. Using the methods of [14, Section 2.3], Lemma 3.3 can be shown for Γ 1 * Γ 2 instead of the infinite free product Γ * ∞ , for arbitrary infinite groups Γ 1 , Γ 2 with given free p.m.p. actions on (X, µ). Such a generalization does not provide a refinement for Theorem 3.1 though, since the proof of Theorem 3.1 involves taking once more an infinite free product.
For the formulation of the following theorem, recall that the automorphism group Aut(N ) of a von Neumann algebra with separable predual is a Polish group under the topology making the maps Aut(N ) → N * : α → ω • α continuous for all ω ∈ N * . Similarly, the group Aut(Y, η) of non-singular isomorphisms of (Y, η) (up to equality almost everywhere) is a Polish group and Centr Aut Y (Γ 2 ) is a closed subgroup whenever Γ 2 (Y, η) is a non-singular action.
Make the following assumptions.
• The restriction of α to Γ 1 is ergodic and rigid.
Then, the following holds.

Define the II
is closed and normal in Aut(N ) and the map is an isomorphism and homeomorphism of Polish groups.
Proof. The proof of point 1 is identical to [29,Lemma 5.1]. It remains to prove point 2.
Write A = L ∞ (X) and B = L ∞ (Y ). We first prove that every automorphism of N preserves the Cartan subalgebra A⊗B up to unitary conjugacy. Together with point 1, this implies 2(a). So, let θ be an automorphism of N := (A⊗B) ⋊ (Γ 1 * Γ 2 ). Take a projection p ∈ A⊗B of finite trace and put q = θ(p). After unitary conjugacy, we may assume that q ∈ A⊗B. By [27, Theorem A.1], it is sufficient to prove that θ(Ap) ≺ qN q (A⊗B)q.
Since θ(Ap) ⊂ qN q is rigid, [17, Theorem 5.1] implies that We finally prove 2(b). Observe that U(N ) is a Polish group in a natural way and that the map U(N ) → Aut(N ) : u → Ad u is a continuous group morphism. Define H as in the formulation of the theorem and note that H is a closed subgroup of Aut(N ). We form the semi-direct product Polish group U(N ) ⋊ H in such a way that π : Note that π is continuous and denote K := (U(N ) ⋊ H)/ Ker π. Again, K is a Polish group. We form the semi-direct product Polish group K ⋊ Centr Aut Y (Γ 2 ) in such a way that is a group morphism. Then, ρ is a continuous and injective group morphism between Polish groups. Moreover, by 2(a), ρ is onto. So, ρ is a homeomorphism. Hence, Inn(N ) · H = ρ(K) is closed and normal in Aut(N ) and the map ∆ → θ ∆ provides an isomorphism and homeomorphism between Centr Aut Y (Γ 2 ) and Aut(N )/(Inn(N ) · H). (5) Note that H is isomorphic to the group of S 1 -valued 1-cocycles for the action Γ1 * Γ2 X × Y . Lemma 3.6. Let Γ * Λ (X, µ) be a free p.m.p. action with the restriction to Γ being ergodic. Let Λ (Y, η) be a free ergodic action preserving the infinite standard measure η. Assume that Λ is amenable. Consider Γ * Λ X × Y as in (3.1). Let Z ⊂ Y be a subset of finite measure and define the II 1 Whenever Σ is an infinite amenable group, there exists a free ergodic p.m.p. action Proof. Denote by R 1 the equivalence relation given by the restriction of R(Λ X × Y ) to X × Z. Note that R 1 need not be ergodic. Since Λ is amenable and almost every equivalence class of R 1 is infinite, the results in [7] and [20] allow to take a free p.m.p. action Σ X × Z whose OE relation is precisely R 1 .
Since the action of Λ on (Y, η) is ergodic, take φ n ∈ [[Λ]] with dom(φ n ) = X × Z and range(φ n ) = X × Z n , where Z n , n ∈ N, forms a partition of Y (up to measure zero). Since the action of Γ leaves every X × Z n globally invariant, we can view φ −1 n Γφ n as a group of automorphisms of X × Z. It is now an exercise to check that R is freely generated by the OE relations of φ −1 n Γφ n , n ∈ N, together with Σ X × Z. This provides us with the required free action of Γ * ∞ * Σ X × Z.
The following is the final ingredient in the proof of Theorem 3.1.
Proof. As explained in [29,Section 2], an ergodic measure ν on R can be associated to any pair (a n ), (b n ) of sequences in N satisfying ∞ n=1 b −1 n < ∞ and b n < a n /2 for all n, in such a way that a n b n a 1 · · · a n−1 x < ∞ where x denotes the distance of x ∈ R to Z ⊂ R. Take a n = 2 2n+2 , b n = 2 2n and associate with it the ergodic measure ν. Take a ′ n = 2 2n+1 , b ′ n = 2 2n−1 and associate with it the ergodic measure ν ′ . First of all, Let now x ∈ R and write x n a 1 · · · a n with x n ∈ {0, . . . , a n − 1} .
Write for every n ∈ N, x n = y n + √ a n z n with y n , z n ∈ {0, . . . , √ a n }. Define y n a 1 · · · a n and z = ∞ n=1 2z n a ′ 1 · · · a ′ n .
One checks that y ∈ H ν , z ∈ H ν ′ and x = y + z. So, Proof of Theorem 3.1. Since Γ * ∞ = (Γ * Γ) * ∞ , we may assume that Γ is an infinite group. Let Σ, Λ be infinite amenable groups and Λ (Y, η) a free ergodic action preserving the infinite standard measure η. Set G := Γ * ∞ * Σ. We prove the existence of a free ergodic p.m.p. action of G Z such that the associated II 1 factor M := L ∞ (Z) ⋊ G and equivalence relation R := R(G Z) have the following properties.
1. The fundamental group of M and the fundamental group of R equal mod(Centr Aut Y (Λ)).
2. The II ∞ factor M ⊗ B(ℓ 2 (N)) admits a strongly continuous trace-scaling action of R + if and only if the group morphism mod : is onto and splits continuously.
to a subset Z of finite measure. By Lemma 3.6, we can take a free ergodic p.m.p. action G Z whose OE relation equals R. By point 1 of Theorem 3.5, If the group morphism (3.2) splits continuously, it is clear that N admits a strongly continuous trace scaling action. The converse follows from point 2(b) of Theorem 3.5.
In order to conclude the proof of Theorem 3.1, we have to construct an action Λ (Y, η) such that mod(Centr Aut Y (Λ)) = R + , but the morphism (3.2) does not split continuously. By Lemma 3.7, we can take ergodic measures [29], we can take amenable groups Λ 1 , Λ 2 and free ergodic infinite measure preserving actions Λ i (Y i , η i ) such that mod(Centr Aut Y i (Λ i )) = exp(H ν i ). Since the homomorphism mod is continuous, we equip exp(H ν i ) with the (Polish) quotient topology. In this way, the H ν i become Polish groups and the embedding H ν i ֒→ R continuous.
We prove that mod : admits no continuous splitting. Assume that it does. Since the left-hand side of the previous formula equals Centr Aut admits a continuous splitting. We then find continuous homomorphisms θ i : R → H ν i such that x = θ 1 (x) + θ 2 (x) for all x ∈ R. Since the embedding H ν i ֒→ R is continuous, there exist λ i ∈ R such that θ i (x) = λ i x for all x ∈ R. But, H ν i = R, forcing λ i = 0 for i = 1, 2, a contradiction.

Groups G for which S factor (G) is trivial
Combining results from [3,13,17], we prove that for the following groups G, S factor (G) is trivial.  Since v * v commutes with ρ(L(Γ 1 )), we may first of all assume that q = v * v. Next, it follows that v * θ(L(Γ))v ⊂ qM n i q. Hence, θ(L(Γ)) ≺ M M i .
Repeating the previous paragraph, we may assume that ρ : L(Γ) → qM n i q, θ(a)v = vρ(a) for all a ∈ L(Γ) and v * v = q. Since Γ is an ICC group, we get that So, vv * ∈ θ(A Γ ). It follows that v * θ(A)v is a Cartan subalgebra of qM n q. Moreover, for all g ∈ Γ, the unitary v * θ(u g )v = ρ(u g ) belongs to qM n i q and normalizes v * θ(A)v. Then, [17, Theorem 1.8] implies that there exists w ∈ qM n such that ww * = q, w * w ∈ A n and w * v * θ(A)vw = w * wA n . It follows that θ(A) and pA are unitarily conjugate. 5 S factor (Z n ⋊ SL(n, Z)) is non-trivial, for all n ≥ 3 When Γ is an ICC property (T) group, all groups in S factor (Γ) are countable (cf. [15, Proof of Theorem 1.7], or [23, Theorem 4.5.1]). Nevertheless, S factor (Γ) can be non-trivial, as shown by the next theorem, in which we show that if Γ = Z n ⋊ SL(n, Z), n ≥ 3, then S factor (Γ) contains "many" subgroups of Q + . It is unclear though whether there exists a free ergodic p.m.p. action Γ (X, µ) of an ICC property (T) group Γ such that F(R(Γ X)) ⊂ Q + .
Theorem 5.1. Let F ⊂ Q + be a subgroup generated by a subset of the prime numbers. Let Γ = Z n ⋊ SL(n, Z) with n ≥ 3. Then, Γ admits a "concrete" free ergodic p.m.p. action Γ (X, µ) such that both the fundamental group of L ∞ (X) ⋊ Γ and of R(Γ X) equal F.
We prove Theorem 5.1 as a consequence of the following more general result.  Before proving Theorem 5.2, we introduce some terminology and an auxiliary result. Recall that a 1-cocycle ω : Γ × X → Λ for an action Γ (X, µ) with values in a countable group Λ, is a measurable map satisfying ω(gh, x) = ω(g, h · x) ω(h, x) for all g, h ∈ Γ and almost all x ∈ X .
• There exists a partition X = n X n of X into measurable subsets X n ⊂ X and there exist measurable subsets X ′ n ⊂ X ′ such that for every n ∈ N, the restriction of ∆ to X n is a non-singular isomorphism between X n and X ′ n .
By ergodicity, every of these non-singular isomorphisms ∆| Xn : X n → X ′ n is measure scaling, with the scaling being independent of n. This scaling factor is called the compression constant of the stable OE ∆ and denoted by c(∆).
The Zimmer 1-cocycle ω : Γ × X → Λ associated with the stable OE ∆ is defined by Note that similar stable OEs give rise to cohomologous 1-cocycles.
Whenever X 0 ⊂ X and X ′ 0 ⊂ X ′ are non-negligible measurable subsets and ∆ 0 : X 0 → X ′ 0 is a non-singular isomorphism satisfying ∆ 0 (Γ·x∩X 0 ) = Λ·∆ 0 (x)∩X ′ 0 for almost all x ∈ X 0 , ergodicity allows to choose a measurable map p : X → X 0 with p(x) ∈ Γ · x for almost all x ∈ X and then, ∆ := ∆ 0 • p defines a stable OE. Another choice of p gives rise to a similar stable OE. It follows that F(R(Γ X)) = {c(∆) | ∆ is a stable OE between Γ X and Γ X} .
Denote by ∆ 1 the restriction of ∆ to X 1 . Then, ∆ 1 is a stable OE between Γ 1 X 1 and Λ Y . By construction, the 1-cocycle associated with ∆ 1 is cohomologous to a homomorphism from Γ 1 to Λ. The conclusion of the proposition now follows from [35,Lemma 4.7].
In order to show the equality of the fundamental groups of the II 1 factor and the II 1 equivalence relation associated with the Γ-actions defined in Theorem 5.2, we need the following result about automatic preservation of Cartan subalgebras.
Proposition 5.5. Let Γ be a countable group having a normal, non virtually abelian subgroup Σ with the relative property (T). Let Γ (X, µ) be a free ergodic p.m.p. action and assume that this action admits a free and profinite quotient: there exists a free profinite p.m.p. action Γ (X 1 , µ 1 ) and a quotient map π : X → X 1 satisfying π(g · x) = g · π(x) almost everywhere.
Let (Y 0 , η 0 ) be a non-trivial standard probability space and set (Y, Since Σ is normal in Γ and since Σ X 1 is profinite, the quasi-normalizer of L(Σ) inside M contains A 1 ⋊ Γ. Since Σ is non virtually abelian, L(Σ) cannot be embedded in an amplification of A. So, by [35,Proposition D.5 We are now ready to prove Theorem 5.2.
Proof of Theorem 5.2. Put (X, µ) = lim ← − Γ/Γ n as in the formulation of the theorem. We assume Γ X to be (essentially) free. Let Y = [0, 1] Γ and denote by η the infinite product of the Lebesgue measure on [0, 1]. We consider the diagonal action Γ X × Y .
By Proposition 5.5, we have Whenever Λ 1 < Γ is a subgroup containing Γ n for large enough n, the action Γ (X, µ) is induced from the action Λ 1 Let ∆ : X × Y → X × Y be a stable OE between Γ X × Y and itself. We have to prove that c(∆) belongs to the set defined in (5.1). Proposition 5.4 provides us with finite index subgroups G 1 , G 2 of Γ, non-negligible measurable subsets Z 1 , Z 2 ⊂ X × Y and a finite normal subgroup H of G 1 such that Γ X × Y is induced from G i Z i and such that G 1 /H Z 1 /H is conjugate with G 2 Z 2 , say through the isomorphism ∆ : Z 1 /H → Z 2 and the group isomorphism δ : Moreover, still because the Bernoulli action is mixing, ∆(x, y) = (∆ 0 (x), . . .), where ∆ 0 : X 1 /H → X 2 is an isomorphism conjugating the actions G 1 /H X 1 /H and G 2 X 2 through the group isomorphism δ : Denote by π n : X → Γ/Γ n the natural quotient map. By [16,Lemma 4.1], we find k ∈ N and g ∈ Γ such that gΓ k g −1 ⊂ G 1 and X 1 = π −1 k (G 1 gΓ k ). Moreover, since Γ X is free, we can take k large enough and assume that H ∩ Γ k = {e}. Replacing G 1 by g −1 G 1 g and X 1 by g −1 · X 1 , we may assume that g = e. Note that G 1 /H H)). Applying as above [16,Lemma 4.11], we find h ∈ Γ such that, after replacing δ by g → hδ(g)h −1 and ∆ 0 by x → h · ∆ 0 (x), we have Γ n ⊂ Λ 2 := δ((Γ k H)/H) for n large enough and Denoting Λ 1 = Γ k , we have constructed finite index subgroups Λ 1 , Λ 2 ⊂ Γ such that Γ n ⊂ Λ 1 ∩ Λ 2 for n large enough and such that the actions Λ i lim ← − Λ i /Γ n are conjugate for i = 1, 2. Tracing back the construction, we also have concluding the proof of the theorem.
Proof of Theorem 5.1. Let F be a subgroup of Q + generated by a non-empty subset P of the prime numbers. The case F = {1} will be discussed at the end of the proof. Denote by R the subring of Q generated by P −1 . Note that R * = F ∪ (−F). Set G = R n ⋊ GL(n, R) and Γ = Z n ⋊ SL(n, Z). Let G = {g 1 , g 2 , · · · } and define the finite index subgroups Γ k < Γ as Γ k = Γ ∩ k i=1 g i Γg −1 i . Define the profinite action Γ (X, µ) := lim ← − Γ/Γ k . We first argue why Γ (X, µ) is essentially free. Let p ∈ P and take k 1 < k 2 < k 3 < · · · such that (0, p l 1) ∈ {g 1 , . . . , g k l }. One checks that Γ k l ⊂ G l := p l Z n ⋊ SL(n, Z) and hence, it suffices to prove freeness of Γ lim ← − Γ/G l . The latter has been shown in [16, discussion before Corollary 5.8].

Property (T) and countability of the fundamental group
In his celebrated "rigidity paper" [4], Connes showed that II 1 factors arising from ICC groups with the property (T) of Kazhdan have countable fundamental group. Using the same ideas, it was later shown that for a separable II 1 factor M to have countable F(M ), it is in fact sufficient that M contains a subfactor with the property (T) in the sense of [9] and having trivial relative commutant, N ′ ∩ M = C (cf. Theorem 4.6.1 in [23]; see also [19] for a more general statement). It was also shown that if M is a separable II 1 factor, then the family of subfactors N i ⊂ M, i ∈ I, having property (T) and trivial relative commutant, is countable modulo conjugacy by unitaries in M (cf. Theorem 4.5.1 in [23]; see also [21] for a related result). In this section, we prove some analogous results for II 1 equivalence relations.
In particular, these results show that given any Kazhdan group Γ, S eqrel (Γ) can only contain countable subgroups of R + , and if in addition Γ is ICC then the same holds true for S factor (Γ). More generally, Part 1 of Theorem 6.4 below shows that this is still the case if the center of Γ "virtually coincides" with its FC radical (as defined before 6.4). However, if one drops this assumption on Γ, then the situation seems to become quite complicated. Thus, Part 2 of Theorem 6.4 shows that if a property (T) group Γ is residually finite and has non-virtually abelian FC radical, then Γ admits free ergodic p.m.p. actions Γ (X, µ) such that L ∞ (X) ⋊ Γ is McDuff and hence, its fundamental group is equal to R + .
We first need some notations. Thus, if R is a II 1 equivalence relation on the standard probability space (X, µ), then we denote by [R] the full group of the equivalence relation R, consisting of all non-singular isomorphisms φ : X → X satisfying (x, φ(x)) ∈ R for almost all x ∈ X. The full pseudogroup of R is denoted by [[R]] and consists of all non-singular partial automorphisms φ between measurable subsets D(φ), R(φ) ⊂ X, satisfying (x, φ(x)) ∈ R for almost all x ∈ D(φ). Theorem 6.1. Let R be a II 1 equivalence relation on the probability space (X, µ).

If [R] contains a property (T) group Γ implementing an ergodic action on
contains a countable group Γ having a subgroup H ⊂ Γ with the relative property (T) implementing an ergodic action on (X, µ), then F(R) is countable.

Let T be the set of property (T) subgroups Γ ⊂ [[R]] acting ergodically on s(Γ). Then T is countable, modulo conjugacy by elements in [[R]].
Note that the ergodicity assumption of the action of Γ on (X, µ) in Part 1 of the above statement is crucial. Indeed, Theorem 3.1 provides examples of free ergodic p.m.p. actions G (X, µ) such that R(G X) has uncountable fundamental group, but nevertheless G contains a subgroup having property (T) (which, a fortiori, acts non-ergodically on (X, µ)). In turn, the existence of a property (T) subgroup of [R] acting freely and ergodically on X, does not insure that the II 1 factor L(R) has countable fundamental group. Indeed, by [8] there exist free ergodic p.m.p. actions of groups of the form G = Γ × Σ, with Γ having property (T) and acting by Bernoulli shifts (thus ergodically), such that M = L ∞ (X) ⋊ G splits off the hyperfinite II 1 factor, and thus F(M ) = R + . In fact, as pointed out in [26], more than being countable, the fundamental group of R(G X) is trivial.
Note also that in the case R comes from a free ergodic action of a property (T) group, Γ (X, µ), Part 1 of Theorem 6.1 was already shown in [15,Corollary 1.8], in the case Γ is ICC, and in [16,Theorem 5.9], in the general case. We will use the above result in [31], to prove that the II 1 equivalence R obtained by restricting the II ∞ equivalence relation implemented by SL(n, Z) R n , n ≥ 4, to a subset of measure 1, has property (T) in the sense of Zimmer, yet cannot be implemented by an action (even non-free) of a property (T) group because F(R) = R + .
We will prove Theorem 6.1 by contradiction, using the property (T) of the subgroups and a "separability" argument, in the spirit of [23]. For more on this strategy of proofs, which grew out of Connes' rigidity paper [4], we send the interested reader to Section 4 in [28]. As a result of this argument, we obtain two copies Γ 1 , Γ 2 ⊂ [[R]] of the same property (T) group, which are uniformly close one to the other. This in turn gives rise to a non-zero intertwiner φ ∈ [[R]] between Γ 1 , Γ 2 . But if the Γ i -actions are assumed ergodic, this forces µ(s(Γ 1 )) = µ(s(Γ 2 )) and the conjugacy of The existence of an intertwiner between uniformly close subgroups in [[R]] is the subject of the next lemma. Recall that [[R]] has a natural metric space structure, inherited from the Hilbert-norm · 2 of the underlying II 1 factor L(R) associated with R, by viewing every φ ∈ [[R]] as a partial isometry in L(R) and using the · 2 -norm on the latter. The metric can be concretely written as where △ denotes the symmetric difference of two sets. We will also need the natural σ-finite measure µ (1) on R ⊂ X × Y , defined by the formula for all measurable subsets U ⊂ R. Lemma 6.2. Let R be a II 1 equivalence relation on the standard probability space (X, µ). Suppose that Γ is a countable group, X 0 , Y 0 ⊂ X and let be group morphisms satisfying d(α g , β g ) ≤ 1/5 for all g ∈ Γ and µ(X 0 ), µ(Y 0 ) ≥ 3/4. Then, there exist non-negligible measurable subsets Proof. Denote by Tr the normal faithful semi-finite trace on L ∞ (R) given by integration along µ (1) . Let p ∈ L ∞ (R) be the projection onto R∩X 0 ×Y 0 and e ≤ p the projection onto {(z, z) | z ∈ X 0 ∩Y 0 }. Set B = L ∞ (R)p. The group Γ acts by automorphisms ρ g of B given by (ρ g F )(x, y) = F (α g −1 (x), β g −1 (y)) for almost all (x, y) ∈ R ∩ X × Y .
Define a ∈ B + as the unique element of minimal · 2,Tr in the weakly closed convex hull conv{ρ g (e) | g ∈ Γ}. It follows that a − e 2,Tr ≤ 1/5 and that ρ g (a) = a for all g ∈ Γ. Note that 0 ≤ a ≤ 1.
Choose for every t ∈ (0, 1) a measurable subset Y t ⊂ X with µ(Y t ) = t and such that Y s ⊂ Y t if s ≤ t. Assume that the fundamental group of R is uncountable. For every t ∈ F(R) ∩ (3/4, 1), choose an isomorphism ∆ t : X → Y t between R and R| Yt . Note that ∆ t scales the measure µ by t. (1) ) and the unitary representation Set ∆ s := {(y, y) | y ∈ Y s } and ξ 0 := s −1/2 χ ∆s . Then, ξ 0 is a unit vector in H and, for all g ∈ F , Since also, given g ∈ F , Since (α h ) h∈H implements an ergodic action on (X, µ), the same holds for (α s h ) h∈H , (α t h ) h∈H and so, Lemma 6.2 provides an element φ ∈ [[R]] with D(φ) = Y s and R(φ) = Y t . Since φ is a measure preserving isomorphism between Y s and Y t and µ(Y s ) = s < t = µ(Y t ), we reached a contradiction.
To prove Part 2 of Theorem 6.1, assume by contradiction that there exist uncountably many ] which have property (T) and are non-conjugate in [[R]]. We continue to use the measurable subsets Y t ⊂ X with µ(Y t ) = t and Y s ⊂ Y t whenever s ≤ t. By the ergodicity of R, we may assume that for every i ∈ I, the support of Γ i is one of the Y s . By Shalom's theorem [33,Theorem 6.7], every property (T) group is the quotient of a finitely presented property (T) group. Since there are only countably many finitely presented groups, we may assume that all Γ i 's are representations α i of the same property (T) group Γ. Finally, we may assume that there exists t ∈ (0, 1) such that µ(s(Γ i )) ∈ (3t/4, t) for all i ∈ I. So, replacing R by R| Yt , we may assume that µ(s(Γ i )) ∈ (3/4, 1) for all i ∈ I.
Part 2 of Theorem 6.1 readily implies that the functor Γ → R Γ , from free ergodic p.m.p. actions of property (T) groups with morphisms given by conjugacy, to the associated equivalence relations with morphisms given by orbital isomorphism, is "countable to one". In other words, there are at most countably many non-conjugate free ergodic p.m.p. actions in each OE class of a free ergodic p.m.p. action of a property (T) group. In fact, even more is true: any free ergodic p.m.p. action of a property (T) group follows "orbit equivalent superrigid, modulo countable classes", in a sense made precise below.
Corollary 6.3. Let Γ X be a free ergodic p.m.p. action of a property (T) group. Let Λ i X i , i ∈ I, be a family of free ergodic p.m.p. actions such that R Γ ≃ R t i Λ i , for some t i > 0. Then the family I is countable, modulo conjugacy of actions.
Proof. We may assume that t i ≥ 1/c for all i ∈ I and some c > 0. Setting R = (R Γ ) c , we can view all Λ i as subgroups of [[R]], with the action of Λ i on s(Λ i ) ⊂ X being conjugate to Λ i X i . By [12,Corollary 1.4], all Λ i have property (T). So, by Part 2 of Theorem 6.1, the family I is countable modulo conjugacy of actions.
When Γ is an ICC property (T) group, all groups in S factor (Γ) are countable (cf. [15, Proof of Theorem 1.7], or [23, Theorem 4.5.1]). The next theorem generalizes this result to Kazhdan groups Γ with the property that the center Z(Γ) has finite index in the FC-radical Γ f of Γ, defined by Γ f := {g ∈ Γ | g has a finite conjugacy class } .
As far as we know, the only examples of Kazhdan groups Γ with infinite FC-radical Γ f that exist in the literature are such that Z(Γ) has finite index in Γ f (see e.g. [2, Example 1.7.13] and [11,Definition 2.4]). Concerning the remaining case, we prove in the second part of the theorem below that if Γ is a residually finite property (T) group such that Γ f is not virtually abelian (i.e., Z(Γ f ) < Γ f has infinite index), then Γ admits a free ergodic p.m.p. action on (X, µ) with L ∞ (X)⋊Γ being McDuff and hence, R + ∈ S factor (Γ). While we were unable to show whether Kazhdan groups with these properties exist or not, after discussing this problem with several specialists, it was indicated to us by Mark Sapir and Denis Osin that such groups probably do exist, but their actual construction may require a substantial effort. Assume first that Z(Γ) has finite index in Γ f . Let Γ (X, µ) be free ergodic p.m.p. Write A := L ∞ (X) and M := A ⋊ Γ. Define Λ := C Γ (Γ f ). Since Z(Γ) has finite index in Γ f , it follows that Λ has finite index in Γ. A fortiori, the subgroup Λ 1 := Λ · Γ f has finite index in Γ. Also, the subalgebra A Λ of Λ-invariant functions in A, is finite dimensional and globally Λ 1 -invariant. By construction, Γ n is a decreasing sequence of normal, finite index subgroups with n Γ n = {e} and such that for all h ∈ Γ f , we have Γ n ⊂ C Γ (h) for all n large enough.
Denote (X, µ) = lim ← − (Γ/Γ n , counting probability measure). Consider the natural free, ergodic, profinite, p.m.p. action Γ (X, µ). Put A = L ∞ (X) and M := A ⋊ Γ. For every s ∈ Γ and n ∈ N, denote by χ sΓn the function equal to 1 on sΓ n and zero elsewhere and interpret χ sΓn as a projection in A. Claim. If for all n ∈ N, we have h n ∈ Γ f ∩ Γ n with h n = e, then (v hn ) is a central sequence in M with τ (v hn ) = 0 for all n. For all n, we have v hn ∈ L(Γ) ′ . So, to prove the claim, it suffices to take k ∈ N, g ∈ Γ and prove that lim n [χ gΓ k , v hn ] 2 = 0 .
But, by construction, χ gΓ k and v hn commute when n ≥ k.
Since Z(Γ f ) < Γ f has infinite index and since Γ f has finite conjugacy classes, it follows that Γ f has no finite index abelian subgroups. So, for every n, the finite index subgroup Γ f ∩ Γ n of Γ f is non-abelian. Therefore, we can choose h n , h ′ n ∈ Γ f ∩ Γ n such that h n h ′ n h −1 n h ′ n −1 = e. By the claim above, v hn and v h ′ n are central sequences. By construction, τ (v hn v h ′ n v * hn v * h ′ n ) = 0 for all n. So, M is McDuff.