AN UNCOUNTABLE FAMILY OF NON ORBIT EQUIVALENT ACTIONS OF 𝔽_{n}

Damien Gaboriau^{∗}  Sorin Popa^{†}

Abstract

For each 2 ≤ n ≤ ∞, we construct an uncountable family of free ergodic measure preserving actions α_{t} of the free group 𝔽_{n} on the standard probability space (X, μ) such that any two are non orbit equivalent (in fact, not even stably orbit equivalent). These actions are all “rigid” (in the sense of [Po01]), with the II_{1} factors L^{∞}(X, μ) ⋊_{α_{t}} 𝔽_{n} mutually non-isomorphic (even nonstably isomorphic) and in the class HT_{s}.

§0.

Recall that two ergodic probability measure preserving (p.m.p.) actions σ_{i} for i = 1, 2 of two countable groups Γ_{i} on probability measure standard Borel spaces (X_{i}, μ_{i}) are orbit equivalent (OE) if they define partitions of the spaces into orbits that are isomorphic; more precisely if there exists a measurable, almost everywhere defined isomorphism f : X_{1} → X_{2} such that f_{∗}μ_{1} = μ_{2} and the Γ_{1}-orbit of μ_{1}-almost every x ∈ X_{1} is sent by f onto the Γ_{2}-orbit of f(x).

The theory of orbit equivalence, although underlying the “group measure space construction” of Murray and von Neumann [MvN36], was born with the work of H. Dye who proved, for example, the following striking result [Dy59]: Any two ergodic p.m.p. free actions of Γ_{1} ≃ ℤ and Γ_{2} ≃ ⊕_{j∈ℕ}ℤ/2ℤ are orbit equivalent.

Through a series of works, the class of groups Γ_{2} satisfying Dye’s theorem gradually increased until it achieved the necessary and sufficient condition: Γ_{2} is infinite amenable [OW80]. In particular, all infinite amenable groups produce one and only one ergodic p.m.p. free action up to orbit equivalence (see also [CFW81] for a more general statement).

∗C.N.R.S.

†Supported in part by a NSF Grant 0100883

By using the notion of strong ergodicity, A. Connes and B. Weiss proved that any non amenable group without Kazhdan property (T) admits at least two non OE p.m.p. free ergodic actions [CW80]. The first examples of groups with uncountably many non OE free ergodic actions was obtained in [BG81], using a somewhat circumstancial construction, based on prior work in [McD69]. Within the circle of ideas brought up by Zimmer’s cocycle super-rigidity [Zi84], certain Kazhdan property (T) lattices of Lie groups such as SL(n, ℤ), n ≥ 3, were shown to admit uncountably many non OE free ergodic actions as well (see [GG88]). It is only recently that we learned that this property was shared by all infinite groups with Kazhdan property (T) [Hj02-b], and thanks to another reason (bounded cohomology) by many torsion free finitely generated direct products Γ_{1} ×Γ_{2}, including non trivial (l ≥ 2) products of free groups like 𝔽_{p_{1}} × 𝔽_{p_{2}} × ⋯ × 𝔽_{p_{l}}, p_{i} ≥ 2 [MS02].

On the other hand, the situation for the free groups themselves or SL(2, ℤ) remained unclear and arouse the interest of producing more non OE free ergodic actions of 𝔽_{n} than just the two given by [CW80] (more precisely in producing ways to distinguish them from the OE point of view), reaching the number of 3 in 2001 [Po01], then 4 in 2002 [Hj02-b], for each finite n ≥ 2.

Meanwhile, a certain rigidity phenomenon has been detected, behind the notion of cost [Ga00] or l^{2} Betti numbers [Ga01], showing in particular that the rank of the free groups is an invariant of OE, as well as the virtual Euler characteristic of groups. As a consequence, all actions of finitely generated free groups were shown to have trivial fundamental group ([Ga00] or [Ga01, Cor. 5.7]).

But these techniques have no hold on various actions of the same group. A new notion of rigidity for actions of a given group has been found in [Po01] (Def. 5.10.1), called relative property (T). It is an OE invariant, is satisfied by certain actions of the free group 𝔽_{n}, 2 ≤ n < ∞, and not by others, and led to the already mentioned third example by way of countability of its outer automorphism group (5.3.3 or 8.7 in [Po01]). This property will be crucial in our results and notice right away that it is satisfied by the natural actions of SL(2, ℤ) on the 2-torus 𝕋^{2} (cf. [Po01]). The main goal of this paper is to prove the following:

Theorem For each 2 ≤ n ≤ ∞, there exist uncountably many non orbit equivalent actions of the free group 𝔽_{n}, each being free ergodic and probability measure preserving on a standard Borel space.

Our result also applies to every virtually free group, and more generally to every group that is virtually a non trivial free product of infinite amenable groups. All actions in our examples have the relative property (T), thus having at most countable outer automorphism groups and countable fundamental group (trivial in the case 𝔽_{n}). Moreover, the uncountable family of actions can be taken so that any two of them are non stably equivalent. Similar results are obtained for the associated II_{1} factors. All along the paper, we consider only standard Borel spaces.

§1.

From the very beginning, the orbit equivalence theory has been strongly connected with operator algebras through the “group measure space construction” [MvN36, pp. 192-209], which was the first general way of producing various infinite dimensional von Neumann algebras (even before the “group von Neumann algebra” [MvN43, Sect. 5.3]): each ergodic p.m.p. free action α of an infinite countable group Γ on a standard Borel space (X, μ) defines a type II_{1} von Neumann factor M together with the abelian subalgebra A = L^{∞}(X, μ) ⊂ L^{∞}(X, μ) ⋊_{α} Γ = M sitting in it as a Cartan subalgebra. Moreover two ergodic actions define isomorphic inclusions (A_{1} ⊂ M_{1}) ≃ (A_{2} ⊂ M_{2}) iff they are orbit equivalent [FM77]. This fact establishes a useful equivalence between the study of actions of groups, up to orbit equivalence, and the study of the inclusions of Cartan subalgebras into II_{1} factors entailed by such actions, up to isomorphism of inclusions.

The notion of relative property (T) for an action of a group in [Po01] is in fact expressed in terms of its associated Cartan subalgebra, and the terminology used in this functional analytical framework is relatively rigid Cartan subalgebra (or rigid inclusion) A ⊂ M. Inspired by the notion of relative property (T) of Kazhdan-Margulis for inclusions of groups [Ma82] (see also [dHV89, p. 16]) and by the Connes-Jones definition of property (T) for II_{1} factors [CJ85], this property requires that whenever a completely positive map on M (the operator algebra analogue of a positive definite function) is close to the identity on some “critical” finite subset of M, then it must be uniformly close to the identity on A.

If a Cartan subalgebra A ⊂ M comes from a group von Neumann algebra construction corresponding to an inclusion of groups H ⊂ G = Γ ⋉ H then A ⊂ M is rigid in the sense of Definition 4.2 in [Po01] iff H ⊂ G has the relative property (T) (cf. 5.1 in [Po01]). Since ℤ^{2} ⊂ SL(2, ℤ) ⋉ ℤ^{2} has the relative property (T) (cf. [Ma82]), this observation gives rise to an important example of an action with the relative property (T), namely the standard action (SL(2, ℤ), 𝕋^{2}). More generally, M. Burger has shown that for any non-amenable subgroup of automorphisms of ℤ^{2}, Γ ⊂ SL(2, ℤ), the induced inclusion ℤ^{2} ⊂ Γ ⋉ ℤ^{2} has the relative property (T). So in fact by (5.1 in [Po01]) all actions (Γ, 𝕋^{2}) have the relative property (T). Equivalently, A ⊂ M is a rigid inclusion of von Neumann algebras, where M is the group von Neumann factor of the natural semi-direct product Γ ⋉ ℤ^{2} and A is the Cartan subalgebra A = L^{∞}(𝕋^{2}), identified via Fourier transform with the von Neumann algebra of the group ℤ^{2}.

For the proof of the Theorem, we start with the above example of rigid inclusion A ⊂ M in which Γ = 𝔽_{n}. Then we use Dye’s theorem to write the equivalence relation R_{1} implemented by the first of the generators of 𝔽_{n} (which one may assume acts ergodically) as a limit of an increasing one parameter family of ergodic hyperfinite equivalence relations R_{1,t}, for t ∈ (0, 1]. Together with 𝔽_{n−1}, each of the R_{1,t} gives rise to a free ergodic action α_{t} of 𝔽_{n} on 𝕋^{2}, with strictly increasing group measure space factors A ⊂ M_{t} ⊂ M and M_{t} ↗ M. By a rigidity property [Po01, 4.8], there is a value c of the parameters after which (c < t ≤ 1) the actions α_{t} have relative property (T), equivalently, A ⊂ M_{t} are rigid.

We then use the rigidity of each of these A ⊂ M_{t} to prove that if an endomorphism of M_{t} is close enough to the identity, then it has to be onto. This is done by analogy with the proof that Out(A ⊂ M_{t}) is countable in [Po01, 4.7], along the lines of Connes’ pioneering rigidity result on the countability of Out(N) for II_{1} factors N with the property (T) [Co80]. Thus, by a separability argument as in [Co80], [Po86], if uncountably many inclusions (A ⊂ M_{t}) are isomorphic then one gets an endomorphism M_{s} ≃ M_{t} ⊂ M_{s} close to the identity, thus onto, contradicting M_{t} ≠ M_{s}. The partition of the interval of parameters [c, 1] into the subsets where the factors M_{t} are isomorphic is then composed of finite or countable subsets. We mention that in fact when proving the statement involving stable orbit equivalence, the above argument becomes slightly more complicated, with “isomorphism” (resp. “endomorphism”) replaced by surjective (resp. non-surjective) isomorphism between amplifications of suitable algebras.

§2.

Set I = (0, 1) ∩ ℚ and I_{t} = I ∩ (0, t], for t ∈ (0, 1]; set G = ⊕_{i∈I}ℤ/2ℤ and G_{t} = ⊕_{i∈I_{t}}ℤ/2ℤ, for 0 < t ≤ 1. This gives a strictly increasing continuum of subgroups of G.

Proposition 1 Let α be a free p.m.p. action of Γ_{1}∗Γ_{2}, with Γ_{1} an infinite amenable group acting ergodically. Then

1^{∘}. The action α is orbit equivalent to a free ergodic action σ_{1} of the group G∗Γ_{2}. 2^{∘}. The restrictions σ_{t} of σ_{1} to the subgroups G_{t} ∗ Γ_{2} define a strictly increasing

family of ergodic equivalence relations S_{t} indexed by t ∈ (0, 1], such that for any s and for any increasing sequence t_{n} ↗ s one has S_{s} = ∪_{n}S_{t_{n}}.

3^{∘}. The action σ_{t} of G_{t} ∗ Γ_{2} is OE to some ergodic free action α_{t} of Γ_{1} ∗ Γ_{2}.

4^{∘}. The associated Cartan subalgebra inclusions define a strictly increasing family of subfactors A ⊂ M_{t} ⊂ M_{1}, 0 < t ≤ 1, such that M_{t} = A ⋊_{α_{t}} (Γ_{1} ∗ Γ_{2}), M_{1} = A ⋊_{α} (Γ_{1} ∗ Γ_{2}), and \overline{∪_{t<s}M_{t}}^{w} = M_{s}, for each s ∈ (0, 1].

Lemma 2 Let Γ = B ∗ D and Λ = C ∗ D be free products such that B and C are infinite amenable groups. Every p.m.p. free action α of Γ, where B acts ergodically, is orbit equivalent to a free action σ of Λ, where (up to a null set) σ_{|C} is any prescribed free ergodic p.m.p. C-action and σ_{|D} ≃ α_{|D}.

Proof. By Ornstein-Weiss’ [OW80] and Dye’s [Dy59] theorems, the free action of B on (X_{1}, μ_{1}) is OE to the prescribed C-action on (X_{2}, μ_{2}), as witnessed by f : X_{1} → X_{2}. Conjugating α_{|D} by f leads to an action of the free product C ∗ D, which is OE to α. The equivalence relations defined by the actions, S_{1} = S_{D} and S_{2} = S_{B} = S_{C}, are in free product in the sense of [Ga00, Déf. IV.9], so that a non trivial reduced word in the free product C ∗ D has obviously a set of fixed point of measure zero: the Λ-action σ is free. Q.E.D.

Proof of the Proposition 1. By Lemma 2, α is orbit equivalent to a free action σ_{1} of G ∗ Γ_{2}, where the G-action is its Bernoulli shift on {0, 1}^{G}. The restriction of σ_{1} to G_{t} remains ergodic, and the restriction σ_{t} of σ_{1} to G_{t} ∗ Γ_{2} is free ergodic. The properties of the family S_{t} are now obvious, and another application of Lemma 2 shows that each σ_{t} is OE to a free action of Γ_{1} ∗ Γ_{2}. The operator algebra statement 4^{∘} follows immediately, due to the above and [FM77]. Q.E.D.

Recall that a Cartan subalgebra A in a factor M is a maximal abelian subalgebra whose normalizer N_{M}(A) =^{def} {u ∈ U(M) | uAu^{∗} = A} generates M as von Neumann algebra (where U(M) is the group of unitary elements of M). Also, recall from 4.2 in [Po01] that a Cartan subalgebra A in a II_{1} factor M is called relatively rigid if for any ε > 0 there exist a finite set F ⊂ M and δ > 0 such that if ϕ : M → M is a completely positive map with τ ∘ ϕ ≤ τ, ϕ(1) ≤ 1 and ∥ϕ(x) − x∥_{2} ≤ δ, ∀x ∈ F, then ∥ϕ(a) − a∥_{2} ≤ ε, ∀a ∈ A, ∥a∥ ≤ 1.

In [Po01, 4.7], it was proved that if M is a type II_{1} factor with A ⊂ M a relatively rigid Cartan subalgebra then the subgroup of automorphisms G_{A} ⊂ Aut(M) that are inner perturbations of automorphisms leaving A pointwise fixed is open and closed in Aut(M), thus giving rise to a countable quotient group Aut(M)/G_{A}. In particular, if A ⊂ M comes from a free ergodic p.m.p. action α of a group Γ and R denotes the corresponding equivalence relation implemented by α on the probability space, then Out(R) is countable. The next Proposition generalizes this result to the case of (not necessarily unital) endomorphisms of M and of its amplifications.

Lemma 3 Let M be a type II_{1} factor with A ⊂ M a relatively rigid Cartan subalgebra. For each x ∈ M denote i(x) = x ⊕ 0 ∈ 𝕄_{2×2}(M) and p_{0} = i(1_{M}) = 1 ⊕ 0. There exist F ⊂ M finite and δ > 0 such that if θ : i(M) → p𝕄_{2×2}(M)p is a unital ∗-isomorphism satisfying ∥θ(i(x)) − i(x)∥_{2} ≤ δ, ∀x ∈ F, for some projection p ∈ 𝕄_{2×2}(M), then there exists p′ ∈ θ(i(A))′ ∩ p𝕄_{2×2}(M)p and a partial isometry u ∈ 𝕄_{2×2}(M) such that

u^{∗}u = p_{0}, uu^{∗} = p′, θ(i(M))p′ = p′𝕄_{2×2}(M)p′, θ(i(a))p′ = ui(a)u^{∗}, ∀a ∈ A.

If in addition θ(i(M))′ ∩p𝕄_{2×2}(M)p = ℂp, then θ is surjective and τ(p) = 1/2, i.e., τ preserves the trace.

Proof. Since θ is a unital ∗-isomorphism, it is completely positive. Thus, if ϕ : M → M is defined by ϕ(x) = p_{0}θ(i(x))p_{0}, x ∈ M, then ϕ is completely positive and for all x ∈ M we have:

∥ϕ(x) − x∥_{2} ≤ ∥θ(i(x)) − i(x)∥_{2} + ∥θ(i(x)) − p_{0}θ(i(x))p_{0}∥_{2}
≤ ∥θ(i(x)) − i(x)∥_{2} + ∥θ(i(x)) − θ(i(x))p_{0}∥_{2}
+∥ θ(i(x))p_{0} − p_{0}θ(i(x))p_{0}∥_{2}
≤ ∥θ(i(x)) − i(x)∥_{2} + 2∥θ(p_{0}) − p_{0}∥_{2}.

By the definition of relative rigidity it follows that there exist F ⊂ M finite, with 1_{M} ∈ F, and δ > 0, with δ ≤ 1/8, such that if a subunital subtracial completely positive map ϕ on M satisfies ∥ϕ(x) − x∥_{2} ≤ δ, ∀x ∈ F, then ∥ϕ(v) − v∥_{2} ≤ 1/4 for all elements v in the unitary group U(A) of A. But then we also have

∥θ(i(v)) − i(v)∥_{2} ≤ ∥θ(i(v)) − θ(i(v))p_{0}∥_{2} + ∥θ(i(v))p_{0} − p_{0}θ(i(v))p_{0}∥_{2}

+∥ϕ(v) − v∥_{2}

≤ 2∥θ(p_{0}) − p_{0}∥_{2} + ∥ϕ(v) − v∥_{2}
≤ 2 · 1/8 + 1/4 = 1/2.

Thus, if h denotes the unique element of minimal Hilbert norm ∥ · ∥_{2} in the weakly compact convex set \overline{co}^{w}\{θ(i(v))i(v)^{∗} | v ∈ U(A)\} ⊂ p𝕄_{2×2}(M)p_{0} then ∥h − p_{0}∥_{2} ≤ 1/2. Also, since ∥θ(i(v))hi(v)^{∗}∥_{2} = ∥h∥_{2}, ∀v ∈ U(A), by the uniqueness of h it follows that θ(i(v))h = hi(v), ∀v ∈ U(A). By a standard functional calculus trick (as in group representations), if u_{0} ∈ 𝕄_{2×2}(M) is the partial isometry in the polar decomposition of h then u_{0} ≠ 0, u_{0}i(a) = θ(i(a))u_{0}, ∀a ∈ A, and u_{0}^{∗}u_{0} ∈ i(A)′ ∩ i(M) = i(A), [u_{0}u_{0}^{∗}, θ(i(A))] = 0.

We can thus apply the above to a = i^{−1}(u_{0}^{∗}u_{0}) ∈ A to get θ(u_{0}^{∗}u_{0})u_{0} = u_{0}u_{0}^{∗}u_{0} = u_{0}. But this implies θ(u_{0}^{∗}u_{0})u_{0}u_{0}^{∗} = u_{0}u_{0}^{∗}, so that θ(u_{0}^{∗}u_{0}) ≥ u_{0}u_{0}^{∗}. Note that in particular this implies that θ doesn’t “shrink” the trace τ, i.e., τ(θ(x)) ≥ τ(x), ∀x ∈ p_{0}𝕄_{2×2}(M)p_{0}.

Since the normalizer N_{M}(A) of A in M acts ergodically on A, there exist partial isometries v_{1}, v_{2}, …, v_{n} in M such that v_{j}v_{j}^{∗}, v_{j}^{∗}v_{j} ∈ A, v_{j}Av_{j}^{∗} = Av_{j}v_{j}^{∗}, v_{j}^{∗}v_{j} ≤ u_{0}^{∗}u_{0} and ∑_{j} v_{j}v_{j}^{∗} = 1. Then u = ∑_{j} θ(i(v_{j}))u_{0}i(v_{j})^{∗} is easily seen to be a partial isometry satisfying ui(a) = θ(i(a))u, ∀a ∈ A, with u^{∗}u = p_{0}, p′ = uu^{∗} ∈ θ(i(A))′∩p𝕄_{2×2}(M)p. By spatiality it follows that θ(i(A))p′ is a Cartan subalgebra (in particular maximal abelian) in p′𝕄_{2×2}(M)p′.

Let B = θ(i(A))′ ∩ p𝕄_{2×2}(M)p. Since θ(i(w)) normalizes θ(i(A)) for w ∈ N (A) it follows that θ(i(w)) normalizes B as well. Also, θ(i(A))p′ maximal abelian implies p′Bp′ = θ(i(A))p′. Thus, if p_{1} ∈ B denotes the maximal abelian central projection of B then p_{1} ≤ p′ and τ(p′ − p_{1}) ≤ τ(p − p′). Since τ(p′) = τ(p_{0}) = 1/2 and

τ(p) − τ(p_{0}) ≤ ∥p − p_{0}∥_{1} ≤ 2∥p − p_{0}∥_{2} ≤ 2δ ≤ 1/4,

it follows that 1/2 − τ(p_{1}) ≤ 1/4, so that p_{1} ≠ 0. But since B is normalized by θ(i(N (A))), p_{1} is left invariant by Ad(w′) for w′ ∈ θ(i(N (A))). Thus, [p_{1}, θ(i(M))] = 0, so in particular a ↦ ap_{1}, a ∈ θ(i(A)) is an isomorphism. But p′Bp′ = θ(i(A))p′ ≃ A as well. Thus, p_{1} = p′.

In particular, this shows that if we denote θ_{0}(x) = u^{∗}θ(x)u, x ∈ M, then θ_{0} is an isomorphism. Moreover, θ_{0}(a) = a, ∀a ∈ A. Then we get:

waw^{∗} = θ_{0}(waw^{∗}) = θ_{0}(w)θ_{0}(a)θ_{0}(w^{∗}) = θ_{0}(w)aθ_{0}(w^{∗})

for all a ∈ A, implying that θ_{0}(w)^{∗}w ∈ A′ ∩ M = A. Hence, w ∈ θ_{0}(w)A. This shows that N_{M}(A) ⊂ θ_{0}(M)A = θ_{0}(M), so that M ⊂ θ_{0}(M), thus θ_{0}(M) = M. But this implies θ(M) = M as well. Q.E.D.

To state the next result, recall from [MvN43] that if M is a II_{1} factor and s > 0 then the s-amplification of M is the II_{1} factor M^{s} = p𝕄_{n×n}(M)p, where n is some integer ≥ s and p is a projection in 𝕄_{n×n}(M) of (normalized) trace equal to s/n. This factor is defined up to isomorphisms implemented by unitary elements in large enough matrix algebras over M. Note also that in case M has a Cartan subalgebra A ⊂ M, then p can be taken in the Cartan subalgebra ⊕^{n}A of 𝕄_{n×n}(M) given by diagonal elements with entries in A and A^{t} = p(⊕^{n}A)p is a Cartan subalgebra of M^{t}.

Two factors of type II_{1}, M_{1}, M_{2} are stably isomorphic if there exist s > 0 such that M_{1} is isomorphic to M_{2}^{s}. We then write M_{1} ≅ M_{2}, while the symbol ≃ will be used to denote usual isomorphism. Recall that the fundamental group of M is the subgroup F(M) := {s : M^{s} ≃ M} of ℝ_{+}^{∗}.

Proposition 4 Let M be a type II_{1} factor with a relatively rigid Cartan subalgebra A ⊂ M. Then the fundamental group of M is at most countable. If in addition A ⊂ M_{t} ⊂ M, t ∈ (0, 1], is a strictly increasing family of subfactors such that M = \overline{∪_{t<1}M_{t}}, then we have:

1^{∘}. There exists c < 1 such that for all c ≤ t ≤ 1, A ⊂ M_{t} are relatively rigid Cartan subalgebras.

2^{∘}. If c satisfies 1^{∘}, then for each t ∈ [c, 1] the class of stable isomorphism {s | M_{s} ≅ M_{t}} is at most countable.

Thus, the quotient set of parameters [c, 1]/ ∼ is uncountable.

Proof. Item 1^{∘} is part of 4.5.4^{∘} in [Po01]. To prove 2^{∘}, assume there exist t_{0} ∈ [c, 1] and an uncountable subset L ⊂ [c, 1] such that θ_{t} : M_{t_{0}} → M_{t}^{s(t)}, ∀t ∈ L, for some s(t) > 0 and some surjective isomorphisms θ_{t}. We may further assume n_{0}^{−1} ≤ s(t) ≤ n_{0}, ∀t ∈ L, for some integer n_{0} ≥ 1.

Let F ⊂ M_{t_{0}}, δ > 0 be so that if θ is a unital isomorphism of i(M_{t_{0}}) = M_{t_{0}} ⊕ 0 into p𝕄_{2×2}(M_{t_{0}})p satisfying θ(i(M_{t_{0}}))′ ∩ p𝕄_{2×2}(M_{t_{0}})p = ℂp and ∥θ(i(x)) − i(x)∥_{2} ≤ δ, ∀x ∈ F, as in Lemma 3. Then θ is onto and preserves the trace (cf. Lemma 3). By isomorphism, it follows that for each t ∈ L we have: If θ is a unital isomorphism of i(M_{t}^{s(t)}) into p𝕄_{2×2}(M_{t}^{s(t)})p with range having trivial relative commutant and satisfying ∥θ(i(x)) − i(x)∥_{2} ≤ δ, ∀x ∈ θ_{t}(F), then θ is surjective and preserves the trace.

We will now regard the isomorphisms θ_{t}, t ∈ L, as isomorphisms of M_{t_{0}} ⊕ 0 ⊂ M ⊕ 0 into 𝕄_{n_{0}×n_{0}}(M), in the obvious way, where M ⊕ 0 is the algebra of n_{0} × n_{0} matrices of the form x ⊕ 0, having x ∈ M as upper left entry and 0 elsewhere.

By the separability of the Hilbert space L^{2}(𝕄_{n_{0}×n_{0}}(M), τ) and the uncountability assumption on L, it follows that there exist t_{1} ≠ t_{2} ∈ L such that ∥θ_{t_{1}}(x)−θ_{t_{2}}(x)∥_{2} ≤ δ/n_{0}^{2}, ∀x ∈ F. Assume t_{1} < t_{2}. Denote θ = θ_{t_{1}} ∘ θ_{t_{2}}^{−1} : M_{t_{2}}^{s(t_{2})} ≃ M_{t_{1}}^{s(t_{1})} ⊂ M_{t_{2}}^{s(t_{1})} and note that for δ ≤ 1/8 we may assume θ is an isomorphism of i(M_{t_{2}}^{s(t_{2})}) into 𝕄_{2×2}(M_{t_{2}}^{s(t_{2})}) satisfying ∥θ(i(x)) − i(x)∥_{2} ≤ δ, ∀x ∈ θ_{t_{2}}(F) ⊂ θ_{t_{2}}(M_{t_{0}}) = M_{t_{2}}^{s(t_{2})}. Note also that since M′_{t_{1}} ∩ M_{t_{2}} = ℂ, the range of θ has trivial relative commutant.

By the above observation (consequence of Lemma 3) this implies θ is onto and trace preserving. Thus s(t_{1}) = s(t_{2}) and M_{t_{1}} = M_{t_{2}}, a contradiction.

Finally, note that the above argument applied to just the factor M and its amplifications shows that F(M) is countable. Q.E.D.

For the next result recall that the fundamental group F(R) of an ergodic equivalence relation R is the set of all t > 0 such that the t-amplification R^{t} of R is orbit equivalent to R. Also, two equivalence relations R_{1}, R_{2} are stably orbit equivalent (R_{1} ≅ R_{2}) if there exists s > 0 such that R_{1}, R_{2}^{s} are orbit equivalent. Note that if M, M_{1}, M_{2} are the II_{1} factors associated to R, R_{1}, R_{2}, respectively, then F(M) ⊃ F(R) and R_{1} ≅ R_{2} implies M_{1} ≅ M_{2}.

Corollary 5 For each 2 ≤ n ≤ ∞ there exists an uncountable family of non stably orbit equivalent free ergodic p.m.p. actions α_{t} of 𝔽_{n} such that moreover the corresponding equivalence relations R_{α_{t},𝔽_{n}} have at most countable fundamental group (trivial in the case n < ∞) and at most countable outer automorphism group.

Proof. For each 2 ≤ n ≤ ∞, let 𝔽_{n} ⊂ SL(2, ℤ) be a non-cyclic free subgroup, and consider its standard action α on the 2-torus 𝕋^{2}. The following lemma 6 gives a base of 𝔽_{n} with at least one ergodically acting element, thus allowing the use of Proposition 1. The family of Cartan subalgebra inclusions given by Proposition 1 satisfies Proposition 4, since α has the relative property (T), i.e. A = L(ℤ^{2}) = L^{∞}(𝕋^{2}, λ) is a relatively rigid Cartan subalgebra of M_{1} = L(ℤ^{2} ⋊ 𝔽_{n}) = L(𝕋^{2}, λ) ⋊_{α} 𝔽_{n} (by [Bu91] and [5.1–5.2, Po01]). But non isomorphic factors necessarily come from non OE actions.

Thus, the uncountable family of non stable equivalent actions, each one with a countable fundamental group, can be taken to be the α_{t} of Proposition 1, for a set of parameters representative of the classes [c, 1]/ ∼ in Proposition 4 (2^{∘}). The triviality of the fundamental group in the case n < ∞ is a consequence of [Ga01, Cor. 5.7]. The countability of the outer automorphism group follows from [Cor. 4.7, Po01]. Q.E.D.

Lemma 6 If Γ is a non cyclic free subgroup of SL(2, ℤ), then it admits a base of elements whose standard actions on the 2-torus 𝕋^{2} are individually ergodic.

Proof. The criterion for an element to act ergodically is that it doesn’t have a root of unity as eigenvalue, or equivalently that it is hyperbolic when looking at its action on the circle, boundary of the hyperbolic disk. Fact: if a and b are two non commuting elements of infinite order, where a is parabolic, then there exists a power k ∈ ℤ such that b^{k}a is hyperbolic: let P_{a} be the unique fixed point of a, let P_{b} ≠ P_{a} the first fixed point of b met on the circle from the attracting side of P_{a} and let I be the corresponding interval between them (the case of a common unique fixed point is excluded by non commutativity, then non solvability, of the subgroup generated by a and b). Let J be a closed interval strictly contained in I with end point P_{b}. Then aJ is contained in the interior of I, and up to taking negative powers P_{b} is attracting so that a big enough powers b^{k} satisfies (b^{k}a)J ⊂ Int(J), which is equivalent to hyperbolicity. Finally, starting from a base of Γ, one can successively transform it according to the fact so as to produce a base satisfying the lemma. Q.E.D.

For next statement, recall that a group is virtually of a certain kind if it contains a finite index subgroup of that kind. The non triviality hypothesis aims to avoid Λ ≃ Λ ∗ {1}.

Corollary 7 Corollary 5 applies equally well with, instead of the group 𝔽_{n}, any countable discrete group Γ that is virtually a non trivial free product Λ of infinite amenable groups. More generally, if Γ = Λ ∗ H admits a free p.m.p. action α with the relative property (T), where H is infinite amenable and acts ergodically, then Γ has an uncountable family of non stably orbit equivalent free ergodic p.m.p. actions α_{t}, such that R_{α_{t},Γ} have at most countable fundamental group (trivial in the case β_{1}(Λ) is finite) and at most countable outer automorphism group.

Note that in order to ensure that the above action α has the relative property (T), it is enough to get this property for some subgroup of Γ. This is, of course, trivial from the definition of the relative property (T) ([Po01, 4.6.2]).

Proof. The first part relies on the fact that actions of Γ may be produced from actions of 𝔽_{n}. Let Λ be a finite index subgroup of Γ, then the usual suspension construction associates, to each free p.m.p. action of Λ on some standard Borel space X, a stably orbit equivalent free p.m.p. action of Γ (given by right multiplication on itself) on the diagonal action quotient Λ\(X × Γ). If the Λ-action is ergodic, then so is the Γ-action. Two non OE Λ-actions lead to two non OE Γ-actions.

By hypothesis, Λ may be chosen to be a free product H_{1} ∗ H_{2} ∗ … of n ∈ {2, 3, …, ∞} infinite amenable groups. Start with an uncountable collection of free p.m.p. actions of 𝔽_{n} given by corollary 5. Observe that in that construction one may assume that 𝔽_{n} is given with a base of elements that act individually in an ergodic manner.

Then up to removing at most countably many times a null set, as in lemma 2, one can replace each cyclic factor in the decomposition of 𝔽_{n} by one H_{i}. This leads to uncountably many free p.m.p actions of Λ.

The last part is an immediate consequence of Propositions 1 and 4. Again, the triviality of the fundamental group in the case β_{1}(Λ) < ∞ follows from [Ga01, Cor. 5.7]. Q.E.D.

We’ll now derive some applications of the above results to type II_{1} factors. To this end, recall from Section 6 in [Po01] that a Cartan subalgebra A of a II_{1} factor M is called HT_{s} if in addition to A ⊂ M being a rigid inclusion, one has that M has the property H relative to A, i.e., there exists a sequence of unital completely positive A − A bimodular maps ϕ_{n} on M that tend to the identity in the point ∥ · ∥_{2}-topology and are compact relative to A (cf. 2.1 in [Po01]). A II_{1} factor M is in the class HT_{s} if it has such a HT_{s} Cartan subalgebra.

The property H of M relative to A is automatically satisfied whenever M comes from a “group measure space construction”, M = L^{∞}(X, μ) ⋊_{α} Γ, with Γ a group satisfying Haagerup’s compact approximation property and α a free ergodic measure preserving action of Γ on the probability space (X, μ). Thus, since all subgroups Γ of SL(2, ℤ) do have Haagerup’s property (cf. [Ha79]) and (Γ ⋉ ℤ^{2}, ℤ^{2}) has the relative property (T) for all non-amenable Γ ⊂ SL(2, ℤ) (cf. [Bu91]), it follows by Lemma 6 that if Γ = 𝔽_{n} for some 2 ≤ n ≤ ∞ then A ⊂ M = L^{∞}(𝕋^{2}, λ) ⋊ Γ is a HT_{s} Cartan subalgebra of the HT_{s} factor M. We then have:

Corollary 8 For each 2 ≤ n ≤ ∞ there exists an uncountable family of mutually non stably isomorphic II_{1} factors M_{t} of the form L^{∞}(X, μ)⋊𝔽_{n}, all in the class HT_{s}. These factors have at most countable fundamental group (trivial in the case n < ∞). Moreover, in the case n = ∞ they can all be taken group von Neumann algebras of the form L(𝔽_{∞} ⋉ ℤ^{2}), with 𝔽_{∞} acting on ℤ^{2} as subgroups of automorphisms via various embeddings 𝔽_{∞} ↪ SL(2, ℤ).

Proof. By the above comments and [Po01], all actions α_{t} of 𝔽_{n}, 2 ≤ n ≤ ∞, considered in the proof of Corollary 5, give rise to factors M_{t} = L^{∞}(𝕋^{2}, λ) ⋊_{α_{t}} 𝔽_{n} in the class HT_{s}. By Proposition 4, not only the actions α_{t} are not stably orbit equivalent (modulo countable sets), but even the factors M_{t} are non stably isomorphic and they all have countable fundamental group. Since for n < ∞ by [8.6.2^{∘}, Po01] they have the same (≠ 0, ∞) first ℓ2_{HT} -Betti number, β1^{HT} (M_{t}) = n − 1, ∀t, the fundamental group F(M_{t}) is in this case trivial, ∀t.

To see that in case n = ∞ the factors M_{t} can be taken group factors, write 𝔽_{∞} as 𝔽_{I}, the free group with a countable set of generators labeled by I = (0, 1) ∩ ℚ, as in Proposition 1. Then let I_{t} = I ∩ (0, t], for t ∈ (0, 1], and denote by 𝔽_{I_{t}} the subgroup of 𝔽_{I} generated by generators with indices in I_{t}. Choose 𝔽_{I} ⊂ SL(2, ℤ) an embedding of 𝔽_{I} as a group of automorphisms of ℤ^{2}. By the remarks above again, the factors L(𝔽_{I_{t}} ⋉ ℤ^{2}) are in the class HT_{s} and by Proposition 4 they all have countable fundamental group and are mutually non-stably isomorphic, modulo countable sets. Q.E.D.

§3.

The families of uncountably many non-equivalent free ergodic p.m.p. actions of 𝔽_{n} in the previous section are all obtained by approximating an initial equivalence relation S implemented by a relative property (T) action α of 𝔽_{n} from bellow by a one parameter family of sub-equivalence relations S_{t} implemented by actions α_{t} of the same group 𝔽_{n}: by the rigidity result (4.5 in [Po01]) the relations S_{t} (and thus the actions α_{t}) have the relative property (T) when close enough to S, and thus Proposition 4 applies to obtain that modulo countable sets the α_{t} are non-equivalent (even non stably orbit equivalent).

In this section we prove two more results in this vein. In both cases, from an initial relative property (T) action of a group Γ we will produce new actions having this property, but of groups which in general are different from the initial one. The first result is based on a construction similar to the one in Proposition 1. The second one gives a general hereditary property, showing that if an action σ of a group Γ_{0} has the relative property (T) and Γ_{0} = Γ ⋊ G with G amenable, then σ|Γ has this property as well.

Proposition 9 Let α be a free p.m.p. action of Γ_{1}∗Γ_{2}, with Γ_{1} an infinite amenable group acting ergodically. There exists a strictly increasing family of ergodic equivalence relations S_{t}, t ∈ (0, 1], implemented by free ergodic p.m.p. actions α_{t} of the group Λ = Γ_{1} ∗ (∗_{j∈ℕ}Γ_{2}), such that the associated Cartan subalgebra inclusions define a strictly increasing family of subfactors A ⊂ M_{t} ⊂ M_{1}, 0 < t ≤ 1, with M_{t} = A ⋊_{α_{t}} Λ, ∀t ∈ (0, 1), M_{1} = A ⋊_{α} (Γ_{1} ∗ Γ_{2}), and \overline{∪_{t<s}M_{t}}^{w} = M_{s}, for each s ∈ (0, 1]. If in addition α has the relative property (T) (equivalently, if A ⊂ M is rigid) then there exists c < 1 such that α_{t} has the relative property (T) (equivalently A ⊂ M_{t} is rigid) for all t ∈ [c, 1], and thus the conclusions in part 2^{∘} of Proposition 4 hold true.

Proof. We prove a series of statements similar to Proposition 1, keeping the hypotheses and the notation I, I_{t}, G, G_{t}, Γ_{1}, Γ_{2}, α, σ_{1} from there. For t ∈ (0, 1), define Λ_{t} to be the kernel of the natural epimorphism G ∗ Γ_{2} → G/G_{t} ≃ ⊕_{i∈I,i∉I_{t}}ℤ/2ℤ. Clearly, G∗Γ_{2} is the strictly increasing union of the normal subgroups Λ_{t}, t ∈ (0, 1). The restrictions σ̃_{t} of the action σ_{1} = σ̃_{1} to the subgroups Λ_{t} are ergodic on G_{t} and define a strictly increasing family of ergodic equivalence relations S̃_{t}, with the same properties as S_{t}.

On the other hand, Λ_{t} ≃ G_{t} ∗ (∗_{j∈ℕ}Γ_{2}): the quotient of the Bass-Serre tree of G ∗ Γ_{2} by Λ_{t} is a graph of groups made of a set of vertices of degree 1, indexed by g ∈ G/G_{t} with vertex groups g^{−1}Γ_{2} g, all connected to one “central” vertex with vertex group G_{t}. By lemma 2, σ̃_{t} is OE to some ergodic free action α̃_{t} of Γ_{1}∗(∗_{j∈ℕ}Γ_{2}). Q.E.D.

Observe that, with the notations in the above proof, by normality of Λ_{t}, the outer automorphism group of R_{α̃_{t},Λ} contains G/G_{t} ≃ ⊕_{n∈ℕ}ℤ/2ℤ for all t ∈ [c, 1). Notice also that if we take Γ_{1} ≃ Γ_{2} ≃ ℤ then one has Λ_{t} ≃ G_{t} ∗ 𝔽_{∞} and Γ_{1} ∗ (∗_{j∈ℕ}Γ_{2}) ≃ 𝔽_{∞}.

Proposition 10 Let Γ̃ be a group of the form Γ̃ = H ⋉ Γ, with H amenable. Let σ be a free p.m.p. action of Γ̃ on (X, μ). Then σ has the relative property (T) iff σ_{|Γ} has this property.

Corollary 11 Let Γ̃ be a group of the form Γ̃ = H ⋉ Γ, with H amenable. Assume Γ̃ acts by outer automorphisms on an abelian group Δ. Then the pair (Δ ⋉ Γ̃, Δ) has the relative property (T) iff (Δ ⋉ Γ, Δ) has this property.

Proof. By (5.1 in [Po01]), it is sufficient to prove Proposition 10. Let A = L^{∞}(X, μ), M̃ = A ⋊_{σ} Γ̃, M = A ⋊_{σ} Γ. If A ⊂ M is rigid then A ⊂ M̃ is clearly rigid.

Conversely, assume A ⊂ M̃ is rigid and let F̃ = F̃(ε/16) ⊂ M̃, δ̃ = δ̃(ε/16) > 0 be so that if ϕ̃ is a completely positive map on M̃ with τ ∘ ϕ̃ ≤ τ, ϕ̃(1) ≤ 1 and ∥ϕ̃(x) − x∥_{2} ≤ δ̃ for any x ∈ F̃ then ∥ϕ̃(a) − a∥_{2} ≤ ε/16, ∀a ∈ A, ∥a∥ ≤ 1.

By the amenability of H, it follows that there exists a finite subset K ⊂ H such that

∥ \frac{1}{|K|} ∑_{g,h∈K}u_{g}E_{M}(u_{g}^{∗}xu_{h})u_{h}^{∗} − x∥_{2} ≤ δ̃/2, ∀x ∈ F̃  (1)

By using again the amenability of H, by the Ornstein-Weiss Theorem [OW80] it follows that for any δ̃ > 0 there exists a finite group H_{0} = ⊕_{j}(ℤ/2ℤ)_{j} and unitary elements {v_{g}}_{g∈H_{0}} in the normalizer of A in A ⋊ H ⊂ M̃ which implement a free action of H_{0} on (X, μ) such that each u_{g}, g ∈ K, is ε/64-close to an element u′_{g} in the normalizer of A in A ∨ {v_{g}}_{g∈H_{0}} ≃ A ⋊ H_{0}.

Thus, if we denote by A_{0} = A ∩ {v_{g}}′_{g} the fixed point algebra of the action of H_{0} on A implemented by the unitaries v_{g}, then A can be decomposed as A = A_{0} ⊗ A_{1} with A_{1} generated by unitary elements w_{h}, h ∈ Ĥ_{0}, satisfying v_{g}w_{h}v_{g}^{∗} = h(g)w_{h}, ∀g ∈ H_{0}, h ∈ Ĥ_{0}.

There exists δ > 0 such that if ϕ is a completely positive map on M with τ ∘ϕ ≤ τ, ϕ(1) ≤ 1 and ∥ϕ(w_{h}) − w_{h}∥_{2} ≤ δ, ∀h ∈ Ĥ_{0}, then ∥ϕ(uw_{h}) − ϕ(u)w_{h}∥_{2} ≤ ε/16, ∀u ∈ U(A), h ∈ Ĥ_{0} (see e.g., Corollary 1.1.2 in [Po01]). Thus, for u ∈ U(A_{0}), h ∈ Ĥ_{0} we have

∥ϕ(uw_{h}) − uw_{h}∥_{2} ≤ ∥ϕ(uw_{h}) − ϕ(u)w_{h}∥_{2} + ∥ϕ(u) − u∥_{2} ≤ ε/16 + ∥ϕ(u) − u∥_{2} (2)

Moreover, δ can be chosen small enough so that we also have

∥ϕ(E_{M}(u_{g}^{∗}xu_{g′})) − E_{M}(u_{g}^{∗}xu_{g′})∥_{2} ≤ (2|K|)^{−1}δ̃, ∀g, g′ ∈ K  (3)

We set F (ε) = {E_{M}(u_{g}^{∗}xu_{g′}) | x ∈ F̃(ε/16), g, g′ ∈ K} ∪ {w_{h} | h ∈ Ĥ_{0}}, δ(ε) = min{δ, δ̃, ε/4} and will show that if ϕ is a completely positive map on M such that τ ∘ϕ ≤ τ, ϕ(1) ≤ 1 and ∥ϕ(x)−x∥_{2} ≤ δ(ε), ∀x ∈ F (ε) then ∥ϕ(a)−a∥_{2} ≤ ε, ∀a ∈ A, ∥a∥ ≤ 1.

To see this, note first that if we put ϕ̃(x) =^{def} |K|^{−1} ∑_{g,g′∈K} u_{g}ϕ(E_{M}(u_{g}^{∗}xu_{g′}))u_{g′}^{∗} for x ∈ M̃ then ϕ̃ is a completely positive map on M̃ with τ ∘ ϕ̃ ≤ τ, ϕ̃(1) ≤ 1. Moreover, by (1) and (3), for each x ∈ F̃(ε/16) we have the inequalities:

∥ϕ̃(x) − x∥_{2} ≤ |K|^{−1} ∑_{g,g′∈K}∥u_{g}ϕ(E_{M}(u_{g}^{∗}xu_{g′}))u_{g′}^{∗} − u_{g}E_{M}(u_{g}^{∗}xu_{g′})u_{g′}^{∗}∥_{2}

+∥|K|^{−1} ∑_{g,g′∈K}u_{g}E_{M}(u_{g}^{∗}xu_{g′})u_{g′}^{∗} − x∥_{2} ≤ δ̃/2 + δ̃/2 = δ̃.

Thus, ∥ϕ̃(a) − a∥_{2} ≤ ε/16, ∀a ∈ A, ∥a∥ ≤ 1. Since for a ∈ A we have

ϕ̃(a) = |K|^{−1} ∑_{g∈K}u_{g}ϕ(E_{M}(u_{g}^{∗}au_{g}))u_{g}^{∗} = |K|^{−1} ∑_{g∈K}u_{g}ϕ(σ_{g}^{−1}(a))u_{g}^{∗}

and ∥u_{g} − u′_{g}∥_{2} ≤ ε/64, this implies that for a_{0} ∈ A_{0}, ∥a_{0}∥ ≤ 1 we have:

∥|K|^{−1} ∑_{g∈K}u′_{g}ϕ(a_{0})u′_{g}^{∗} − a_{0}∥_{2} ≤ ∥ϕ̃(a_{0}) − a_{0}∥_{2} + 4∥K∥^{−1} ∑_{g∈K}∥u_{g} − u′_{g}∥_{2}

≤ ε/16 + ε/16 = ε/8.

By convexity, since ∥u′_{g}ϕ(a_{0})u′_{g}^{∗}∥_{2} = ∥ϕ(a_{0})∥_{2} ≤ ∥a_{0}∥, it follows that for each a_{0} ∈ A_{0}, ∥a_{0}∥ ≤ 1, there exists g = g(a_{0}) ∈ K such that

∥ϕ(a_{0}) − a_{0}∥_{2} = ∥u′_{g}ϕ(a_{0})u′_{g}^{∗} − a_{0}∥_{2} ≤ ε/8.

Thus, with the notations in (1.1.2 of [Po01]), by using part 4^{∘} of (Lemma 1.1.2 in [Po01]), in the Hilbert M-bimodule H_{ϕ} we have ∥[u_{0}, ξ_{ϕ}]∥_{2} ≤ 2∥ϕ(u_{0}) − u_{0}∥_{2} ≤ ε/4, ∀u_{0} ∈ U(A_{0}) and ∥[w_{h}, ξ_{ϕ}]∥_{2} ≤ ε/4, ∀h ∈ Ĥ_{0}, as well. This implies ξ_{ϕ} is ε/2-close to a vector ξ ∈ H_{ϕ} that commutes with both U(A_{0}) and the group {w_{h}}_{h}.

Since U(A_{0}) and {w_{h}}_{h} generate A, it follows that ξ commutes with all the elements in A. Thus, ∥[a, ξ_{ϕ}]∥_{2} ≤ ε/2, ∀a ∈ A, ∥a∥ ≤ 1. By (part 2^{∘} of Lemma 1.1.3 in [Po01]), this implies ∥ϕ(a) − a∥_{2} ≤ ε, ∀a ∈ A, ∥a∥ ≤ 1. Q.E.D.

Final Remarks. 1^{∘}. Note that Propositions 9 and 4, taken together, provide another construction of uncountable families of non stably orbit equivalent actions of 𝔽_{∞}. As concerning orbit equivalence, observe that it is in fact enough, by [Ga00, Prop. II.6] and [Hj02-a], to get one free ergodic p.m.p. action of 𝔽_{∞} with countable fundamental group to get uncountably many non OE actions by restriction to Borel subsets of various measures. However, these actions are stably orbit equivalent.

2^{∘}. Note that in Proposition 10 we have actually proved a slightly more general statement, namely: Let M be a finite von Neumann algebra with A ⊂ M a relatively rigid Cartan subalgebra. Let A ⊂ N ⊂ M be an intermediate von Neumann subalgebra such that M = N ⋊_{σ} H for some free action of an amenable group H on N, with the property that σ_{h}(A) = A, ∀h ∈ H. Then A is relatively rigid in N as well. Related to this statement, it would be interesting to investigate the following question: Assuming B ⊂ N ⊂ M is an inclusion of finite von Neumann algebras such that M is amenable relative to N, in the sense of Definition 3.5.2^{∘} of [Po01], and B ⊂ M is rigid, does it follow that B ⊂ N is rigid as well ?

3^{∘}. Let S be the equivalence relation implemented by the action of SL(2, ℤ) on (𝕋^{2}, λ). By [Bu91] and [Po01], any sub-equivalence relation S_{0} of S implemented by a nonamenable subgroup Γ_{0} ⊂ SL(2, ℤ) has the relative property (T). By the rigidity result 5.4 in [Po01], Propositions 1, 4, 9 and 10, the class of subequivalence relations S_{0} ⊂ S for which this still holds true gets considerably larger. The following problem thus seems quite natural: Do all nonamenable ergodic subequivalence relations S_{0} of S have the relative property (T) ?

Acknowledgments. We are grateful to É. Ghys for helpful indications concerning Lemma 6 and to B. Bekka for pointing out to us the results on relative property (T) in [Bu91].

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D. G.: UMPA, UMR CNRS 5669, ENS-Lyon, F-69364 Lyon Cedex 7 S. P.: Math.Dept., UCLA, LA, CA 90095-155505, USA

gaboriau@umpa.ens-lyon.fr,  popa@math.ucla.edu