We prove that a type II 1 factor M can have at most one Cartan subalgebra A satisfying a combination of rigidity and compact approximation properties. We use this result to show that within the class HT of factors M having such Cartan subalgebras A ⊂ M, the Betti numbers of the standard equivalence relation associated with A ⊂ M ([G2]), are in fact isomorphism invariants for the factors M, β HT n (M), n ≥ 0. The class HT is closed under amplifications and tensor products, with the Betti numbers satisfying β HT n (M t ) = β HT n (M)/t, ∀t > 0, and a Künneth type formula. An example of a factor in the class HT is given by the group von Neumann factor M = L(Z 2 ⋊ SL(2, Z)), for which β HT 1 (M) = β 1 (SL(2, Z)) = 1/12. Thus, M t ≃ M, ∀t = 1, showing that the fundamental group of M is trivial. This solves a long standing problem of R.V. Kadison. Also, our results bring some insight into a recent problem of A. Connes and answer a number of open questions on von Neumann algebras.

This is a continuation of our previous paper studying the structure of Cartan subalgebras of von Neumann factors of type II 1 . We provide more examples of II 1 factors having either zero, one or several Cartan subalgebras. We also prove a rigidity result for some group measure space II 1 factors.

Abstract. We prove that any isomorphism θ : M 0 ≃ M of group measure space II 1 factors, M 0 = L ∞ (X 0 , µ 0 ) ⋊ σ 0 G 0 , M = L ∞ (X, µ) ⋊ σ G, with G 0 an ICC group containing an infinite normal subgroup with the relative property (T) of KazhdanMargulis (i.e. G 0 w-rigid) and σ a Bernoulli shift action of some group G, essentially comes from an isomorphism of probability spaces which conjugates the actions with respect to some identification G 0 ≃ G. Moreover, any isomorphism θ of M 0 onto a "corner" pMp of M, for p ∈ M an idempotent, forces p = 1. In particular, all group measure space factors associated with Bernoulli shift actions of w-rigid ICC groups have trivial fundamental group and any isomorphism of such factors comes from an isomorphism of the corresponding groups. This settles a "group measure space version" of Connes rigidity conjecture, shown in fact to hold true in a greater generality than just for ICC property (T) groups. We apply these results to ergodic theory, establishing new strong rigidity and superrigidity results for orbit equivalence relations. Introduction.We continue in this paper the study of rigidity properties of isomorphisms θ : M 0 ≃ M of crossed product II 1 factors initiated in ([Po4]), concentrating here on the "group measure space" case, when the factors M 0 , M involved come from free ergodic measure preserving (m.p.) actions of groups on probability spaces. Similarly to ([Po4]), we assume the "source" factor M 0 comes from an arbitrary free ergodic measure preserving action σ 0 of a "mildly rigid" group G 0 (i.e. having a "large" subgroup with the relative property (T) of Kazhdan-Margulis), while the "target" factor M comes from an action σ satisfying good "deformation+mixing"

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