STRONG RIGIDITY OF II1 FACTORS ARISING FROM MALLEABLE ACTIONS OF w-RIGID GROUPS, I

We consider crossed product II1 factors M = N ⋊σ G, with G discrete ICC groups that contain infinite normal subgroups with the relative property (T) and σ trace preserving actions of G on finite von Neumann algebras N that are “malleable” and mixing. Examples are the actions of G by Bernoulli shifts (classical and non-classical), and by Bogoliubov shifts. We prove a rigidity result for isomorphisms of such factors, showing the uniqueness, up to unitary conjugacy, of the position of the group von Neumann algebra L(G) inside M . We use this result to calculate the fundamental group of M , F (M), in terms of the weights of the shift σ, for G = Z ⋊SL(2,Z) and other special arithmetic groups. We deduce that for any subgroup S ⊂ R∗+ there exist II1 factors M (separable if S is countable or S = R ∗ +) with F (M) = S. This brings new light to a long standing open problem of Murray