On a class of type II1 factors with Betti numbers invariants

We prove that a type II1 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 subset of M, the Betti numbers of the standard eqilivalence relation associated with A subset of M ([G2]), are in fact isomorphism invariants for HT the factors M, beta n(HT) (M), n >= 0. The class HT is closed under amplifications and tensor products, with the Betti numbers satisfying beta(HT)(n) (M-t) = beta(HT)(n) (M)/t, for all t > 0, and a Kunneth type formula. An example of a factor in the class HT is given by the group von Neumann factor M = L(Z(2) x SL(2, Z)), for which beta(HT)(1)(M) = beta(1)(SL(2,Z)) = 1/12. Thus, M-t not similar or equal to M, for all t , 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.