On a Class of Type II$_1$ Factors with Betti Numbers Invariants

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 $\Cal H \Cal T$ of factors $M$ having such Cartan subalgebras $A \subset M$, the Betti numbers of the standard equivalence relation associated with $A \subset M$ ([G2]), are in fact isomorphism invariants for the factors $M$, $\beta^{^{HT}}_n(M), n\geq 0$. The class $\Cal H\Cal T$ is closed under amplifications and tensor products, with the Betti numbers satisfying $\beta^{^{HT}}_n(M^t)= \beta^{^{HT}}_n(M)/t, \forall t>0$, and a K{\"u}nneth type formula. An example of a factor in the class $\Cal H\Cal T$ is given by the group von Neumann factor $M=L(\Bbb Z^2 \rtimes SL(2, \Bbb Z))$, for which $\beta^{^{HT}}_1(M) = \beta_1(SL(2, \Bbb Z)) = 1/12$. Thus, $M^t \not\simeq M, \forall t \neq 1$, 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.