ON A CLASS OF TYPE II_{1} FACTORS WITH BETTI NUMBERS INVARIANTS

SORIN POPA

Abstract. We prove that a type II_{1} factor M has at most one Cartan subalgebra A satisfying a combination of rigidity and compact approximation properties. In particular, this shows that the Betti numbers of A ⊂ M, as defined by Gaboriau (2001), are isomorphism invariants for the factors M having such Cartan subalgebras. Examples of factors with this property are the group measure space algebras associated with certain actions of the free groups with finitely many generators on the probability space, for which the Betti numbers are calculated by Gaboriau. As a consequence, we obtain the first examples of type II_{1} factors M with trivial fundamental group and solve a problem formulated by Kadison. Also, our results bring some insight into a recent problem of A. Connes.