On a class of type ${\rm II}_1$ factors with Betti numbers invariants
Sorin Popa
We prove that a type ${\rm II}_1$ 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 $A\subset M$, 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 ${\rm II}_1$ 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.