Strong Rigidity of II${}_1$ Factors Arising from Malleable Actions of w-Rigid Groups, I

We consider cross-product II${}_1$ factors $M=N{⋊}_{\sigma }G$, with $G$ discrete ICC groups that contain infinite normal subgroups with the relative property (T) and $\sigma :G\to \text{Aut}N$ trace preserving actions of $G$ on finite von Neumann algebras $N$ that are ``malleable’’ and mixing. Examples are the weighted Bernoulli and Bogoliubov shifts. We prove a rigidity result for such factors, showing the uniqueness of the position of $L(G)$ inside $M$. We use this to calculate the fundamental group $\text{mycal}F(M)$ in terms of the weights of the shift, for certain arithmetic groups $G$ such as $G={\mathbb{Z}}^2⋊SL(2,\mathbb{Z})$. We deduce that for any countable group $S \subset \Bbb R_+^*$thereexistII$_1$factors$M$with$\mycal F(M)=S$, thus bringing new light to a longstanding problem of Murray and von Neumann.