Strong rigidity of II1 factors arising from malleable actions of w-rigid groups, I

Abstract

We consider crossed product II1 factors \(M = N\rtimes_{\sigma}G\), with G discrete ICC groups that contain infinite normal subgroups with the relative property (T) and σ trace preserving actions of G on finite von Neumann algebras N that are “malleable” and mixing. Examples are the actions of G by Bernoulli shifts (classical and non-classical) and by Bogoliubov shifts. We prove a rigidity result for isomorphisms of such factors, showing the uniqueness, up to unitary conjugacy, of the position of the group von Neumann algebra L(G) inside M. We use this result to calculate the fundamental group of M, \(\mathcal{F}(M)\), in terms of the weights of the shift σ, for \(G=\mathbb{Z}^2\rtimes SL(2,\mathbb{Z})\) and other special arithmetic groups. We deduce that for any subgroup S⊂ℝ+ * there exist II1 factors M (separable if S is countable or S=ℝ+ *) with \(\mathcal{F}(M)=S\). This brings new light to a long standing open problem of Murray and von Neumann.