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

TL;DR

This paper establishes strong rigidity results for certain II$_1$ factors arising from malleable, mixing actions of w-rigid groups, enabling the explicit calculation of their fundamental groups and constructing factors with prescribed fundamental groups.

Contribution

It proves the uniqueness of the position of group von Neumann algebras inside these factors and computes their fundamental groups for specific groups, advancing understanding of II$_1$ factor invariants.

Findings

Proved rigidity results for II$_1$ factors from malleable actions.

Calculated fundamental groups for factors associated with specific groups.

Constructed factors with arbitrary countable fundamental groups.

Abstract

We consider cross-product II$_1$ factors $M = N\rtimes_{\sigma} G$, with $G$ discrete ICC groups that contain infinite normal subgroups with the relative property (T) and $\sigma: G \to {\text{\rm Aut}}N$ 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 $\mycal F(M)$ in terms of the weights of the shift, for certain arithmetic groups $G$ such as $G=\Bbb Z^2 \rtimes SL(2, \Bbb Z)$. We deduce that for any countable group $S \subset \Bbb R_+^*$ there exist II$_1$ factors $M$ with $\mycal F(M)=S$, thus bringing new light to a longstanding problem of Murray and von Neumann.