Maximal rigid subalgebras of deformations and L2-cohomology

In the past two decades, Sorin Popa's breakthrough deformation/rigidity theory has produced remarkable rigidity results for von Neumann algebras $M$ which can be deformed inside a larger algebra $\widetilde M \supseteq M$ by an action $\alpha: \mathbb{R} \to {\rm Aut}(\widetilde M)$, while simultaneously containing subalgebras $Q$ {\it rigid} with respect to that deformation, is, such $\alpha_t id}$ uniformly on unit ball of as $t 0$. However, it remained unclear how exploit interplay between distinct rigid not in specified relative position. We show fact, any diffuse subalgebra is mixing s-malleable deformation contained uniquely maximal being rigid. particular, generated family intersect diffusely must itself deformation. The case where this members was motivation work, showing example if $G$ countable group $\beta^{1}_{(2)}(G) > 0$, then $L(G)$ cannot property $(T)$ intersection; however, result most striking when infinite.