  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, that
is, such that $\alpha_t \to {\rm id}$ uniformly on the unit ball of $Q$ as $t
\to 0$. However, it has remained unclear how to exploit the interplay between
distinct rigid subalgebras not in specified relative position.
  We show that in fact, any diffuse subalgebra which is rigid with respect to a
mixing s-malleable deformation is contained in a subalgebra which is uniquely
maximal with respect to being rigid. In particular, the algebra generated by
any family of rigid subalgebras that intersect diffusely must itself be rigid
with respect to that deformation. The case where this family has two members
was the motivation for this work, showing for example that if $G$ is a
countable group with $\beta^{1}_{(2)}(G) > 0$, then $L(G)$ cannot be generated
by two property $(T)$ subalgebras with diffuse intersection; however, the
result is most striking when the family is infinite.
