  A discrete countable group G is matricially stable if the finite dimensional
approximate unitary representations of G are perturbable to genuine
representations in the point-norm topology. For large classes of groups G, we
show that matricial stability implies the vanishing of the rational cohomology
of G in all nonzero even dimensions. We revisit a method of constructing almost
flat K-theory classes of BG which involves the dual assembly map and
quasidiagonality properties of G. The existence of almost flat K-theory classes
of BG which are not flat represents an obstruction to matricial stability of G
due to continuity properties of the approximate monodromy correspondence.
