Inapproximability of actions and Kazhdan's property (T)

We construct p.m.p. group actions that are not local-global limits of sequences of finite graphs. Moreover, they do not weakly contain any sequence of finite labeled graphs. Our methods are based on the study of almost automorphisms of sofic approximations: We show that the set of epsilon-automorphisms of a sufficiently good sofic approximation of a Kazhdan group by expanders form a group in a natural way.