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

Inapproximability of actions and Kazhdan's property (T)