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We show that for every countable group, any sequence of approximatehomomorphisms with values in permutations can be realized as the restriction ofa sofic approximation of an orbit equivalence relation. Moreover, this orbitequivalence relation is uniquely determined by the invariant random subgroup ofthe approximate homomorphism. We record applications of this result to recovervarious known stability and conjugacy characterizations for almosthomomorphisms of amenable groups.

Approximate homomorphisms and sofic approximations of orbit equivalence relations