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Essential embedded surfaces in an irreducible 3-manifold correspond to non-trivial splittings of its fundamental group. In joint work with Khánh Lê, we give some conditions on the fundamental group of a Haken hyperbolic 3-manifold which guarantee that any other 3-manifold group with the same set of finite quotients must have a non-trivial splitting. Using one of these conditions, we show that every finite regular cover of an aspherical integer homology three sphere with positive first Betti number will have first Betti number at least four and this is optimal. We will also discuss examples of Haken 3-manifolds to which the theorems in this work apply.