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In this paper we shall be interested in the existence of maps of nonzero degree between compact 3-mamfolds, and epimorphisms between their fundamental groups. We shall be particularly interested in the case when the manifolds are finite volume hyperbolic 3-manifolds. Throughout, by a hyperbolic n-manifold we shall always mean an n-manifold admitting a complete Riemannian metric of constant curvature —1. In what follows, all manifolds are assumed connected and orient able unless indicated. There is a rich history to the study of maps of non-zero degree (see for example [14], [15] and references therein), and of particular motivation to us is the suggestion of Gromov and others that one can use maps of nonzero degree to define a partial order on the set of homeomorphism classes of compact n-manifolds. More precisely, say that a compact n-manifold M dominates (resp. d-dominates) a compact n-manifold AT, and write M > N (resp. M >£ N) if there is a map / : M —> N of non-zero degree (resp. degree d). For n ^ 3, by H. C. Wang's famous theorem that for bounded volume there are at most finitely many closed hyperbolic n-manifolds, together with the work of Gromov (see [24]), each closed hyperbolic n-manifold dominates at most finitely many closed hyperbolic n-manifolds. In dimension 3, it was shown by Thurston ([24]) that there exists a V > 0 such that there are infinitely many closed hyperbolic 3-manifolds of volume bounded above by V. The difference between dimension 3 and dimension n 7^ 3 is further emphasized by an example in [1] of a closed non-orientable

Non-Haken 3-manifolds are not large with respect to mappings of non-zero degree