Let G be the fundamental group of a compact 3manifold which supports on its interior a complete non positively curved Riemannian metric and is a cilinder near de metric. Is G hyperbolic?
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Yes, this follows from the geometrization theorem. A negatively curved complete manifold is atoroidal, and the manifold is irreducible, so there is a complete hyperbolic metric on the interior by geometrization. In your comments, you ask about zero curvature. Then it is just a Euclidean manifold, with fundamental group a Bieberbach group. 

