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We show that the existence of a maximal embedded tube in a hyperbolic n-manifold implies the existence of a certain conical region. One application is to establish a lower bound on the volume of the region outside the tube, thereby improving estimates on volume and estimates on lengths of geodesics in small volume hyperbolic 3-manifolds. We also provide new bounds on the injectivity radius and diameter of an n-manifold.

Cones Embedded in Hyperbolic Manifolds