On the geometric and topological rigidity of hyperbolic 3-manifolds

Theorem 0.1. Let N be a closed hyperbolic 3-manifold containing an embedded hyperbolic tube of radius (log 3)/2 = .549306... about a closed geodesic. Then i) Iff : M N is a homotopy equivalence where M is an irreducible 3-manifold, then f is homotopic to a homeomorphism. ii) If f, g: M -s N are homotopic homeomorphisms, then f is isotopic to g. iii) The space of hyperbolic metrics on N is path connected.