The Mostow rigidity theorem implies that if a manifold of dimension at least 3 has a hyperbolic structure of finite volume, then it is essentially unique.
Thurston's geometrization theorem in this special case states that if M is a 3-manifold that fibers over the circle and whose monodromy is a pseudo-Anosov diffeomorphism, then the interior of M has a complete hyperbolic metric of finite volume.
Otal (1998) and Kapovich (2009) gave proofs of Thurston's theorem for the generic case of manifolds that do not fiber over the circle.
The core of the proof of the geometrization theorem is to prove that if N is not an interval bundle over a surface and M is an atoroidal then the skinning map has a fixed point.
(If N is an interval bundle then the skinning map has no fixed point, which is why one needs a separate argument when M fibers over the circle.)