TITLE: A strong form of Mostow rigidity without geometrization?
QUESTION [9 upvotes]: Mostow rigidity theorem says that two closed hyperbolic manifolds with isomorphic fundamental groups are isometric.
Here is my question: suppose that $M$ and $N$ are two closed 3-manifolds such that $M$ and $N$ are homotopy equivalent and such that $N$ is hyperbolic. Is it possible to prove that $M$ and $N$ are homeomorphic (diffeomorphic) without using geometrization theorem?

REPLY [10 votes]: Gabai proved that homotopy hyperbolic 3-manifolds are virtually hyperbolic, in the paper of that name:

Gabai, David, Homotopy hyperbolic 3-manifolds are virtually
hyperbolic.
J. Amer. Math. Soc. 7 (1994), no. 1, 193–198.

I suspect this is the best you can do without geometrisation.