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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?

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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.

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    $\begingroup$ There is anoher paper "Homotopy Hyperbolic 3-Manifolds Are Hyperbolic" by Gabai, Meyerhoff, and N. Thurston, jstor.org/stable/3597207?seq=1 saying that a homotopy equivalence from a hyperbolic 3-manifold to a closed irreducible 3-manifold is homotopic to a diffeomorphism. Getting rid of "irreducability" assumption is impossible without Poincare conjecture (just take the connected sum of a closed hyperbolic 3-manifold with a homotopy sphere). $\endgroup$
    – Igor Belegradek
    Commented Oct 25, 2020 at 17:03
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    $\begingroup$ PS Virtually hyperbolic = has a finite cover admitting a hyperbolic structure $\endgroup$
    – YCor
    Commented Oct 25, 2020 at 17:03

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