Mostow rigidity theorem

Remarks

Here is the idea of Mostow's proof. A homotopy equivalence $M \to N$ inducing $\phi$ lifts to a quasi-isometry of the universal covers, both equal to hyperbolic space $\mathbb{H}^n$. This quasi-isometry extends to a quasiconformal homeomorphism of the sphere at infinity $S^{n-1}$, equivariant for the two actions of $\pi_1$. Ergodicity of the action of $\pi_1(M)$ on $S^{n-1}$ forces this map to be conformal, so it comes from an isometry of $\mathbb{H}^n$.

Consequence: the volume of a closed hyperbolic manifold of dimension at least $3$ depends only on its fundamental group. So the volume is a topological invariant.

The hypothesis $n \ge 3$ is necessary. For a closed orientable surface of genus $g \ge 2$, the hyperbolic metrics, up to isometries isotopic to the identity, form the Teichmüller space of real dimension $6g - 6$. Its quotient by the mapping class group still has positive dimension, so two hyperbolic surfaces of genus $g$ need not be isometric.

The theorem extends to complete hyperbolic manifolds of finite volume, by work of Marden for $n = 3$ and of Prasad in general.

Going deeper

For a student

In the language of Lie groups, the theorem says the following. Two cocompact lattices in $\mathrm{SO}(n, 1)$, $n \ge 3$, that are isomorphic as abstract groups are conjugate in the isometry group of $\mathbb{H}^n$. Mostow extended it to irreducible cocompact lattices in semisimple Lie groups not locally isomorphic to $\mathrm{SL}_2(\R)$. In higher rank, the Margulis superrigidity theorem gives a much stronger form. By contrast, surfaces have flexible moduli, which leads to Teichmüller theory.