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  "body": "\n## Statement\n\nLet $n \\ge 3$, and let $M$ and $N$ be closed connected hyperbolic $n$-manifolds.\nThat is, they are compact connected [[Riemannian manifolds|7CBT4SM83Y]] without boundary, complete, with constant [[sectional curvature|1X42JAA349]] $-1$.\nLet $\\phi \\colon \\pi_1(M) \\to \\pi_1(N)$ be an [[isomorphism|08MG7H8JW5]] of [[fundamental groups|S1XK3HXHFX]].\nThen there is a unique [[isometry|02HPP5VCZH]] $f \\colon M \\to N$ that induces $\\phi$, up to the choice of base points.\n\n## Remarks\n\nHere is the idea of Mostow's proof.\nA homotopy equivalence $M \\to N$ inducing $\\phi$ lifts to a quasi-isometry of the universal covers, both equal to hyperbolic space $\\mathbb{H}^n$.\nThis quasi-isometry extends to a quasiconformal homeomorphism of the sphere at infinity $S^{n-1}$, equivariant for the two actions of $\\pi_1$.\nErgodicity of the action of $\\pi_1(M)$ on $S^{n-1}$ forces this map to be conformal, so it comes from an isometry of $\\mathbb{H}^n$.\n\nConsequence: the volume of a closed hyperbolic manifold of dimension at least $3$ depends only on its fundamental group.\nSo the volume is a topological invariant.\n\nThe hypothesis $n \\ge 3$ is necessary.\nFor 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 $6g - 6$.\nIts quotient by the mapping class group still has positive dimension, so two hyperbolic surfaces of genus $g$ need not be isometric.\n\nThe theorem extends to complete hyperbolic manifolds of finite volume, by work of Marden for $n = 3$ and of Prasad in general.\n\n## Explanations\n\n### L1 human\n\nA hyperbolic manifold is a space that looks everywhere like the hyperbolic plane or its higher-dimensional analogues.\nThe fundamental group records how loops in the space can be deformed.\nIn dimension $3$ and higher, Mostow proved that this purely topological information determines the whole geometry.\nSo the shape and the size of such a manifold cannot be changed without changing its topology.\n\n### L2 human\n\nIn the language of Lie groups, the theorem says the following.\nTwo cocompact lattices in $\\mathrm{SO}(n, 1)$, $n \\ge 3$, that are isomorphic as abstract groups are conjugate in the isometry group of $\\mathbb{H}^n$.\nMostow extended it to irreducible cocompact lattices in semisimple Lie groups not locally isomorphic to $\\mathrm{SL}_2(\\R)$.\nIn higher rank, the [[Margulis superrigidity theorem|DSKV65NHZZ]] gives a much stronger form.\nBy contrast, surfaces have flexible moduli, which leads to Teichmüller theory.\n",
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