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  "body": "\n## Statement\n\nLet $M$ be a [[hyperbolic 3-manifold|GHVW1Z35BS]] with finitely generated [[fundamental group|S1XK3HXHFX]].\nEach end of $M$ carries an *end invariant*.\nFor a geometrically finite end, it is a point of a Teichmüller space.\nFor a degenerate end, it is an ending lamination on a surface.\nThen $M$ is determined up to isometry by its topological type together with its end invariants.\n\n## Remarks\n\nThurston conjectured the theorem.\nBrock, Canary and Minsky proved it, first for manifolds homeomorphic to $S \\times \\R$ for a surface $S$, building on earlier work of Minsky.\nThe general case uses the [[tameness theorem|DE1Z7R9T05]], which makes the ends of $M$ products of a surface and a half-line, so that end invariants make sense.\nThe proof builds a combinatorial model manifold from the end invariants, using the curve complex, and shows that $M$ is bilipschitz to this model.\nFor example, a quasi-Fuchsian manifold $S \\times \\R$ has two geometrically finite ends.\nThe theorem then recovers Bers simultaneous uniformization: the manifold is determined by a pair of points in the [[Teichmüller space|ZZGC6BZNJ5]] of $S$.\n\n## Explanations\n\n### L1 human\n\nA hyperbolic $3$-manifold with finitely generated fundamental group may have ends that open out to infinity.\nEach end leaves a trace at infinity, either a shape of surface or a pattern of curves called a lamination.\nThe theorem says that the topology and these traces at infinity pin down the whole geometry.\n\n### L2 human\n\nThe theorem completes the classification of hyperbolic $3$-manifolds with finitely generated fundamental group.\nTogether with tameness, it was used to prove the density theorem: every finitely generated Kleinian group is a limit of geometrically finite ones.\nIt also makes rigorous Thurston's picture of doubly degenerate surface groups, such as the fiber group of a fibered hyperbolic $3$-manifold.\n",
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