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  "body": "\n## Statement\n\nLet $M$ be a [[hyperbolic 3-manifold|GHVW1Z35BS]] whose [[fundamental group|S1XK3HXHFX]] is finitely generated.\nThen $M$ is homeomorphic to the interior of a compact $3$-manifold.\n\n## Remarks\n\nThe statement was conjectured by Marden and is also called the Marden conjecture.\nIt was proved in 2004 by Agol and, independently, by Calegari and Gabai.\nBoth proofs approximate the ends of $M$ by surfaces of controlled geometry, building on work of Canary and others on tame ends.\nFor example, if $M = \\mathbb{H}^3 / \\Gamma$ with $\\Gamma$ a quasi-Fuchsian group isomorphic to the fundamental group of a closed surface $S$, then $M$ is homeomorphic to $S \\times \\R$, the interior of $S \\times [0, 1]$.\nThe hypothesis that $\\pi_1(M)$ is finitely generated is needed: there are hyperbolic $3$-manifolds with infinitely generated fundamental group that are not the interior of any compact manifold.\nAmong its consequences is the Ahlfors measure conjecture: the limit set of a finitely generated Kleinian group is either the whole sphere or has Lebesgue measure zero.\n\n## Explanations\n\n### L1 human\n\nA hyperbolic $3$-dimensional space may be infinite, with ends that stretch off to infinity in complicated ways.\nThe theorem says that, when the fundamental group is finitely generated, these ends are as simple as possible topologically.\nThe whole space is a compact manifold with its boundary removed.\n\n### L2 human\n\nTameness is the topological input for the classification of hyperbolic $3$-manifolds with finitely generated fundamental group.\nTogether with the ending lamination theorem and the density theorem, it gives a complete picture of the deformation space of Kleinian groups.\nIt also implies that such a manifold has finitely many ends, each homeomorphic to a surface times a half-line.\n",
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  "name": "Tameness theorem",
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