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  "body": "\n## Statement\n\nLet $M$ be a compact connected [[orientable|702SNFQG0W]] irreducible [[3-manifold|R20TB2KEQQ]] with infinite [[fundamental group|S1XK3HXHFX]], where irreducible means that every embedded $2$-sphere bounds a $3$-ball.\nThen $M$ has a finite [[covering space|3XZRRE165N]] that is a [[Haken manifold|7DP7H8M5B2]].\n\n## Remarks\n\nAgol completed the proof in 2012.\nBy the [[geometrization theorem|B27HB43K86]], the main case is that of a closed hyperbolic $3$-manifold.\nKahn and Markovic showed that its fundamental group contains many quasi-Fuchsian surface subgroups, and Bergeron and Wise deduced that it acts properly and cocompactly on a CAT(0) cube complex.\nAgol then proved that such a cubulated hyperbolic group is virtually special in the sense of Haglund and Wise.\nWise's theory then gives a finite cover with positive first Betti number, which is Haken.\nFor example, Burton, Rubinstein and Tillmann showed that the [[Seifert–Weber space|FBVEG8HVTP]] is not Haken, so the theorem gives it a finite cover that is Haken.\n\n## Explanations\n\n### L1 human\n\nMany closed $3$-dimensional spaces contain no essential surface, so they cannot be cut into simpler pieces.\nThe theorem says that one can always unwrap such a space a finite number of times to get a bigger space that does contain one.\nSo the cutting techniques for Haken manifolds apply, after passing to a finite cover.\n\n### L2 human\n\nAgol's work proves much more: every closed hyperbolic $3$-manifold has a finite cover that fibers over the circle, and its fundamental group is large and linear over $\\Z$.\nThese results answer several of Thurston's questions on $3$-manifolds from 1982.\nThey also show that the fundamental groups of such manifolds are residually finite in a strong form, being virtually subgroups of right-angled Artin groups.\n",
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  "msc": [
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