Virtually Haken conjecture

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In topology, an area of mathematics, the virtually Haken conjecture states that every compact, orientable, irreducible three-dimensional manifold with infinite fundamental group is virtually Haken. That is, it has a finite cover (a covering space with a finite-to-one covering map) that is a Haken manifold.

After the proof of the geometrization conjecture by Perelman, the conjecture was only open for hyperbolic 3-manifolds.

The conjecture is usually attributed to Friedhelm Waldhausen in a paper from 1968,[1] although he did not formally state it. This problem is formally stated as Problem 3.2 in Kirby's problem list.

A proof of the conjecture was announced on March 12, 2012 by Ian Agol in a seminar lecture he gave at the Institut Henri Poincaré. The proof has now been written up, and is published in the journal Documenta Mathematica.[2] The proof built on results of Kahn and Markovic[3][4] in their proof of the surface subgroup conjecture and results of Daniel Wise in proving the Malnormal Special Quotient Theorem[5] and results of Bergeron and Wise for the cubulation of groups.[6]

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  1. ^ Waldhausen, Friedhelm (1968). "On irreducible 3-manifolds which are sufficiently large". Annals of Mathematics. 87 (1): 56–88. doi:10.2307/1970594. MR 0224099.
  2. ^ Agol, Ian (2013). With an appendix by Ian Agol, Daniel Groves, and Jason Manning. "The virtual Haken Conjecture". Doc. Math. 18: 1045–1087. MR 3104553.
  3. ^ Kahn, Jeremy; Markovic, Vladimir (2012). "Immersing almost geodesic surfaces in a closed hyperbolic three manifold". Annals of Mathematics. 175 (3): 1127–1190. arXiv:0910.5501. doi:10.4007/annals.2012.175.3.4. MR 2912704.
  4. ^ Kahn, Jeremy; Markovic, Vladimir (2012). "Counting essential surfaces in a closed hyperbolic three-manifold". Geometry & Topology. 16 (1): 601–624. arXiv:1012.2828. doi:10.2140/gt.2012.16.601. MR 2916295.
  5. ^ Daniel T. Wise, The structure of groups with a quasiconvex hierarchy, https://docs.google.com/file/d/0B45cNx80t5-2NTU0ZTdhMmItZTIxOS00ZGUyLWE0YzItNTEyYWFiMjczZmIz/edit?pli=1
  6. ^ Bergeron, Nicolas; Wise, Daniel T. (2012). "A boundary criterion for cubulation". American Journal of Mathematics. 134 (3): 843–859. arXiv:0908.3609. doi:10.1353/ajm.2012.0020. MR 2931226.

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