Virtually Haken conjecture
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.
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. The proof built on results of Kahn and Markovic in their proof of the surface subgroup conjecture and results of Daniel Wise in proving the Malnormal Special Quotient Theorem and results of Bergeron and Wise for the cubulation of groups.
- Waldhausen, Friedhelm (1968). "On irreducible 3-manifolds which are sufficiently large". Annals of Mathematics. 87 (1): 56–88. doi:10.2307/1970594. MR 224099.
- 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.
- Kahn, Jeremy; Markovic, Vladimir (2012). "Immersing almost geodesic surfaces in a closed hyperbolic three manifold". Annals of Mathematics. 175 (3): 1127–1190. arXiv: . doi:10.4007/annals.2012.175.3.4. MR 2912704.
- Kahn, Jeremy; Markovic, Vladimir (2012). "Counting essential surfaces in a closed hyperbolic three-manifold". Geometry & Topology. 16 (1): 601–624. arXiv: . doi:10.2140/gt.2012.16.601. MR 2916295.
- Daniel T. Wise, The structure of groups with a quasiconvex hierarchy, https://docs.google.com/file/d/0B45cNx80t5-2NTU0ZTdhMmItZTIxOS00ZGUyLWE0YzItNTEyYWFiMjczZmIz/edit?pli=1
- Bergeron, Nicolas; Wise, Daniel T. (2012). "A boundary criterion for cubulation". American Journal of Mathematics. 134 (3): 843–859. arXiv: . doi:10.1353/ajm.2012.0020. MR 2931226.
- Dunfield, Nathan; Thurston, William (2003), "The virtual Haken conjecture: experiments and examples", Geometry and Topology, 7: 399–441, doi:10.2140/gt.2003.7.399, MR 1988291.
- Kirby, Robion (1978), "Problems in low dimensional manifold theory.", Algebraic and geometric topology (Proc. Sympos. Pure Math., Stanford Univ., Stanford, Calif., 1976), 7, pp. 273–312, MR 0520548.
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