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 was subsequently outlined in three lectures March 26 and 28th at the Workshop on Immersed Surfaces in 3-Manifolds at the Institut Henri Poincaré. A preprint of the claimed proof has been posted on the ArXiv. The proof built on results of Kahn and Markovic in their proof of the Surface subgroup conjecture and results of Wise in proving the Malnormal Special Quotient Theorem and results of Bergeron and Wise for the cubulation of groups.
- virtually fibered conjecture
- virtually positive Betti number conjecture
- Friedhelm Waldhausen, On irreducible 3-manifolds which are sufficiently large. Ann. of Math. (2) 87 1968 56–88.,
- Ian Agol, Daniel Groves, and Jason Manning, A boundary criterion for cubulation. http://arxiv.org/abs/1204.2810
- Kahn and Markovic, Immersing almost geodesic surfaces in a closed hyperbolic manifold http://arxiv.org/abs/0910.5501, Counting essential surfaces in a closed hyperbolic 3-manifold, http://arxiv.org/abs/1012.2828
- Daniel T. Wise, The structure of groups with a quasiconvex hierarchy, https://docs.google.com/file/d/0B45cNx80t5-2NTU0ZTdhMmItZTIxOS00ZGUyLWE0YzItNTEyYWFiMjczZmIz/edit?pli=1
- Nicolas Bergeron and Daniel T. Wise, A boundary criterion for cubulation, http://arxiv.org/abs/0908.3609
- 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.
- Kirby, Robion (1978), Problems in low dimensional manifold theory. 7, pp. 273–312 Unknown parameter
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