Virtually Haken conjecture explained

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 appeared shortly thereafter in a preprint which was eventually published in Documenta Mathematica.[2] The proof was obtained via a strategy by previous work of Daniel Wise and collaborators, relying on actions of the fundamental group on certain auxiliary spaces (CAT(0) cube complexes, also known as median graphs)[3] It used as an essential ingredient the freshly-obtained solution to the surface subgroup conjecture by Jeremy Kahn and Vladimir Markovic.[4] [5] Other results which are directly used in Agol's proof include the Malnormal Special Quotient Theorem of Wise[6] and a criterion of Nicolas Bergeron and Wise for the cubulation of groups.[7]

In 2018 related results were obtained by Piotr Przytycki and Daniel Wise proving that mixed 3-manifolds are also virtually special, that is they can be cubulated into a cube complex with a finite cover where all the hyperplanes are embedded which by the previous mentioned work can be made virtually Haken.[8] [9]

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Notes and References

  1. Friedhelm. Waldhausen. On irreducible 3-manifolds which are sufficiently large. Annals of Mathematics. 87. 1968. 1. 56–88. 224099. 10.2307/1970594. 1970594.
  2. Ian. Agol. The virtual Haken Conjecture. With an appendix by Ian Agol, Daniel Groves, and Jason Manning. Doc. Math.. 2013. 1045–1087. 18. 10.4171/dm/421 . 3104553. 255586740 . free.
  3. Haglund. Frédéric. Wise. Daniel. A combination theorem for special cube complexes. . 2012 . 176. 3. 1427–1482. 2979855. 10.4007/annals.2012.176.3.2. free.
  4. Kahn. Jeremy. Markovic. Vladimir. Immersing almost geodesic surfaces in a closed hyperbolic three manifold. 0910.5501. . 2012 . 175. 3. 1127–1190. 2912704. 10.4007/annals.2012.175.3.4. 32593851.
  5. Kahn. Jeremy. Markovic. Vladimir. Counting essential surfaces in a closed hyperbolic three-manifold. 2012. 16. 1. 601–624. 1012.2828. Geometry & Topology . 2916295. 10.2140/gt.2012.16.601.
  6. Daniel T. Wise, The structure of groups with a quasiconvex hierarchy, https://docs.google.com/file/d/0B45cNx80t5-2NTU0ZTdhMmItZTIxOS00ZGUyLWE0YzItNTEyYWFiMjczZmIz/edit?pli=1
  7. Nicolas. Bergeron. Daniel T.. Wise. A boundary criterion for cubulation. 2012. 134. 3. 843–859. American Journal of Mathematics. 0908.3609. 2931226. 10.1353/ajm.2012.0020. 14128842.
  8. Przytycki. Piotr. Wise. Daniel. 2017-10-19. Mixed 3-manifolds are virtually special. Journal of the American Mathematical Society. en. 31. 2. 319–347. 10.1090/jams/886. 39611341. 0894-0347. free. 1205.6742.
  9. Web site: Piotr Przytycki and Daniel Wise receive 2022 Moore Prize. American Mathematical Society. en.