virtually Haken conjecture

{{short description|Every compact, irreducible 3-manifold with infinite fundamental group is virtually Haken}}

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,{{cite journal|first=Friedhelm|last=Waldhausen|title=On irreducible 3-manifolds which are sufficiently large|journal=Annals of Mathematics|volume=87|year=1968|issue=1|pages=56–88|mr=224099|doi=10.2307/1970594|jstor=1970594|url=https://pub.uni-bielefeld.de/record/1782185}} 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.{{cite journal|first1=Ian|last1=Agol|title=The virtual Haken Conjecture|others=With an appendix by Ian Agol, Daniel Groves, and Jason Manning|journal=Doc. Math.|year=2013|pages=1045–1087|volume=18|doi=10.4171/dm/421 |mr=3104553|s2cid=255586740 |url=https://www.math.uni-bielefeld.de/documenta/vol-18/33.html|doi-access=free}} 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){{cite journal| last1=Haglund|first1=Frédéric|last2=Wise|first2=Daniel|title=A combination theorem for special cube complexes|journal=Annals of Mathematics |year=2012 |volume=176|issue=3|pages=1427–1482|mr=2979855|doi=10.4007/annals.2012.176.3.2|doi-access=free}}

It used as an essential ingredient the freshly-obtained solution to the surface subgroup conjecture by Jeremy Kahn and Vladimir Markovic.{{cite journal|last1=Kahn|first1=Jeremy|last2=Markovic|first2=Vladimir|title=Immersing almost geodesic surfaces in a closed hyperbolic three manifold|arxiv=0910.5501|journal=Annals of Mathematics |year=2012 |volume=175|issue=3|pages=1127–1190|mr=2912704|doi=10.4007/annals.2012.175.3.4|s2cid=32593851}}{{cite journal|last1=Kahn|first1=Jeremy|last2=Markovic|first2=Vladimir|title=Counting essential surfaces in a closed hyperbolic three-manifold|year=2012|volume=16|issue=1|pages=601–624|arxiv=1012.2828|journal=Geometry & Topology |mr=2916295|doi=10.2140/gt.2012.16.601}}

Other results which are directly used in Agol's proof include the Malnormal Special Quotient Theorem of WiseDaniel T. Wise, The structure of groups with a quasiconvex hierarchy, https://docs.google.com/file/d/0B45cNx80t5-2NTU0ZTdhMmItZTIxOS00ZGUyLWE0YzItNTEyYWFiMjczZmIz/edit?pli=1 and a criterion of Nicolas Bergeron and Wise for the cubulation of groups.{{cite journal|first1=Nicolas|last1=Bergeron|first2=Daniel T.|last2=Wise|title=A boundary criterion for cubulation| year=2012|volume=134|issue=3|pages=843–859|journal=American Journal of Mathematics|arxiv=0908.3609|mr=2931226| doi=10.1353/ajm.2012.0020|s2cid=14128842}}

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.{{Cite journal|last1=Przytycki|first1=Piotr|last2=Wise|first2=Daniel|date=2017-10-19|title=Mixed 3-manifolds are virtually special|url=https://www.ams.org/jams/2018-31-02/S0894-0347-2017-00886-5/|journal=Journal of the American Mathematical Society|language=en|volume=31|issue=2|pages=319–347|doi=10.1090/jams/886|s2cid=39611341|issn=0894-0347|doi-access=free|arxiv=1205.6742}}{{Cite web|title=Piotr Przytycki and Daniel Wise receive 2022 Moore Prize|url=https://www.ams.org/news?news_id=6854|access-date=|website=American Mathematical Society|language=en}}