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.

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]