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, 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. 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.

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