---
tag: GYB411M966
kind: statement
role: theorem
name: Virtually Haken conjecture
msc: [57K30, 57K32, 20F65]
uses: [R20TB2KEQQ, 702SNFQG0W, 7DP7H8M5B2, 3XZRRE165N, S1XK3HXHFX]
about: [7DP7H8M5B2]
sources:
  - url: https://en.wikipedia.org/wiki/Virtually_Haken_conjecture
same_as:
  wikidata: Q7935203
  wikipedia: Virtually Haken conjecture
---

## Statement

Let $M$ be a compact connected [[orientable|702SNFQG0W]] irreducible [[3-manifold|R20TB2KEQQ]] with infinite [[fundamental group|S1XK3HXHFX]], where irreducible means that every embedded $2$-sphere bounds a $3$-ball.
Then $M$ has a finite [[covering space|3XZRRE165N]] that is a [[Haken manifold|7DP7H8M5B2]].

## Remarks

Agol completed the proof in 2012.
By the [[geometrization theorem|B27HB43K86]], the main case is that of a closed hyperbolic $3$-manifold.
Kahn and Markovic showed that its fundamental group contains many quasi-Fuchsian surface subgroups, and Bergeron and Wise deduced that it acts properly and cocompactly on a CAT(0) cube complex.
Agol then proved that such a cubulated hyperbolic group is virtually special in the sense of Haglund and Wise.
Wise's theory then gives a finite cover with positive first Betti number, which is Haken.
For example, Burton, Rubinstein and Tillmann showed that the [[Seifert–Weber space|FBVEG8HVTP]] is not Haken, so the theorem gives it a finite cover that is Haken.

## Explanations

### L1 human

Many closed $3$-dimensional spaces contain no essential surface, so they cannot be cut into simpler pieces.
The theorem says that one can always unwrap such a space a finite number of times to get a bigger space that does contain one.
So the cutting techniques for Haken manifolds apply, after passing to a finite cover.

### L2 human

Agol's work proves much more: every closed hyperbolic $3$-manifold has a finite cover that fibers over the circle, and its fundamental group is large and linear over $\Z$.
These results answer several of Thurston's questions on $3$-manifolds from 1982.
They also show that the fundamental groups of such manifolds are residually finite in a strong form, being virtually subgroups of right-angled Artin groups.