Worked solution: Agol's proof of the Virtual Haken conjecture via cube complexes (2012)

Step 8 of 8: Conclusion: the virtual Haken and virtual fibering theorems

In plain words

Every piece of the argument now clicks into place: a hyperbolic 3-manifold's fundamental group acts on a cube complex (Kahn-Markovic plus Sageev), that action is virtually special (Agol's theorem), and virtual specialness hands over both an incompressible surface in a finite cover and, remarkably, a way to fiber that same finite cover over a circle.

Since Perelman's geometrization already handles every closed 3-manifold that is not hyperbolic (spherical, Euclidean, Seifert fibered, or split by an essential torus), the virtual Haken theorem extends to closed aspherical 3-manifolds. The virtual-fibering theorem remains the hyperbolic case, resolving Thurston's 1982 question.

M = \mathbb{H}^3/\Gamma\ \implies\ \exists\, \tilde M \to M \text{ finite-sheeted}:\ \tilde M \text{ Haken and fibers over } S^1

Detailed analysis

Combining the reduction to hyperbolic 3-manifolds (Step 2), Sageev's cubulation via Kahn-Markovic surfaces (Step 4), and Agol's cube-complex theorem (Step 6): for every closed hyperbolic 3-manifold $M$ , $\pi_1(M)$ is virtually special; Agol's 3-manifold consequences (Step 7) give LERF and largeness, hence $M$ has a finite-sheeted cover $\tilde M$ containing an embedded incompressible surface — proving Waldhausen's Virtual Haken Conjecture (Agol 2012, Theorem 9.1).

Wise's book-length development of special cube complexes shows, moreover, that a virtually special hyperbolic 3-manifold group is virtually fibered, i.e. $\tilde M$ can be chosen to fiber over the circle (Wise, cited as Agol 2012, Theorem 9.2), resolving Thurston's Virtual Fibering Question; combined with Step 7, $\pi_1(M)$ is in addition LERF and large for every closed hyperbolic 3-manifold $M$ .

Terms in this step

Fibered manifold (over the circle): A 3-manifold built by gluing the two ends of (surface) $\times [0,1]$ via a "monodromy" homeomorphism of the surface; equivalently, one admitting a fibre bundle map to $S^1$ whose fibre is a surface.

Knowledge used in this step

Smooth manifolds

Common mistake. Agol's cube-complex theorem alone needs

\pi_1(M)

to already act on some CAT(0) cube complex; producing that action required the separately deep Kahn-Markovic and Sageev-Bergeron-Wise constructions of Steps 3-4, so no single one of these results by itself would have solved the conjecture.