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

Step 1 of 8: Setting the scene: Haken manifolds and cube complexes

In plain words

A Haken manifold is one you can cut open along a genuinely embedded surface that does not fold back on itself — like slicing a loaf of bread along one flat cut that goes all the way through cleanly. In the 1960s Waldhausen asked whether every hyperbolic 3-manifold has some finite cover admitting such a cut, even when the original manifold does not.

Agol's 2012 proof answers this by translating the question into a completely different world: nonpositively curved cube complexes, built by gluing unit cubes along their faces, whose combinatorics can be analysed with tools from geometric group theory rather than 3-manifold topology directly.

M \text{ Haken} \iff M \text{ contains an embedded } \pi_1\text{-injective surface}

Detailed analysis

Waldhausen's Virtual Haken Conjecture asks whether every closed hyperbolic 3-manifold $M$ has a finite-sheeted cover $\tilde M$ that is Haken, meaning $\tilde M$ contains an embedded, $\pi_1$ -injective (incompressible) surface (Agol 2012, Theorem 9.1, resolving Waldhausen 1968).

Agol's strategy runs through a purely group-theoretic statement, his main Theorem 1.1: every word-hyperbolic group $G$ acting properly and cocompactly on a CAT(0) cube complex $X$ has a finite-index subgroup acting specially on $X$ (§1). Turning the topological question into this algebraic one is what lets the proof use combinatorial techniques (walls, hyperplanes, colorings) that have no obvious 3-manifold analogue.

The remaining steps assemble the two halves needed to invoke this theorem for a hyperbolic 3-manifold: first, producing a suitable cube complex on which $\pi_1(M)$ acts (via Kahn-Markovic surfaces and a construction of Sageev), and then applying the cube-complex theorem itself.

Terms in this step

Haken manifold: A compact, orientable, irreducible 3-manifold containing an embedded, two-sided surface that is $\pi_1$ -injective (its inclusion introduces no extra loop relations); such manifolds are unusually tractable topologically.
CAT(0) cube complex: A simply connected space built by gluing unit Euclidean cubes of various dimensions along their faces so that the result has nonpositive curvature in a combinatorial sense; group actions on such complexes are a central tool of modern geometric group theory.

Knowledge used in this step

Smooth manifolds