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

Step 4 of 8: Sageev cubulation: turning surfaces into a cube complex

In plain words

Each of Kahn-Markovic's immersed surfaces lifts, in the universal cover $\mathbb{H}^3$ , to a family of embedded copies — think of them as huge, curved walls slicing through space. Sageev showed back in 1995 that any sufficiently rich collection of such walls determines a combinatorial 'shadow': a cube complex whose vertices record every consistent choice of side for every wall.

Bergeron and Wise combined this construction with Kahn-Markovic's surfaces to show that a closed hyperbolic 3-manifold group always acts properly and cocompactly on such a cube complex — turning geometric information about surfaces into the exact algebraic input Agol's theorem needs.

\{\Sigma_g\} \text{ walls in } \mathbb{H}^3 \ \xrightarrow{\text{Sageev}}\ \pi_1(M) \curvearrowright X \text{ (proper, cocompact CAT(0) cube complex)}

Detailed analysis

Sageev's dual cube complex construction (1995) associates to a space equipped with a sufficiently separating family of codimension-1 walls a CAT(0) cube complex $X$ , with one hyperplane for each wall, on which the ambient symmetry group acts.

Bergeron and Wise (cited as Agol 2012, Theorem 9.3, building on Kahn-Markovic and on prior cubulation criteria of Sageev) prove that for a closed hyperbolic 3-manifold $M$ , the $\mathbb{H}^3$ -lifts of enough immersed quasi-fuchsian surfaces from the previous step give a wall structure rich enough that $\pi_1(M)$ acts properly and cocompactly on the resulting dual cube complex $X$ .

This is precisely the hypothesis of Agol's Theorem 1.1 (a word-hyperbolic group acting properly and cocompactly on a CAT(0) cube complex), so the remaining task, taken up in the next steps, is to prove that theorem itself.

Terms in this step

Wall / hyperplane: In a cube complex, a hyperplane is obtained by gluing together the "midcubes" cut out by bisecting each cube along one direction; a wall is the analogous separating hypersurface in the space a cube complex is built from.
Proper and cocompact action: An action is proper if only finitely many group elements move any given compact set to overlap itself, and cocompact if the quotient of the space by the action is compact; together they mean the group and the space are geometrically "the same size".

Knowledge used in this step

Smooth manifolds