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

Step 6 of 8: Agol's theorem: every cubulated hyperbolic group is virtually special

In plain words

Agol's proof builds a "hierarchy": starting from an infinite cover with embedded walls, it labels the walls with finitely many colours so same-coloured walls never cross, then cuts along one colour at a time, like peeling an onion, until the whole complex is reduced to simple polyhedral pieces.

A random-looking colouring will not glue back together correctly, so Agol constructs a carefully invariant measure on the space of colourings, refines it stage by stage to respect earlier cuts, and shows a system of "gluing equations" has a solution — this lets each stage of the hierarchy be reassembled, after passing to a further finite cover, using a general virtual-gluing theorem, until the whole complex is rebuilt as a finite-sheeted special cover.

G \curvearrowright X \text{ (proper, cocompact, } G \text{ word-hyperbolic)} \ \implies\ \exists\, F \le_{f.i.} G:\ F \curvearrowright X \text{ specially}

Detailed analysis

Given a word-hyperbolic group $G$ acting properly and cocompactly on a CAT(0) cube complex $X$ , Agol (2012, §§4-8) first uses a weak separability result to find an infinite regular cover $\mathcal{X}$ of $X/G$ with embedded, pairwise non-crossing compact walls admitting a finite hierarchy: labelling walls with finitely many colours so like-coloured walls are disjoint, and cutting successively by colour to obtain an infinite collection of "cubical polyhedra" (§4).

An invariant colouring measure on the wall graph (§5), refined to track how each stage's cuts interact with earlier ones (§6), is used to solve a system of polyhedral gluing equations (§7) providing the base case of an inductive construction; a virtual gluing theorem (Theorem 3.1) then lets each stage of the hierarchy be assembled, after passing to a further finite-sheeted cover, into the next (§8).

Running the induction down through every colour eventually produces a finite-sheeted cover $\mathcal{V}_0 \to X/G$ with trivial hierarchy and a genuinely embedded wall structure — one on which the deck group acts specially — proving Theorem 1.1: $G$ has a finite-index subgroup acting specially on $X$ .

Terms in this step

Hierarchy: A finite sequence of cuts along embedded walls, one colour at a time, that reduces a cube complex step by step to simple polyhedral pieces whose combinatorics can be analysed directly.
Virtual gluing theorem: A technical tool (Agol 2012, Theorem 3.1) guaranteeing that, given compatible combinatorial data on two sides of a cut, a further finite cover exists on which the two sides can be honestly reglued.

Knowledge used in this step

Smooth manifolds