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

Step 5 of 8: Special cube complexes: Wise's embedding criterion

In plain words

Among all CAT(0) cube complexes, some are especially well-behaved: their hyperplanes never cross themselves, never touch themselves at a single point without crossing, and never touch a neighbouring hyperplane twice in incompatible ways. Wise, building on Haglund-Wise (2008), called these complexes "special".

The payoff is enormous: a special complex's fundamental group embeds cleanly into a right-angled Artin group — a very simple, well-understood kind of group built directly from a graph — which brings a whole toolbox of algebra (linearity, separability, and more) along for free.

X \text{ special (Haglund-Wise)} \iff X \hookrightarrow \text{Salvetti}(\Gamma) \implies \pi_1(X) \hookrightarrow A(\Gamma)

Detailed analysis

A nonpositively curved (NPC) cube complex $X$ is special (Haglund-Wise 2008) when its hyperplanes avoid several kinds of local pathology: self-intersection, one-sidedness, self-osculation, and inter-osculation between distinct hyperplanes.

Haglund and Wise proved that $X$ is special exactly when there is a combinatorial local isometry from $X$ into the Salvetti complex of some right-angled Artin group $A(\Gamma)$ , which realises $\pi_1(X)$ as a subgroup of $A(\Gamma)$ (cited as Agol 2012, Theorem 2.7, attributed to Wise).

Right-angled Artin groups are residually finite and linear, and their quasi-convex subgroups are separable, so any group that is virtually the fundamental group of a special cube complex automatically inherits all of these properties — which is why Agol's programme targets virtual specialness rather than anything weaker.

Terms in this step

Special cube complex: An NPC cube complex whose hyperplanes are embedded, two-sided, and free of self- and inter-osculation pathologies; equivalently, one admitting a combinatorial local isometry into the Salvetti complex of a right-angled Artin group.
Right-angled Artin group: A group presented by generators, one per vertex of a graph $\Gamma$ , subject only to commutation relations between generators joined by an edge; the simplest and most flexible family of groups built directly from combinatorial data.

Knowledge used in this step

Smooth manifolds