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

Step 7 of 8: From virtual specialness to separability, linearity, and largeness

In plain words

Haglund–Wise's embedding of the group of a special cube complex into a right-angled Artin group gives a powerful package of structure: linearity, residual finiteness, and separability of quasiconvex subgroups. This package should not be confused with LERF for every right-angled Artin group.

For the hyperbolic 3-manifold group, Agol's theorem supplies the additional 3-manifold consequences needed below: virtual specialness gives the separability and largeness properties used in the virtual Haken and virtual fibering arguments.

G \text{ virtually special} \implies G \text{ linear, large; quasi-convex subgroups of } G \text{ separable}

Detailed analysis

Haglund-Wise's embedding of a special cube complex's group into a right-angled Artin group $A(\Gamma)$ transports several structural properties for free: $A(\Gamma)$ is linear (embeds in $\mathrm{GL}_n(\mathbb{Z})$ for some $n$ ) and residually finite, and its quasi-convex subgroups are separable. This is not equivalent to LERF for the whole RAAG.

Since $G$ is only virtually (i.e. up to finite index) the fundamental group of a special complex, these properties pass to $G$ itself: Agol's Corollary 1.2 concludes that a nonelementary word-hyperbolic group acting properly and cocompactly on a CAT(0) cube complex is linear, large (has a finite-index subgroup surjecting onto a nonabelian free group), and has separable quasi-convex subgroups.

For $\pi_1(M)$ of a closed hyperbolic 3-manifold this yields, in particular, that $\pi_1(M)$ is LERF — the property Waldhausen needed decades earlier to attack the Haken conjecture directly, and which Agol later uses to deduce that all finite-covolume Kleinian groups are LERF, resolving a question of Thurston.

Terms in this step

LERF (subgroup separable): A group in which every finitely generated subgroup $H$ is the intersection of the finite-index subgroups containing it; equivalently, for every $g \notin H$ there is a finite quotient of $G$ in which the images of $g$ and $H$ are disjoint.

Knowledge used in this step

The fundamental group