General Theory of Cube Complexes

In Sect. 5.1 Sageev’s definition of a “pocset” or “abstract half-space system” is given and Sageev’s theorem that one can reconstruct a CAT(0) cube complex from such a half-space system is stated. For example, any Coxeter group naturally has an associated half-space system on which it acts and we get the theorem of Niblo–Reeves that any Coxeter group acts properly on a CAT(0) cube complex. (Indeed, many of the ideas in this chapter originate with classical work on the theory of reflection groups.) The fundamental groups of many closed hyperbolic n-manfolds can act freely and cocompactly on CAT(0) cube complexes. For 3-manifolds this is proved by Bergeron–Wise by making use of work of Kahn–Marković (see Theorem 5.21). For certain arithmetic n-manifolds it is a result of Bergeron-Haglund-Wise (see Theorem 5.24). Section 5.2 deals with the Haglund–Wise theory of special cube complexes where “special” is defined by forbidding certain types of intersections of hyperplanes. The principal result is that the fundamental group of any NPC special cube complex virtually embeds in some right-angled Artin group or Coxeter group. Using the fact that RAAGs and RACGs have good separation properties and that these properties descend to subgroups, Agol was able to prove, among other things, that Thurston’s Virtual Haken Conjecture holds for hyperbolic 3-manifolds. In Sect. 5.3 we prove some of these separation properties for RACGs. In particular, we explain Haglund’s result that word quasi-convex subgroups of RACGs are virtual retracts.