Hyperbolization

“Hyperbolization” refers to various functorial constructions of Gromov for converting a cell complex J into an NPC polyhedron \(\mathcal H (J)\) . Several such procedures are discussed in this chapter. The result is usually an NPC cube complex. In Sect. 6.1.1 we discuss some properties which a hyperbolization procedure might satisfy. In Sect. 6.3 the product with interval procedure is explained, while in Sect. 6.4 Gromov’s construction is explained. A procedure for constructing locally \(\operatorname {CAT}(-1)\) spaces, called “strict hyperbolization,” is explained in Sect. 6.5. Relative hyperbolization procedures are explained in Sects. 6.1.2 and 6.3.1. Applications to constructions of NPC manifolds are given in Sect. 6.2.