Cubulation of Hyperbolic Groups and Rips Constructions in Small Cancellation Theory

This monograph explores the structural interplay between Gromov-hyperbolic groups, small cancellation theory, and the geometry of non-positively curved (NPC) cube complexes. We analyze the seminal Rips construction, which produces hyperbolic groups with pathological subgroups via small cancellation techniques, and contrast this with the rigidity imposed by the Agol-Wise cubulation theorem. Specifically, we investigate conditions under which Rips-constructed groups admit a proper, cocompact action on a CAT(0) cube complex. We detail Sageev's construction of dual cube complexes from codimension-1 subgroups (walls) and examine the Haglund-Wise theory of special cube complexes. Furthermore, we discuss recent developments in "cubical small cancellation theory," which generalizes classical metric small cancellation to higher-dimensional cubical presentations, thereby providing a mechanism to construct virtually special hyperbolic groups with prescribed quotients. The synthesis of these theories resolves open questions regarding the residual finiteness and virtual fibering of specific classes of hyperbolic groups.