Deforming cubulations of hyperbolic groups

We describe a procedure to deform cubulations of hyperbolic groups by "bending hyperplanes". Our construction is inspired related constructions like Thurston's Mickey Mouse example, walls in fibred $3$-manifolds and free-by-$\mathbb Z$ groups, Hsu-Wise turns. As an application, we show that every cocompactly cubulated Gromov-hyperbolic group admits proper, cocompact, essential action on ${\rm CAT}(0)$ cube complex with single orbit hyperplanes. This answers (in the negative) question Wise, who proved result case free groups. also study those general $G$ are not susceptible trivial deformations. name these "bald cubulations" observe at least one bald cubulation. then apply hyperplane-bending prove infinitely many cubulations, provided virtually Out}(G)$ finite. By contrast, Burger-Mozes examples each admit unique