Deforming cubulations of hyperbolic groups

Abstract

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