Deforming cubulations of hyperbolic groups

TL;DR

This paper introduces a method to deform cubulations of hyperbolic groups via hyperplane bending, demonstrating the existence of multiple bald cubulations and addressing questions about group actions on CAT(0) cube complexes.

Contribution

It develops a new hyperplane-bending technique for deforming cubulations and proves the existence of infinitely many bald cubulations for most hyperbolic groups.

Findings

Every cocompactly cubulated hyperbolic group admits infinitely many bald cubulations.

Cocompactly cubulated groups have a single orbit of hyperplanes in certain actions.

Burger-Mozes examples have a unique bald cubulation.

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-$\mathbb Z$ 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 ${\rm CAT}(0)$ 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 $G$ 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 $G$ admits infinitely many bald cubulations, provided $G$ is not a virtually free group with ${\rm Out}(G)$ finite. By contrast, we show that the Burger-Mozes examples each admit a unique bald cubulation.