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-$\mathbb{Z}$ 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 $\mathrm{CAT}(0)$ 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 $\mathrm{Out}(G)$ finite. By contrast, we show that the Burger-Mozes examples each admit a unique bald cubulation.

44 pages, 4 figures; to appear on Journal of Topology

References in corpus (2)

• Coarse decompositions of boundaries for CAT(0) groups
• Large facing tuples and a strengthened sector lemma

Cited by in corpus (3)

• On automorphisms and splittings of special groups
• Homotopy equivalent boundaries of cube complexes
• Coarse cubical rigidity