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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.