A combination theorem for special cube complexes

Abstract

We prove that certain compact cube complexes have special finite covers. This means they have finite covers whose fundamental groups are quasiconvex subgroups of right-angled Artin groups. As a result we obtain linearity and the separability of quasiconvex subgroups for the groups we consider. Our result applies, in particular, to a compact negatively curved cube complex whose hyperplanes do not self-intersect. For a cube complex with word-hyperbolic fundamental group, we show that it is virtually special if and only if its hyperplane stabilizers are separable. In a final application, we show that the fundamental groups of every simple type uniform arithmetic hyperbolic manifolds are cubical and virtually special.