A Combination Theorem for Special Cube Complexes

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 compact negatively curved cube complexes whose hyperplanes don’t self-intersect. For cube complexes with word-hyperbolic fundamental group, we are able to show that they are virtually special if and only if the hyperplanes 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.