Cocompactly cubulated crystallographic groups

We prove that the simplicial boundary of a CAT(0) cube complex admitting a proper, cocompact action by a virtually Z group is isomorphic to the hyperoctahedral triangulation of Sn−1, providing a class of groups G for which the simplicial boundary of a G-cocompact cube complex depends only on G. We also use this result to show that the cocompactly cubulated crystallographic groups in dimension n are precisely those that are hyperoctahedral. We apply this result to answer a question of Wise on cocompactly cubulating virtually free abelian groups.