I've been interested in abstract polyhedral decompositions of 3-manifolds for a long time. One thing I've tried to do a lot is to get nice polyhedral decompositions of manifolds with Nil geometry. It has been difficult, however, to find ones without some kind of irregularity (e.g. the boundary graph contains vertices of valence 2).
Cubulation is a strong from of regularity for a group. Can the integer Heisenberg group be cubulated? If so, is there a simple, exact description of nonpositively curved cube complex with fundamental group quasi-isometric to Nil?