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?



Can the integer Heisenberg group be cubulated?
From the comments:
A discrete group with a proper action on an arbitrary CAT(0) cube complex has no distorted cyclic subgroups. Thus Heisenberg (which has a quadratically distorted Z) has no such proper action.


It follows from Pansu's results that if G is a discrete f.g. group QI to Heisenberg, then it has a finite index subgroup isomorphic to Nil. Thus it also has a distorted Z and cannot be the fundamental group of a locally CAT(0) cubing, nor even act properly on a CAT(0) cubing.


Another obstruction is the Dehn function, which is cubic. Any CAT(0) group has quadratic Dehn function. 


Finally, Nil group is not hyperbolic; it is not even semihyperbolic. Thus, integer Heisenberg group cannot be cubulated in any meaningful way. 


