Polynomial Crystallographic Actions on the Plane

We study polynomial crystallographic actions on the plane. These are properly discontinuous and cocompact actions, which are expressed by polynomial functions. We prove that any such action of a polycyclic-by-finite group is of bounded degree and conversely that any polynomial crystallographic action of bounded degree comes from a polycyclic-by-finite group. This last result is a generalization of the well known Auslander conjecture.