Any virtually polycyclic group admits a NIL-affine crystallographic action

In this paper, we prove that any virtually polycyclic group admits a properly discontinuous and cocompact action on a simply connected, connected nilpotent Lie group N. This action is affine in the sense that the group acts as a subgroup of the affine group Aff(N)=N⋊Aut(N) of connection preserving automorphisms of N. As a consequence, we also obtain that any virtually polycyclic group admits a polynomial crystallographic action of degree bounded above by its Hirsch length.