Any virtually polycyclic group admits a NIL-affine crystallographic action.
The author proves that every 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 \(\text{Aff}(N)=N\rtimes\Aut(N)\) of connection preserving automorphisms of \(N\). This result is interesting in view of the existing results, conjectures and counterexamples for crystallographic actions on ordinary real affine space. From the main theorem of the paper follows that every virtually polycyclic group admits a polynomial crystallographic action of degree bounded above by its Hirsch length.
scientific article; zbMATH DE number 1900972