Communications in algebra vol:22 issue:7 pages:2547-2558
In this paper we pre-sent a new algebraic characterization of (almost-)crystallographic groups. In fact we prove that a virtually abelian (resp. nilpotent) group E is crystallographic (re-sp. almost-crystallographic) if and only if E has no finite normal subgroups. Thereafter, we investigate more general groups sharing this property and refer to them as the almost-torsion free groups. We show that a virtually polycyclic group GAMMA is almost-torsion free if and only if GAMMA is (almost-crystallographic)-by-crystallographic. Thereafter, we take a closer look at the topological properties of almost-torsion free groups.