A geometric proof of Bieberbach's theorems on crystallographic groups (Q1068943)

In 1910 \textit{L. Bieberbach} [Math. Ann. 70, 297-336 (1911) and ibid. 72, 400-412 (1912)] proved that every discrete group of isometries acting on n-dimensional euclidean space with compact fundamental domain contains n linearly independent translations, and for each fixed n there are only finitely many isomorphism classes of n-dimensional crystallographic groups. The author uses a new approach to provide much simpler proofs.