On Bieberbach’s analysis of discrete Euclidean groups

Abstract:

For a subgroup G of the euclidean group ${E_n} = {O_n} \cdot {{\mathbf {R}}^n}$ (semidirect product) and a real number $r > 0$, let ${G^ \ast }$ denote the translation subgroup of G, ${G_r}$ the group generated by all (A, a) in G with $\left \| {1 - A} \right \| < r$ (operator norm), and ${k_n}(r)$ the maximum number of elements of ${O_n}$ with mutual distances $\geqslant r$ relative to the metric $d(A,B) = \left \| {A - B} \right \|$. We give an elementary, largely geometrical proof of the following results of Bieberbach: Let G be a subgroup of ${E_n}$. (1) If G is discrete, then ${G_{1/2}}$ is abelian, ${G_{1/2}} \triangleleft G$, and $[G:{G_{1/2}}] \leqslant {k_n}(1/2)$. (2) G is discrete if and only if $G \subset {O_{n - k}} \times {E_k}$, where ${p_2}G$ is discrete, ${({p_2}G)^ \ast }$ spans ${{\mathbf {R}}^k}$, and $G \cap \ker {p_2}$ is finite. (Here ${p_2}$ is the projection on the second factor.) (3) G is crystallographic if and only if G is discrete and ${G^ \ast }$ spans ${{\mathbf {R}}^n}$. Moreover, if G is crystallographic, then $[G:{G^ \ast }] \leqslant {k_n}(1/2)$.