Bieberbach theorems for solvable Lie groups
A discrete cocompact subgroup of a Lie group \(G\) is called lattice of \(G\). In this terminology, an \(n\)-dimensional space group \(\pi\) is a lattice of the Euclidean group \(E(n)\), and the subgroup \(\Gamma\) of the translations of \(\pi\) is a lattice of \(\mathbb{R}^n\). For space groups L. Bieberbach proved three classical theorems, which state that \(\Gamma\) has finite index in \(\pi\) (first Bieberbach), that isomorphic space groups are conjugate subgroups of the affine group\(Aff(\mathbb{R}^n) =\mathbb{R}^n \ltimes\Aut (\mathbb{R}^n)\) (second Bieberbach) and that there are only finitely many isomorphism classes of \(n\)-dimensional space groups (third Bieberbach). The aim of the authors is to investigate the generalization of these theorems for \(G\) a solvable Lie group, the case of a nilpotent Lie group having already been treated by several authors. In general, these theorems are not valid in the solvable case. Therefore, one has to impose a number of restrictions. This leads to the concept of type \((R)\) (or completely solvable) for \(G\) connected solvable, such that \(ad (X):{\mathcal G}\to{\mathcal G}\) has only real eigenvalues for each \(X\in{\mathcal G}\), with \({\mathcal G}\) the Lie algebra of \(G\); and of type\((E)\) (or exponential) if exp: \({\mathcal G}\to G\) is surjective. Moreover a strong lattice property is introduced in terms of the nilradical of \(G\) and of tori. It is then proved that the first Bieberbach theorem is true if and only if \(G\) is of type\((E)\) with strong lattice property. The second and the third theorems, which had already been generalized, are reformulated for \(G\) of type\((R)\). Examples of solvable \(G\) of type\((R)\) and of not type\((E)\) are given for which the first Bieberbach theorem does not hold. A whole class of groups of type\((R)\) is indicated satisfying the conditions required for the validity of the theorem which generalizes first Bieberbach.
scientific article; zbMATH DE number 1818541