On the uniqueness of some affinely flat infra-nilmanifolds
The general Seifert fiber space construction investigates geometric realizations of certain group extensions \(1\to {\mathbb{Z}}^ k\to E\to G\to 1\) as fundamental groups of aspherical manifolds, by methods of cohomological group theory. There is an existence and uniqueness aspect of the theory. In the present paper, a uniqueness result is proved for certain realizations of virtually nilpotent groups by affine maps of \({\mathbb{R}}^ n\) showing that 2 realizations of such groups by affine maps will be conjugate by a polynomial diffeomorphism of \({\mathbb{R}}^ n\) of a certain bounded degree, thus giving a rigidity theorem for infranilmanifolds (the torsion-free case). The proof uses some vanishing result for the first cohomology group of virtually nilpotent groups, with coefficients in a module of polynomial maps between euclidean spaces. See also a paper of \textit{D. Fried} and \textit{W. M. Goldman} [Adv. Math. 47, 1- 49 (1983; Zbl 0571.57030); Theorem 1.20].
geometric realizations of group extensions
general Seifert fiber space construction
fundamental groups of aspherical manifolds
scientific article; zbMATH DE number 4176031