On the uniqueness of some affinely flat infra-nilmanifolds
type: Item
instance of: scholarly article
title: On the uniqueness of some affinely flat infra-nilmanifolds
zbMATH Open document ID: 0714.57027
DOI: 10.1007/BF02570895
author: Paul Igodt
author: Kyung Bai Lee
published in: Mathematische Zeitschrift
publication date: 1990-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/183792
review text: 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].
Mathematics Subject Classification ID: 57S30
Mathematics Subject Classification ID: 20F34
Mathematics Subject Classification ID: 57R19
Mathematics Subject Classification ID: 57R22
zbMATH DE Number: 4176031
zbMATH Keywords: geometric realizations of group extensions
zbMATH Keywords: general Seifert fiber space construction
zbMATH Keywords: fundamental groups of aspherical manifolds
zbMATH Keywords: virtually nilpotent groups
zbMATH Keywords: rigidity theorem for infranilmanifolds
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
cites work: Q5670826
cites work: Three-dimensional affine crystallographic groups
cites work: Q3216808
cites work: Applications of Group Cohomology to Space Constructions
cites work: THE SEIFERT CONSTRUCTION AND ITS APPLICATIONS TO INFRANILMANIFOLDS
cites work: Aspherical Manifolds with Virtually 3-Step Nilpotent Fundamental Group
description: scientific article; zbMATH DE number 4176031