Finite extensions and unipotent shadows of affine crystallographic groups
type: Item
instance of: scholarly article
title: Finite extensions and unipotent shadows of affine crystallographic groups
zbMATH Open document ID: 1023.20025
DOI: 10.1016/S1631-073X(02)02562-1
author: Oliver Baues
published in: Comptes Rendus. Mathématique. Académie des Sciences, Paris
publication date: 2003-06-16T00:00:00Z
review text: Let \(\Gamma\) be a discrete and cocompact subgroup of \(\text{Aff}(\mathbb{R}^n)\), where \(\mathbb{R}^n\) is \(n\)-dimensional Euclidean space. \(\Gamma\) is called an affine crystallographic group. The paper under review considers the question which abstract virtually polycyclic groups are isomorphic to an affine crystallographic group. A homomorphism \(\rho\colon\Gamma\to A\subset\text{Aff}(\mathbb{R}^n)\) is crystallographic if \(\rho\) is injective and the image \(\rho(\Gamma)\) is a crystallographic subgroup. Let us denote by \(\Hom(\Gamma,A)=\{\rho\colon\Gamma\to A\mid\rho\) crystallographic\}. The group \(A\) acts by conjugation on the above space. Put \(D(\Gamma,A)=\Hom(\Gamma,A)/A\). Moreover, let \(\Delta\) be a finite extension of \(\Gamma\). It is effective, if the homomorphism \(\Delta/\Gamma\to\text{Out}(\Gamma)\) is injective. The main result is the following: Let \(D(\Gamma,A)^{\Delta/\Gamma}\) denote the set of fixed points of the action of \(\Delta/\Gamma\). Let \(\Delta\) be a finite effective extension group of \(\Gamma\). Then the restriction map from \(D(\Delta,A)\) to \(D(\Gamma,A)^{\Delta/\Gamma}\) is a continuous bijection. As corollary the author obtains: \(\Delta\) is isomorphic to an affine crystallographic group of type \(A\) if and only if the induced action of \(\Delta/\Gamma\) on \(D(\Gamma,A)\) has a fixed point. The proofs are not given.
Mathematics Subject Classification ID: 20H15
Mathematics Subject Classification ID: 53A15
Mathematics Subject Classification ID: 20G15
Mathematics Subject Classification ID: 20F16
zbMATH DE Number: 1850800
zbMATH Keywords: affine crystallographic groups
zbMATH Keywords: virtually polycyclic groups
zbMATH Keywords: effective extension groups
zbMATH Keywords: induced actions
reviewed by: Andrzej Szczepański
MaRDI profile type: Publication
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full work available at URL: s1631-073x(02)02562-1
OpenAlex ID: W2002817771
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description: scientific article; zbMATH DE number 1850800