Any virtually polycyclic group admits a NIL-affine crystallographic action.
type: Item
instance of: scholarly article
title: Any virtually polycyclic group admits a NIL-affine crystallographic action.
zbMATH Open document ID: 1036.20044
DOI: 10.1016/S0040-9383(02)00030-7
author: Karel Dekimpe
published in: Topology
publication date: 2003-04-27T00:00:00Z
review text: The author proves that every virtually polycyclic group admits a properly discontinuous and cocompact action on a simply connected, connected nilpotent Lie group \(N\). This action is affine in the sense that the group acts as a subgroup of the affine group \(\text{Aff}(N)=N\rtimes\Aut(N)\) of connection preserving automorphisms of \(N\). This result is interesting in view of the existing results, conjectures and counterexamples for crystallographic actions on ordinary real affine space. From the main theorem of the paper follows that every virtually polycyclic group admits a polynomial crystallographic action of degree bounded above by its Hirsch length.
Mathematics Subject Classification ID: 20H15
Mathematics Subject Classification ID: 57S30
zbMATH DE Number: 1900972
zbMATH Keywords: crystallographic actions
zbMATH Keywords: nilpotent Lie groups
zbMATH Keywords: affine groups
zbMATH Keywords: polynomial diffeomorphisms
zbMATH Keywords: virtually polycyclic groups
zbMATH Keywords: solvable Lie groups
reviewed by: Herbert Abels
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1900972