The linear part of a discontinuously acting Euclidean semigroup
type: Item
instance of: scholarly article
title: The linear part of a discontinuously acting Euclidean semigroup
zbMATH Open document ID: 0991.22011
author: G. A. Soifer
published in: Journal of Algebra
publication date: 2002-07-04T00:00:00Z
review text: A subsemigroup \(\Gamma\) of the group \(O(n)\ltimes\mathbb R^n\) of isometries of \(\mathbb R^n\) will be called a Euclidean semigroup. \(\Gamma\) is called properly discontinuous if the set \(\{\gamma\in\Gamma\); \(\gamma K\cap K\neq \emptyset\}\) is finite for every compact subset \(K\) of \(\mathbb R^n\). And \(\Gamma\) is called crystallographic if it is properly discontinuous and \(\bigcup_{\gamma\in\Gamma} \gamma K =\mathbb R^n\) for some compact subset \(K\) of \(\mathbb R^n\). Margulis conjectured that every Euclidean crystallographic semigroup is actually a group. The paper under review is a first step towards proving this conjecture. Its main result says that for a Euclidean properly discontinuous semigroup \(\Gamma\) the connected component of the closure of the linear part \(\ell(\Gamma)\) of \(\Gamma\) is a reducible group. Here the linear part is the natural homomorphism \(\ell : O(n)\ltimes\mathbb R^n\to O(n)\). In the proof an important role is played by the convex cone \(C(S)\) positively spanned by the translational parts of elements \(\gamma\in\Gamma\) with regular \(\ell(\gamma)\) and a Tits type alternative, proved by using dynamical properties of linear maps.
reviewed by: Herbert Abels
Mathematics Subject Classification ID: 22E40
Mathematics Subject Classification ID: 57S30
Mathematics Subject Classification ID: 20H15
zbMATH DE Number: 1763550
zbMATH Keywords: Euclidean semigroup
zbMATH Keywords: properly discontinuous Euclidean semigroup
zbMATH Keywords: crystallographic semigroup
MaRDI profile type: Publication
full work available at URL: jabr.2001.9054
OpenAlex ID: W1991620318
cites work: Semigroups containing proximal linear maps
cites work: Properly discontinuous groups of affine transformations with orthogonal linear part
cites work: Q4777527
cites work: Q4206155
cites work: Auslander's theorem on discontinuous semigroups and crystallographic semigroups
cites work: Free subgroups in linear groups
DOI: 10.1006/JABR.2001.9054
description: scientific article; zbMATH DE number 1763550