Discontinuous motion groups of different degrees in metric spaces
type: Item
instance of: scholarly article
title: Discontinuous motion groups of different degrees in metric spaces
zbMATH Open document ID: 0821.51012
DOI: 10.1007/BF01406833
author: Benno Klotzek
published in: Journal of Geometry
publication date: 1995-03-30T00:00:00Z
review text: This is an exhaustive survey-style analysis of the equivalencies, dependencies and independencies between 16 conditions for discontinuous groups of isometries of two- and three-dimensional Euclidean space. The conditions are variations on the local finiteness condition for orbits of a group \(\mathbf B\) of isometries of a metric space \(({\mathcal X}, d)\) (the set \(S(X,P,r) := \{\tau(P) \mid d(\tau(P), X) < r\), \(\tau \in {\mathbf B}\}\) is finite, for all \(X\), \(P\) in \(\mathcal X\) and \(r > 0\)) and the condition of discreteness of points in their orbit (for all \(P \in {\mathcal X}\), \(S(P, P, r)\) is a singleton for sufficiently small \(r > 0\)).
Mathematics Subject Classification ID: 51F99
Mathematics Subject Classification ID: 51-02
Mathematics Subject Classification ID: 57S30
Mathematics Subject Classification ID: 20H15
zbMATH DE Number: 739185
zbMATH Keywords: survey
zbMATH Keywords: equivalence
zbMATH Keywords: dependence
zbMATH Keywords: independence
zbMATH Keywords: discontinuous groups
zbMATH Keywords: isometries
zbMATH Keywords: Euclidean space
zbMATH Keywords: local finiteness
zbMATH Keywords: metric space
reviewed by: Victor V. Pambuccian
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 739185