The piecewise projective group as group of projectivities
type: Item
instance of: scholarly article
title: The piecewise projective group as group of projectivities
zbMATH Open document ID: 0871.51008
DOI: 10.1007/BF03322185
published in: Results in Mathematics
publication date: 1997-09-23T00:00:00Z
review text: A flat Minkowski plane consists of a system of circles which are graphs of homeomorphisms of the 1-sphere \(\mathbb{S}^1\) on the torus \(\mathbb{S}^1\times\mathbb{S}^1\) such that three mutually non-parallel points can uniquely be joined by a circle and such that the axiom of touching is satisfied. The set of circles is a locally compact topological space with two connected components, one consisting of all the graphs of orientation preserving homeomorphisms the other of all the graphs of orientation reversing homeomorphisms. Given the same basic representation of two Minkowski planes one can exchange the same kind of connected component and obtain again flat Minkowski planes as was shown by the author and the reviewer [Can. Math. Bull. 38, 469-480 (1995; Zbl 0844.51005)]. The paper under review contains a new construction of flat Minkowski planes from two given flat Minkowski planes which refines the above mentioned method. In this construction one considers a flat Minkowski plane relative to a point \(p\). Circles in the component that are made up of graphs of orientation preserving homeomorphisms are of two kinds. These are the circles that pass through \(p\) and the other circles. Assuming that all translations of the derived affine plane at the distinguished point \(p\) are automorphisms of the Minkowski plane, one obtains relative to this derived affine plane a picture of straight lines and certain curves that look like hyperbolas. Each of these curves has two branches, one convex and the other concave. The convex branches taken together form, what the author calls a quarter of the Minkowski plane, and similarly for the concave branches. The author shows that the convex branches in one quarter of such a Minkowski plane and the concave branches in a quarter of another Minkowski plane plus the lines that come from graphs of orientation preserving homeomorphisms can be put together via a pasting function, i.e., an increasing homeomorphism of \(\mathbb{R}^+\), to…
reviewed by: Günter F. Steinke
Mathematics Subject Classification ID: 51H15
Mathematics Subject Classification ID: 51B20
zbMATH DE Number: 933261
zbMATH Keywords: flat Minkowski plane
zbMATH Keywords: group of projectivities
author: Burkard Polster
MaRDI profile type: Publication
cites work: Laguerre and Minkowski planes produced by dilatations
cites work: Die Projektivitätengruppe der Moulton-Ebenen
cites work: Q3694162
cites work: The groups of projectivities of \(B\)-ovals
cites work: Invertible sharply \(n\)-transitive sets
cites work: Cut and Paste in 2-Dimensional Projective Planes and Circle Planes
cites work: Some Minkowski planes with 3-dimensional automorphism group
cites work: Q4764160
Recommended article: Groups of piecewise projective homeomorphisms
Recommended article: Projective \(G\)-groups
Recommended article: Q3481940
Recommended article: Projectice subgroups for grouping
Recommended article: The groups of projectivities in finite planes
Recommended article: Q4695835
Recommended article: On the projectives of a group algebra
Recommended article: Projective lines as groupoids with projection structure
Recommended article: Q3358858
Recommended article: Q5692657
description: scientific article; zbMATH DE number 933261