Innovations in Incidence Geometry

Topological affine quadrangles

Nils Rosehr

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Abstract

The affine derivation of a generalized quadrangle is the geometry induced on the vertices at distance 3 or 4 of a given point. We characterize these geometries by a system of axioms which can be described as a modified axiom system for affine planes with an additional parallel relation and parallel axiom. A second equivalent description which makes it very easy to verify that, for example, ovoids and Laguerre planes yield generalized quadrangles is given. We introduce topological affine quadrangles by requiring the natural geometric operations to be continuous and characterize when these geometries have a completion to a compact generalized quadrangle. In the connected case it suffices to assume that the topological affine quadrangle is locally compact. Again this yields natural and easy proofs for the fact that many concrete generalized quadrangles such as those arising from compact Tits ovoids are compact topological quadrangles. In an appendix we give an outline of the theory of stable graphs which is fundamental to this work.

Article information

Source
Innov. Incidence Geom., Volume 1, Number 1 (2005), 143-169.

Dates
Received: 7 October 2004
Accepted: 16 January 2005
First available in Project Euclid: 26 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.iig/1551206818

Digital Object Identifier
doi:10.2140/iig.2005.1.143

Mathematical Reviews number (MathSciNet)
MR2213956

Zentralblatt MATH identifier
1104.51005

Subjects
Primary: 51E12: Generalized quadrangles, generalized polygons 51H10: Topological linear incidence structures

Keywords
generalized quadrangle affine quadrangle parallel axiom topological geometry completion

Citation

Rosehr, Nils. Topological affine quadrangles. Innov. Incidence Geom. 1 (2005), no. 1, 143--169. doi:10.2140/iig.2005.1.143. https://projecteuclid.org/euclid.iig/1551206818


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