Innovations in Incidence Geometry

Topological affine quadrangles

Nils Rosehr

Full-text: Open access


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

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

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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

generalized quadrangle affine quadrangle parallel axiom topological geometry completion


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

Export citation


  • S. Boekholt and M. Stroppel, Independence of axioms for fourgonal families, J. Geom. 72 (2001), 37–46.
  • T. Buchanan, Ovale und Kegelschnitte in der komplexen projektiven Ebene, Math.-Phys. Semesterber. 26 (1979), 244–260.
  • T. Buchanan, H. Hähl and R. L öwen, Topologische Ovale, Geom. Dedicata 9 (1980), 401-424.
  • H. Groh, Geometric lattices with topology, J. Combin. Theory Ser. A 42 (1986), 111–125.
  • T. Grundh öfer, Ternary fields of compact projective planes, Abh. Math. Sem. Univ. Hamburg 57 (1987), 87–101.
  • T. Grundh öfer and N. Knarr, Topology in generalized quadrangles, Topology Appl. 34 (1990), 139–152.
  • T. Grundh öfer and R. L öwen, Linear topological geometries, In: Handbook of incidence geometry, North-Holland (1995), 1255–1324.
  • T. Grundh öfer and H. Van Maldeghem, Topological polygons and affine buildings of rank three, Atti Sem. Mat. Fis. Univ. Modena 38 (1990), 459–479.
  • M. Joswig, Translationsvierecke, Ph.D. Thesis, Tübingen, 1994.
  • R. L öwen, Central collineations and the parallel axiom in stable planes, Geom. Dedicata 10 (1981), 283–315.
  • R. L öwen and G. F. Steinke, Nonclassical $4$-dimensional Minkowski planes obtained as brothers of semiclassical $4$-dimensional Laguerre planes, Geom. Dedicata 72 (1998), 143–169.
  • M. Margraf, Topologische Laguerreräume und topologische verallgemeinerte Vierecke, Ph.D. Thesis, Kiel, 2001.
  • S. E. Payne and J. A. Thas, Generalized quadrangles with symmetry, Simon Stevin 49 (1975/76), 3–32, 81–103.
  • H. Pralle, Affine generalized quadrangles –- an axiomatization, Geom. Dedicata 84 (2001), 1–23.
  • N. Rosehr, Compactification of stable planes. Preprint.
  • ––––, Pseudo-isotopic contractions and compactness, Results Math. 44 (2003), 157–158.
  • ––––, Stable Graphs and Polygons, Habilitationsschrift, Würzburg, 2003.
  • H. Salzmann, D. Betten, T. Grundh öfer, H. Hähl, R. L öwen and M. Stroppel, Compact Projective Planes, de Gruyter, 1995.
  • A. E. Schroth, Generalized quadrangles constructed from topological Laguerre planes, Geom. Dedicata 46 (1993), 339–361.
  • ––––, Topological Circle Planes and Topological Quadrangles, Longman, 1995.
  • G. Steinke, Eine Klassifikation $4$-dimensionaler Laguerre-Ebenen mit großer Automorphismengruppe, Habilitationsschrift, Kiel, 1988.
  • ––––, Semiclassical $4$-dimensional Laguerre planes, Forum Math. 2 (1990), 233–247.
  • ––––, Topological circle geometries, In: Handbook of incidence geometry, North-Holland (1995), 1325–1354
  • B. Stroppel, Point-affine quadrangles, Note Mat. 20 (2000/01), 21–31.
  • J. Tits, Sur la trialité et certains groupes qui s'en déduisent, Publ. Math. IHES 2 (1959), 14–60.
  • H. Van Maldeghem, Generalized Polygons. Birkhäuser, 1998.
  • D. Wagner, Ovale und ebene algebraische Kurven mit unendlicher Kollineationsgruppe, Ph.D. Thesis, Erlangen, 2004.