Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-25T01:23:19.369Z Has data issue: false hasContentIssue false
  • English
  • Français

MOUFANG QUADRANGLES OF MIXED TYPE

Published online by Cambridge University Press:  01 January 2008

KOEN STRUYVE  and
HENDRIK VAN MALDEGHEM
KOEN STRUYVE
Affiliation:
Ghent University, Department of Pure Mathematics and Computer Algebra, Krijgslaan 281, S22, B-9000Gent Belgium e-mail: kstruyve@cage.UGent.be, hvm@cage.UGent.be
HENDRIK VAN MALDEGHEM
Affiliation:
Ghent University, Department of Pure Mathematics and Computer Algebra, Krijgslaan 281, S22, B-9000Gent Belgium e-mail: kstruyve@cage.UGent.be, hvm@cage.UGent.be
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In this paper, we present some geometric characterizations of the Moufang quadrangles of mixed type, i.e., the Moufang quadrangles all the points and lines of which are regular. Roughly, we classify generalized quadrangles with enough (to be made precise) regular points and lines with the property that the dual nets associated to the regular points satisfy the Axiom of Veblen-Young, or a very weak version of the Axiom of Desargues. As an application we obtain a geometric characterization and axiomatization of the generalized inversive planes arising from the Suzuki-Tits ovoids related to a polarity in a mixed quadrangle. In the perfect case this gives rise to a characterization with one axiom less than in a previous result by the second author.

Keywords

51E12
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 50 , Issue 1 , January 2008 , pp. 143 - 161
Copyright
Copyright © Glasgow Mathematical Journal Trust 2008

References

REFERENCES

1.Buekenhout, F., Handbook of incidence geometry, buildings and foundations (North-Holland, Elsevier, 1995).Google Scholar
2.Johnson, N., The classification of subplane covered nets, Bull. Belg. Math. Soc. Simon Stevin 2 (1995), 487–508.Google Scholar
3.Payne, S. E. and Thas, J. A., Finite generalized quadrangles (Pitman, 1984).Google Scholar
4.Schroth, A. E., Characterizing symplectic quadrangles by their derivations, Arch. Math. 58 (1992), 98–104.Google Scholar
5.Tent, K., Half Moufang implies Moufang for generalized quadrangles, J. Reine Angew. Math. 566 (2004), 231–236.Google Scholar
6.Thas, J. A. and Maldeghem, H. Van, Generalized quadrangles and the axiom of Veblen, Geometry, combinatorial designs and related structures (ed. Hirschfeld, J. W. P.), London Math. Soc. Lecture Note Ser. No. 245 (Cambridge University Press, 1997), 241–253.Google Scholar
7.Tits, J., Ovoï des et groupes de Suzuki, Arch. Math. 13 (1962), 187–198.CrossRefGoogle Scholar
8.Tits, J., Buildings of spherical type and finite BN-pairs, Lecture Notes in Math. No. 386 (Springer-Verlag, 1974).Google Scholar
9.Tits, J. and Weiss, R., Moufang polygons, (Springer-Verlag, 2002).Google Scholar
10.Van Maldeghem, H., A geometric characterization of the perfect Suzuki-Tits ovoids, J. Geom. 58 (1997), 192–202.Google Scholar
11.Van Maldeghem, H., Generalized polygons, Monographs in Math. No. 93 (Birkhauser-Verlag, 1998).CrossRefGoogle Scholar
12.Van Maldeghem, H., On a question of Arjeh Cohen: a characterization of Moufang projective planes, Bull. Inst. Combin. Appl. 35 (2002), 11–13.Google Scholar
13.Van Maldeghem, H., Moufang lines defined by (generalized) Suzuki groups, Discrete Math., Proceedings of GAC3 (ed. W. Haemers et al.) to appear.Google Scholar
14.Veblen, O. and Young, J. W., Projective geometry (Blaisdell, 1910).Google Scholar