Notes on elation generalized quadrangles

Let be a finite generalized quadrangle of order (s,t),s,t>1. An “elation about a point p” of is an automorphism fixing p linewise and fixing no point which is not collinear with p. An elation that generates a cyclic group of elations is called a “standard elation”. One of the problems already considered in Payne and Thas (Finite Generalized Quadrangles (1984)) is to determine just when the set of elations about the point (∞) is a group. The purpose of this paper is to provide an example where this is not the case, and then to show that for a flock generalized quadrangle the usual group of elations about (∞) is the complete set of standard elations about (∞).     

Determination of generalized quadrangles with distinct elation points

In this paper, we classify the finite generalized quadrangles of order (s,t), s,t > 1, which have a line L of elation points, with the additional property that there is a line M not meeting L for which {L, M} is regular. This is a first fundamental step towards the classification of those generalized quadrangles having a line of elation points. Mathematics Subject Classification (2000): 51E12, 51E20, 20B25, 20E42     

Hemisystems of small flock generalized quadrangles

In this paper, we describe a complete computer classification of the hemisystems in the two known flock generalized quadrangles of order (5², 5) and give numerous further examples of hemisystems in all the known flock generalized quadrangles of order (s ², s) for s ≤ 11. By analysing the computational data, we identify two possible new infinite families of hemisystems in the classical generalized quadrangle H(3, s ²).     

Slanted symplectic quadrangles

By slanting symplectic quadrangles W(F) over fieldsF, we obtain very simple examples of non-classical generalized quadrangles. We determine the collineation groups of these slanted quadrangles and their groups of projectivities. No slanted quadrangle is a topological quadrangle.     

Linearly independent homoclinic bifurcations parameterized by a small function

In this paper we discuss a small nonautonomous perturbation of an autonomous system on Rn which has a homoclinic solution. Regarding the small perturbation as a parameter in an appropriate space of functions we discuss various situations of co-existence of homoclinic orbits. Those conditions of various co-existence actually define bifurcation manifolds in the space of functions for linearly independent homoclinic bifurcations.     

Translation generalized quadrangles

The author gratefully acknowledges the financial support which he received from the Austrian Fonds zur Förderung der wissenschaftlichen Forschung during parts of the work on this paper.     

Extended generalized quadrangles

Extended generalized quadrangles (roughly, connected structures whose every residue is a generalized quadrangle) are studied in some detail, especially those which are uniform or strongly uniform. Much basic structure theory is developed, many examples are given, and something approaching characterization is given for many types.Dedicated to Professor Jacques Tits for his sixtieth birthday     

Extending generalized quadrangles

Extended generalized quadrangles (EGQ) are the geometries associated with the Buekenhout diagram , where is the diagram for generalized quadrangles. In this paper we survey the two cases where an (EGQ) is either a 2-design or a locally polar space of polar rank 2.     

Finite Moufang generalized quadrangles

This paper gives a review of several equivalent Moufang conditions and the corresponding geometric classification in the finite case. __________ Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 22, Algebra and Geometry, 2004.     

Triangular extended generalized quadrangles

In this paper, we continue the study from [2] of what are now called triangular extended generalized quadrangles. In particular, we determine all parameter sets such that the point graph is strongly regular with intersection number =2(t+1).     

Classical Klingenberg generalized quadrangles

Research supported by I.W.O.N.L. grant no. 840037.     

Generalized quadrangles with valuation

We show that the class of generalized quadranges with valuation (as defined in [13]) coincides with the class of the generalized quadrangles associated with the building at infinity of affine buildings of type (up to duality).Dedicated to Prof. Dr J. Tits on the occasion of his sixtieth birthdayThe author is Research Associate at the National Fund of Scientific Research (Belgium).     

Nets and generalized quadrangles

There is a new method of constructing generalized quadrangles (GQs) given by S. Löwe, which is based on covering of nets; all GQs with a regular point can be represented in this way. Here we give a method of constructing GQs with a regular point using the so-called content functions on nets. In the last part of the paper we lay the foundations for a research project aiming to use the more general notion of content to classify GQs and maybe to construct new ones.Both authors acknowledge the financial support by CRUI and DAAD in the frame of Programma Vigoni, which made this work possible.     

On the elation structure of specific designs

Consider symmetric 2-designs D which have an automorphism group G containing sufficiently many elations. This paper proves that all elations in G have the same order and classifies the elation structure (i.e. centre-axis pairs) of a well-defined subgroup of G. For one class, D is necessarily a projective space.The results in this paper form part of a PhD thesis submitted by the author to the University of London. The author gratefully acknowledges the support of the Commonwealth Scholarship Commission.     

Local half Moufang quadrangles

In this article, we show that if every root of a finite generalized quadrangle containing a fixed point x is Moufang, then every dual root containing x in its interior is also Moufang. As a corollary, we obtain a new proof of the half Moufang theorem. This says that finite half Moufang quadrangles are Moufang.