3,256 research outputs found
Generalized quadrangles of orrder (s, s2), I
AbstractIn this paper generalized quadrangles of order (s, s2), s > 1, satisfying property (G) at a line, at a pair of points, or at a flag, are studied. Property (G) was introduced by S. E. Payne (Geom. Dedicata32 (1989), 93–118) and is weaker than 3-regularity (see S. E. Payne and J. A. Thas, “Finite Generalized Quadrangles,” Pitman, London, 1984). It was shown by Payne that each generalized quadrangle of order (s2, s), s > 1, arising from a flock of a quadratic cone, has property (G) at its point (∞). In particular translation generalized quadrangles satisfying property (G) are considered here. As an application it is proved that the Roman generalized quadrangles of Payne contain at least s3 + s2 classical subquadrangles Q(4, s). Also, as a by-product, several classes of ovoids of Q(4, s), s odd, are obtained; one of these classes is new. The goal of Part II is the classification of all translation generalized quadrangles satisfying property (G) at some flag ((∞), L)
Central aspects of skew translation quadrangles, I
Except for the Hermitian buildings , up to a combination
of duality, translation duality or Payne integration, every known finite
building of type satisfies a set of general synthetic
properties, usually put together in the term "skew translation generalized
quadrangle" (STGQ). In this series of papers, we classify finite skew
translation generalized quadrangles. In the first installment of the series, as
corollaries of the machinery we develop in the present paper, (a) we obtain the
surprising result that any skew translation quadrangle of odd order is
a symplectic quadrangle; (b) we determine all skew translation quadrangles with
distinct elation groups (a problem posed by Payne in a less general setting);
(c) we develop a structure theory for root-elations of skew translation
quadrangles which will also be used in further parts, and which essentially
tells us that a very general class of skew translation quadrangles admits the
theoretical maximal number of root-elations for each member, and hence all
members are "central" (the main property needed to control STGQs, as which will
be shown throughout); (d) we solve the Main Parameter Conjecture for a class of
STGQs containing the class of the previous item, and which conjecturally
coincides with the class of all STGQs.Comment: 66 pages; submitted (December 2013
Singer quadrangles
[no abstract available
Classification of skew translation generalized quadrangles, I
We describe new classification results in the theory of generalized quadrangles (= Tits-buildings of rank 2 and type B-2), more precisely in the (large) subtheory of skew translation generalized quadrangles ("STGQs"). Some of these involve, and solve, long-standing open problems
Elation generalised quadrangles of order (s,p), where p is prime
We show that an elation generalised quadrangle which has p+1 lines on each
point, for some prime p, is classical or arises from a flock of a quadratic
cone (i.e., is a flock quadrangle).Comment: 14 page
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