LAX EMBEDDINGS OF GENERALIZED QUADRANGLES IN FINITE PROJECTIVE SPACES

Abstract

A generalized quadrangle $\cal S$ is laxly embedded in a (finite) projective space {\bf PG}$(d,q)$ if $\cal S$ is a subgeometry of the geometry of points and lines of {\bf PG}$(d,q)$, with the only condition that the points of $\cal S$ generate the whole space {\bf PG}$(d,q)$ (which one can always assume without loss of generality). In this paper, we classify thick laxly embedded quadrangles satisfying some additional hypotheses. The hypotheses are (a combination of) arestriction on the dimension $d$, a restriction on the parameters of $\cal S$, and an assumption on theisomorphism class of $\cal S$. In particular, theclassification is complete in the following cases: \begin{enumerate}\item[(1)] for $d\geq 5$;\item[(2)] for $d=4$ and $\cal S$ having `known' order$(s,t)$ with $t\not= s^2$; \item[(3)] for $d\geq 3$ and $\cal S$ isomorphic to a finiteMoufang quadrangle distinct from $W(s)$ with $s$ odd.\end{enumerate}

As a by-product, we obtain a new characterization theorem of the classical quadrangle $H(4,s^2)$, and we also show that every generalized quadrangle of order $(s,s+2)$, with $s>2$, has at least one non-regular line.

2000 Mathematics Subject Classification: 51E12.