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) a restriction on the dimension $d$, a restriction on the parameters of $\cal S$, and an assumption on the isomorphism class of $\cal S$. In particular, the classification 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 finite Moufang 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.