Generalized quadrangles with a thick hyperbolic line weakly embedded in projective space (Q1281153)

Let \(\Gamma\) be a generalized quadrangle and let \(P(\Gamma)\) be the set of points of \(\Gamma\). For some subset \(A \subseteq P(\Gamma)\) let \(A^\perp\) be the set of all points of \(\Gamma\) that are collinear with any point of \(A\). An automorphism of \(\Gamma\) that fixes any point collinear with some point \(p\) is called a central collineation of \(\Gamma\). The generalized quadrangle \(\Gamma\) is said to be weakly embedded in the projective space \(P\) (of dimension at least 3) if there exists an injective map \(\iota : P(\Gamma) \to P\) such that (1) the image of \(\iota\) generates \(P\), (2) any subspace of \(P\) spanned by \(\iota(l)\), where \(l\) is an arbitrary line of \(\Gamma\), is a line of \(P\), and (3) if \(x,y \in P(\Gamma)\) such that \(\iota(y)\) is contained in the subspace of \(P\) generated by \(\iota(x^\perp)\), then \(y \in x^\perp\). The map \(\iota\) is then called a \textit{weak embedding}. Weak embeddings (for polar spaces) have first been studied by \textit{C. Lefèvre-Percsy} [Bull. Cl. Sci., V. Ser., Acad. R. Belg. 67, 45-50 (1981; Zbl 0487.51004)]. The author and Van Maldeghem classified weakly embedded generalized quadrangles (in a paper to appear) under the hypothesis that each secant line of \(P\) contains a third point of \(\Gamma\). The proof of this classification result is based on the fact that \(\Gamma\) admits non-trivial central collineations, see Lefèvre-Percsy (loc. cit.). Using this, they show that \(\Gamma\) is a Moufang quadrangle. The classification ends with a case by case study of the Moufang quadrangles in question. In the paper under review the author generalizes the classification result as described above by weakening the hypothesis to the following condition: any set \(\{a,b\}^{\perp\perp}\) contains a third point. The proof of this generalization runs along the same lines as before thus proving a generalization of the theorem of Lefèvre-Percsy. Moreover, it turns out that the subgroup of Aut\((\Gamma)\) generated by all central collineations is induced by PSL\((V)\), where \(V\) is the vector space associated to the projective space \(P\).