Compact connected translation generalized quadrangles
Classical examples of compact connected translation generalized quadrangles are obtained from quadrics of projective Witt index 1 in compact connected projective spaces (that is, projective spaces over the reals, the complex numbers, or Hamilton's quaternions). In the general case, these incidence structures are described by the translation group \(T\) together with a suitable collection of subgroups. As in the case of translation planes, this collection of subgroups defines a subring of the endomorphism ring of \(T\), the so-called kernel of the translation quadrangle. The paper under review contains a proof that the kernel of a compact connected translation quadrangle is isomorphic to the field of reals, or to the field of complex numbers (3.5, 5.4). Consequently, the important part of the automorphism group is a linear Lie group. A complex kernel characterizes the dual of the symplectic quadrangle over the complex numbers (5.6). Theorem 3.7 establishes an intimate connection with locally compact connected elation Laguerre planes. In Section 4, translation quadrangles of Tits type are characterized by the fact that the stabilizers of lines have dimension 1 over the kernel. In the last section, it is shown that each compact connected translation quadrangle belongs to exactly one of the following classes: (a) there is exactly one translation center, (b) the translation centers form a point row, (c) each point is a translation center. In any case, there is no translation axis. In case (c), we have a Moufang quadrangle, isomorphic to an orthogonal quadrangle over the reals or the complex numbers. The author also proves that possibility (b) does not occur in the smallest case, where the point space has dimension \(3\).
scientific article; zbMATH DE number 1498579
scientific article