Translation generalized quadrangles
A translation generalized quadrangle (TGQ) is a generalized quadrangle admitting a (uniquely determined) abelian group \(T\) of collineations fixing each line passing through some point \(\infty\) and acting transitively (and therefore regularly) on the set of points opposite to \(\infty\). Finite TGQ have been introduced by \textit{J. A. Thas} in Atti Accad. Naz. Lincei, VIII. Ser., Rend., Cl. Sci. fis. mat. nat. 56, 303-314 (1974; Zbl 0327.05028)]. In this paper arbitrary TGQ are investigated. It is shown that there is an entirely algebraic description for TGQ which plays the same rôle as spreads do for translation (projective) planes. In particular, it is proved that the kernel of a TGQ, a certain subring of \(\text{End} (T)\) which is an invariant of the geometry, is in fact a skew field. Furthermore, TGQ with multiple translation centers and axes are examined.
scientific article; zbMATH DE number 938601