### MOUFANG QUADRANGLES OF MIXED TYPE

KOEN STRUYVE, HENDRIK VAN MALDEGHEM
2008 Glasgow Mathematical Journal
In this paper, we present some geometric characterizations of the Moufang quadrangles of mixed type, i.e., the Moufang quadrangles all the points and lines of which are regular. Roughly, we classify generalized quadrangles with enough (to be made precise) regular points and lines with the property that the dual nets associated to the regular points satisfy the Axiom of Veblen-Young, or a very weak version of the Axiom of Desargues. As an application we obtain a geometric characterization and
more » ... omatization of the generalized inversive planes arising from the Suzuki-Tits ovoids related to a polarity in a mixed quadrangle. In the perfect case this gives rise to a characterization with one axiom less than in a previous result by the second author. 2000 Mathematics Subject Classification. 51E12. In 1974, Jacques Tits  introduced what he called groups of mixed type, as a certain generalization of algebraic groups. This was motivated by the fact that certain spherical buildings arise from such groups and Tits classified all spherical buildings of rank at least three in  . Introduction. Roughly, the groups of mixed type of rank 2 arise when the weight of the edge of the rank 2 Coxeter diagram is equal to the characteristic of the underlying field. Indeed, in the commutation relation of the root groups, the weight w of the edge turns up as a coefficient, and as a power. If the corresponding term does not vanish (i.e., if in the underlying field w is not equal to 0), then we are in the generic case where we are able to distinguish long and short roots (by the commutation relations, but also by the geometry of the corresponding building). However, if w = 0, i.e., if the characteristic of the underlying field is equal to w (if the diagram is included in a rank 3 diagram, then only the cases w ∈ {1, 2, 3} occur), then the commutation relations become much more symmetric, allowing for diagram automorphisms. If the field is perfect, not much extra happens since the symmetry is then up to the field Frobenius automorphism x → x w , and we only obtain an extra group automorphism (diagram automorphism). However, if the field is not perfect, then this "duality" is not surjective anymore, and we obtain the peculiar situation in which the rank 2 geometry looks symmetric, but isn't. Technically, the duality maps the geometry into itself, but not onto. In other words, the geometry (building) is isomorphic to the dual of a subgeometry.