Moufang quadrangles of mixed type

TL;DR

This paper provides geometric characterizations and classifications of Moufang quadrangles of mixed type, focusing on regular points and lines, and explores their connection to generalized inversive planes and Suzuki-Tits ovoids.

Contribution

It introduces new geometric characterizations of mixed type Moufang quadrangles and simplifies axiomatizations of related structures.

Findings

Classified generalized quadrangles with regular points and lines.

Connected regular points to the Axiom of Veblen-Young and weak Desargues.

Characterized generalized inversive planes from Suzuki-Tits ovoids.

Abstract

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 net 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 axiomatization 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 an already known result by the second author.