Quadrangles embedded in metasymplectic spaces

TL;DR

This paper investigates conditions under which quadrangles embedded in metasymplectic spaces are Moufang quadrangles, contributing to the classification and understanding of these geometric structures.

Contribution

It provides criteria to determine when embedded quadrangles in metasymplectic spaces are Moufang, enhancing the classification of such geometric configurations.

Findings

Characterization of Moufang quadrangles within metasymplectic spaces

Conditions for embedding quadrangles as points and hyperlines

Extension of classification results for Moufang quadrangles

Abstract

During the final steps in the classification of the Moufang quadrangles by Jacques Tits and Richard Weiss a new class of Moufang quadrangles unexpectedly turned up. Subsequently Bernhard Muhlherr and Hendrik Van Maldeghem showed that this class arises as the fixed points and hyperlines of certain involutions of a metasymplectic space (or equivalently a building of type F_4). In the same paper they also showed that other types of Moufang quadrangles can be embedded in a metasymplectic space as points and hyperlines. In this paper, we reverse the question: given a (thick) quadrangle embedded in a metasymplectic space as points and hyperlines, when is such a quadrangle a Moufang quadrangle?