Moufang sets related to polarities in exceptional Moufang quadrangles of type F_4

TL;DR

This paper constructs a new geometry from a Moufang set related to a polarity in an exceptional Moufang quadrangle of type F_4, linking automorphisms of the geometry and the quadrangle to deepen understanding of their symmetries.

Contribution

It introduces a rank three geometry associated with such Moufang sets, enabling geometric analysis of their automorphisms and providing a key result on automorphisms stabilizing polarities.

Findings

Automorphism group of the new geometry matches that of the Moufang set

Every automorphism stabilizing the polarity centralizes it

Completes the understanding of automorphisms in all polarities of Moufang n-gons

Abstract

Departing from a Moufang set related to a polarity in an exceptional Moufang quadrangle of type F_4, we construct a rank three geometry. The main property of this new geometry is that its automorphism group is identical to the one of the underlying Moufang set, providing a tool to study this Moufang set in a geometrical way. As a corollary we obtain that every automorphism of an exceptional Moufang quadrangle of type F_4 stabilizing the absolute points of a polarity, also centralizes the polarity. This handles the final case of a similar result for all polarities of Moufang n-gons.