# 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.

## Key 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.