Tits Endomorphisms and Buildings of Type $F_4$

TL;DR

This paper explores the relationship between fixed point buildings of certain automorphisms in Moufang quadrangles and octagons of type F4, establishing their equivalence and properties of associated groups.

Contribution

It demonstrates that two classes of Moufang sets are identical and can be constructed via fixed point buildings of groups acting on type F4 buildings, revealing their structural properties.

Findings

The two classes of Moufang sets are the same.

Each Moufang set can be realized as a fixed point building of a group of order 4.

The group generated by root groups in these Moufang sets is simple.

Abstract

The fixed point building of a polarity of a Moufang quadrangle of type $F_4$ is a Moufang set, as is the fixed point building of a semi-linear automorphism of order $2$ of a Moufang octagon that stabilizes at least two panels of one type but none of the other. We show that these two classes of Moufang sets are, in fact, the same, that each member of this class can be constructed as the fixed point building of a group of order $4$ acting on a building of type $F_4$ and that the group generated by all the root groups of any one of these Moufang sets is simple.