Ree geometries

TL;DR

This paper introduces a new geometric framework for Ree groups over arbitrary fields, demonstrating that their automorphism groups are fully captured by associated geometries, with applications to Moufang hexagons.

Contribution

It constructs a rank 3 geometry for Ree groups over any field and proves it characterizes the automorphism group, extending understanding of Ree group symmetries.

Findings

Ree geometries are fully determined by their automorphism groups.

A polarity in Moufang hexagons is uniquely determined by absolute points or lines.

The geometries apply to Ree groups over not necessarily perfect fields.

Abstract

We introduce a rank 3 geometry for any Ree group over a not necessarily perfect field and show that its full collineation group is the automorphism group of the corresponding Ree group. A similar result holds for two rank 2 geometries obtained as a truncation of this rank 3 geometry. As an application, we show that a polarity in any Moufang generalized hexagon is unambiguously determined by its set of absolute points, or equivalently, its set of absolute lines.