Ree geometries

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

Some characterizations of Moufang generalized quadrangles

We prove the longstanding conjecture that the 3-Moufang condition for generalized quadrangles is equivalent to the Moufang condition. We mention some other characterizations of Moufang quadrangles that follow from this result. We also provide a short proof of Tent's recent result that every half Moufang quadrangle is necessarily a Moufang quadrangle