Connectedness of Opposite-flag Geometries in Moufang Polygons

A generalized n-gon, n 2, is a rank 2 geometry whose incidence graph has diameter n and girth 2n, and each vertex has valency 3. If the latter condition is not satisfied , then we have a weak generalized n-gon. In this paper, we will always consider generalized n-gons with n 3 (generalized 2-gons are trivial geometries). They are the irreducible spherical buildings of rank 2. A generalized polygon is a generalized n-gon, for some n 2. We will view generalized polygons as geometries of rank 2 whose elements are points and lines. The dual is obtained by interchanging these names. A flag is an incident point-line pair and hence a chamber in the corresponding spherical rank 2 building. Generalized polygons were introduced by Tits [10] and are the basic rank 2 incidence geometries.