One-point extensions of generalized hexagons and octagons

Abstract

In this note, we prove the uniqueness of the one-point extension $\mathcal{S}$ of a generalized hexagon of order 2 and prove the non-existence of such an extension $\mathcal{S}$ of any other finite generalized hexagon of classical order, different from the one of order 2, and of the known finite generalized octagons provided the following property holds: for any three points x, y and z of $\mathcal{S}$, the graph theoretic distance from y to z in the derived generalized hexagon ${\mathcal{S}}_x$ is the same as the distance from x to z in ${\mathcal{S}}_y$.