Sharp homogeneity in some generalized polygons

Abstract.

We show that, if a collineation group G of a generalized (2n + 1)-gon $\Gamma$ has the property that every symmetry of any apartment extends uniquely to a collineation in G, then $\Gamma$ is the unique projective plane with 3 points per line (the Fano plane) and G is its full collineation group. A similar result holds if one substitutes “apartment” with “path of length 2k ≤ 2n + 2”.