A diameter bound for extensions of the F4(2)-building

Abstract

We consider extensions of the F 4(2)-building with the diagram such that the residue of every element of the rightmost type is a one-point extension of the corresponding C 3(2)-residue in the building. Four flag-transitive such geometries are known with the automorphism groups isomorphic to 2 E 6(2):2, 3· 2E 6(2):2 , E 6(2):2 and 2 26: F 4(2). The first example is a folding of the second one. We show that the last three examples are simply connected. This brings us close to the complete classification of the flag-transitive c-extensions of the F 4(2)-building with the local one-point extension property.