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

We consider extensions of the F4(2)-building with the diagram such that the residue of every element of the rightmost type is a one-point extension of the corresponding C3(2)-residue in the building. Four flag-transitive such geometries are known with the automorphism groups isomorphic to 2E6(2):2, 3· 2E6 (2):2, E6(2):2 and 226:F4(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 F4(2)-building with the local one-point extension property. © 2003 Elsevier Ltd. All rights reserved.