Flag-transitive C2.Ln geometries

AbstractIn this paper we prove that the only locally finite, thick flag-transitive Cn.L geometries with n ⩾ 3 are truncations of polar spaces. We recall that for n = 2 an example of thick flag-transitive geometry which is not a truncated polar space has been given by Ronan (1980, 1986). Moreover, we prove that no flag-transitive thick C2 . A f . An − 2 . L geometry exists with classical generalized quadrangles as lower residues of elements of type 2, except possibly when q = 3 or 4. However there are examples of flag-transitive thick C2 . A f . An − 2 . L geometries where the lower residue of a plane is isomorphic to the generalized quadrangle dual of T∗(O)