On three families of extended tilde geometries

Let Γ be an extended tilde geometry of rank n > 2 such that there exists a 1-covering γ : Γ → Φ where Φ is a c . C n − 1 -geometry with orders 1 , 2 , … , 2 . Suppose that the normalizer in Aut ( Γ ) of the deck group of γ acts flag-transitively on Γ . We prove that, under these hypotheses, only three possibilities exist for the universal cover of Γ .