A theorem on Tits geometries of type Cn

In this paper we extend a result of [2] to cover a more general situation. In [2] it was shown that a finite Cn geometry (n ≥4) in which all lines are thick and all C3 residues are either buildings or flat has to be a building. Here we observe that the finiteness assumption of [2] was unnecessary in order to achieve the major part of the result. If we drop the finiteness assumption we can still prove that such a geometry is either a quotient of a building or flat. Flat Cn geometries for n ≥4 are seen to be degenerate in a certain sense. In the finite case with thick lines such degenerate geometries are easily shown not to exist, while finite buildings of type Cn with thick lines do not admit non-trivial quotients (Brouwer and Cohen, [1]). Thus the result of [2] follows as an immediate corollary of this more general case. The result does not hold when we drop the assumption that all lines are thick. In Section 3 we produce some examples of geometries of this type. © 1987 Birkhäuser Verlag.