Point-line characterizations of Lie geometries

There are two basic theorems. Let G be a strong parapolar space with these three properties: (1) For each point x and symplecton S, x is collinear to some point of S. (2) The set of points at distance at most two from a point forms a geometric hyperplane. (3) If every symplecton has rank at least three, every maximal singular subspace has finite projective rank. Then G is either D6; 6;A5; 3 or E7; 1, a classical dual polar space of rank three, or a product geometry L P where P is a polar space and L is a line. The second theorem concerns parapolar spaces S of symplectic rank at least three whose point-collinearity diameter is at least three such that for every point-symplecton pair, ðx;SÞ, x? VS is never just a point. With a mild local condition, one can show that such a geometry has point-diameter three and has a simply connected point-collinearity graph. If singular spaces have finite projective rank, one can show that S is E6; 4, E7; 7, E8; 1, a metasymplectic space, or a polar Grassmannian of type Bn; 2, Dn; 2, nd 4. All of these geometries are truncations of buildings. The last case can be modified so that the assumption that singular spaces have finite projective rank can be discarded.