Symplectic geometries, transvection groups, and modules

We show that any connected partial linear space in which there is a line with at least four points and that has the property that any pair of intersecting lines is contained in a subspace isomorphic to a symplectic plane is isomorphic to the geometry of hyperbolic lines in some symplectic geometry. As a corollary to this result we obtain a characterization of the subgroups of the symplectic groups that are generated by transvection subgroups. Also a characterization of the natural modules for these groups is obtained.