Compact 16-Dimensional Projective Planes with Large Collineation Groups. IV

Let be a topological projective plane with compact point set P of finite (covering) dimension. In the compact-open topology (of uniform convergence), the group Σ of continuous collineations of is a locally compact transformation group of P.THEOREM. If dim Σ > 40, thenis isomorphic to the Moufang plane 6 over the real octonions (and dim Σ = 78).By [3] the translation planes with dim Σ = 40 form a one-parameter family and have Lenz type V. Presumably, there are no other planes with dim Σ = 40, cp. [17].

Products of Reflections in an Affine Moufang Plane

Let be a Moufang plane. By specializing one line ω, the line at infinity, weobtain an affine Moufang plane . The group generated by the shears of is called the equiaffine group. Veblen [9, § 52] asked whether every equiaffinity is a product of two affine reflections. He gave a proof which will work in an affine Pappian plane, using the following two properties.Property 1. If an equiaffinity fixes two distinct proper points of , it fixes every point collinear with them.Property 2. Let e be an equiaffinity and P a point such that ppe2pe3 is a triangle. Then the lines PePe2 and PPe3 are parallel.Without using these properties, it will be proved that the answer to Veblen's question is “yes“ if and only if the Moufang plane is Pappian.