Projectivities and Projective Embeddings

Abstract

In this chapter, we aim to prove some of the main achievements in the theory of generalized polygons. First, we want to show what the little projective group and the groups of projectivities of some Moufang polygons (in particular, all finite classical polygons) look like; the latter generalizes a result of Knarr [1988]. Secondly, we want to classify all embeddings of a generalized quadrangle in a finite-dimensional projective space. In particular, such a quadrangle is a classical Moufang quadrangle. This is due to Dienst [1980a], [1980b], who extended a result of Buekenhout & Lefèvre-Percsy [1974] on projectively embedded finite generalized quadrangles to the infinite case. We take a somewhat different approach, avoiding the theory of semi-quadratic sets. We also mention some related results about embeddings of octagons, and about weak and lax embeddings of quadrangles and hexagons. These are due to Thas & Van Maldeghem [1996], [19**a], [19**b], [19**c], Steinbach [1996] and Steinbach & Van Maldeghem [19**a], [19**b]. The proofs where hexagons are involved use some very technical results about certain point sets in finite projective spaces, and this is beyond the scope of this book. Where finite generalized quadrangles are involved, one again uses the results of Payne & Thas [1984]. Since this area is still developing quickly, and since there seems to be a lot still to be done, we occasionally give a rough outline of a proof, sometimes a geometric proof of some parts.