Polar Spaces

Polar spaces are the natural geometries for the classical groups. Due to the stunning simplicity of an axiom system found by Buekenhout and Shult, they play a central role in incidence geometry and also appear as combinatorial objects in many disciplines such as discrete mathematics, graph theory, finite geometry and coding theory. They can also be viewed as a class of spherical Tits buildings and as such were classified by Jacques Tits – using pseudo-quadratic forms and octonion algebras. Polar spaces bridge the areas of group theory, algebra, combinatorics and incidence geometry.

These lecture notes arose from a master’s course in Ghent, Belgium, taught annually between 2010 and 2024. Besides many basic and general geometric properties of polar spaces, it contains a complete algebraic description of all polar spaces of rank at least 3, linking them with polarities in projective spaces. The discussion of the related classical groups is limited to the study of axial and central elations. The classification of top-thin polar spaces is included in detail. Triality in top-thin polar spaces of rank 4 is explained both geometrically and algebraically. The last chapter introduces parapolar spaces, which are geometric structures using polar spaces as building blocks. This opens the door for exploring geometries related to the exceptional groups. An appendix explains composition algebras, which are used to describe the so-called non-embeddable polar spaces and triality.