Ebook: Tits Polygons

A generalized polygon is the same thing as a spherical building of rank 2. An arbitrary thick irreducible spherical building of rank at least 3 is uniquely determined by the substructure spanned by the irreducible rank 2 residues containing a single chamber. These residues all satisfy the Moufang condition. Moufang polygons (that is, generalized polygons satisfying the Moufang condition) were classified and this classification was used to give a new proof of the classification of irreducible spherical buildings of rank greater than 2.

In the first few chapters, the study of Moufang polygons is reduced to the study of “root group sequences” and the commutator relations that define them. In the course of the classification, it is shown in each case that the root group sequence in question can be coordinatized by a suitable algebraic structure. In this way, composition algebras, quadratic spaces, pseudo-quadratic spaces, quadratic Jordan algebras of degree 3 (also known as cubic norm structures) and more exotic algebraic structures come into play. In every case, the algebraic structure that arises must be anisotropic in a suitable sense.