Generalized polygons, SCABs and GABs

E 8 root lattices. Miscellaneous problems and examples. Connections with finite group theory. References. Introduction Recently there has been a great deal of activity in the geometric and group theoretic study of "building-like geometries" One of the directions of this activity has concerned GABs ("geometries that are almost buildings") and SCABs ("chamber systems that are almost buildings"). (Other names for these are "chamber systems of type M" and "geometries of type M" in [Ti 7], and "Tits geometries of type M" or "Tits chamber systems of type M" in [AS; Tim 1,2].) The main goal of this paper is to survey these developments with special emphasis on their relationships with finite geometries. The study of SCABs is primarily based upon the work of Tits, and especially on the seminal paper [Ti 7]. While our Chapter B briefly describes his results, few proofs are given. (In fact, relatively few proofs will be provided throughout this survey.) On the other hand, our point of view will be somewhat elementary (especially when compared with [Ti 7] and Tits' paper in these Proceedings), in the sense that a great deal of space will be devoted to ways in which familiar geometric objects (e.g. hyper-ovals, difference sets or root systems) can be easily used in order to construct new or less familiar ones. Consequently, it will be clear both that the subject is still in its infancy and that there are many opportunities to enter into this area. 81 Generalized polygons are the building bricks from which SCABs and GABs are obtained. A great deal has been written about these: they are rank 2 buildings, they include projective planes, and they are very natural from a variety of geometric, group theoretic and combinatorial viewpoints. In Chapter A we will construct almost all of the known finite examples that are not projective planes, and indicate their group theoretic origins when appropriate. However, we will not discuss the large number of characterizations presently known-especially of finite generalized quadrangles; that is outside the scope (and size) of this paper. Instead, Chapter A is intended to function as the source for generalized polygons arising later. Moreover, Chapter A highlights one basic question concerning the constructions in Chapter C: what nonclassical finite generalized polygons can appear in finite SCABs? (In fact, the only such SCABs that seem to be known are similar to those in (C.6.11).) While Chapter B contains …