From buildings to point-line geometries and back again

Abstract

A chamber system is a particular type of edge-labeled graph. We discu ss when such chamber systems are or are not associated with a geometry, and when they are buildings. Buildings can give rise to point-line geometries under constraints imposed by how a line should behave with respect to the point-shadows of the other geometric objects (Pasini (P)). A recent theorem o f Kasikova (K) shows that Pasini's choice is the right one. So, in a general w ay, one has a procedure for getting point-line geometries from buildings. In the other direction, we describe how a class of point-line geometries with elementary local axioms (certain parapolar spaces) successfully char- acterize many buildings and their homomorphic images. A recent result of K. Thas (KT) makes this theory free of Tits' the classification of polar spaces o f rank three (T1). One notes that parapolar spaces alone will not cover all o f the point-line geometries arising from buildings by the Pasini-Kasikova construction, so the door is wide open for further research with points and lines.