From buildings to point-line geometries and back again

Abstract

A chamber system is a particular type of edge-labeled graph. We discuss 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 [24]). A recent theorem of Kasikova [21] shows that Pasini's choice is the right one. So, in a general way, 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 characterize many buildings and their homomorphic images. A recent result of K. Thas [32] makes this theory free of Tits' the classi cation of polar spaces of rank three [35]. One notes that parapolar spaces alone will not cover all of 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.