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). A recent theorem of Kasikova 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 makes this theory free of Tits’ classification of polar spaces of rank three. 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.