Vol. 10, No. 1, 2009

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From buildings to point-line geometries and back again

Ernest E. Shult

Vol. 10 (2009), No. 1, 93–119
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.

Keywords
buildings, chamber systems, point-line geometries, parapolar spaces, Lie incidence geometry
Mathematical Subject Classification 2000
Primary: 51E24
Milestones
Received: 18 September 2007
Accepted: 18 March 2008
Authors
Ernest E. Shult