Innovations in Incidence Geometry

From buildings to point-line geometries and back again

Ernest Shult

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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.

Article information

Source
Innov. Incidence Geom., Volume 10 (2009), 93-119.

Dates
Received: 18 September 2007
Accepted: 18 March 2008
First available in Project Euclid: 28 February 2019

Permanent link to this document
https://projecteuclid.org/euclid.iig/1551323105

Digital Object Identifier
doi:10.2140/iig.2009.10.93

Mathematical Reviews number (MathSciNet)
MR2665196

Zentralblatt MATH identifier
1262.51011

Subjects
Primary: 51E24: Buildings and the geometry of diagrams

Keywords
buildings chamber systems point-line geometries parapolar spaces Lie incidence geometry

Citation

Shult, Ernest. From buildings to point-line geometries and back again. Innov. Incidence Geom. 10 (2009), 93--119. doi:10.2140/iig.2009.10.93. https://projecteuclid.org/euclid.iig/1551323105


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