Near hexagons with four points on every line
by
Bart De Bruyn
Ghent University, Ghent, Belgium

A near polygon is a partial linear space with the property that for every point x and every line L there exists a unique point on L nearest to x (nearest with respect to the distance in the collinearity graph \Gamma of the partial linear space). If d is the diameter of \Gamma, then the near polygon is called a near 2d-gon. A near 0-gon consists of one point, a near 2-gon is a line with a number of points on it, and the class of the near quadrangles coincides with the class of the generalized quadrangles. Near polygons were introduced by Shult and Yanushka while studying tetrahedrally closed system of lines in Euclidean spaces. If a near polygon satisfies some mild conditions then the existence of nice substructures can be proved. Examples are quads which are nondegenerate generalized quadrangles. Brouwer et al. determined all near hexagons with three points on every line and with a quad through every two points at distance 2. This talk treats the case where all lines have four points. We give the known examples and summarize the open cases of the incomplete classification.

Date received: March 6, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cagg-16.