Near polygons and Fischer spaces

A. E. Brouwer, A. M. Cohen, H. A. Wilbrink, J. J. Hall
1994 Geometriae Dedicata  
In this paper we exploit the relations between near polygons with lines of size 3 and Fischer spaces to classify near hexagons with quads and with lines of size three. We also construct some infinite families of near polygons. N E A R P O L Y G O N S A near polygon is a connected partial linear space (X, L) such that given a point x and a line L there is a unique point on L closest to x (where distances are measured in the collinearity graph: two distinct points are adjacent when they are
more » ... ear). A near polygon of diameter n is called a near 2n-gon, and for n --3 a near hexagon. The concept of near polygon was introduced in Shult and Yanushka [20] as a tool in the study of systems of lines in a Euclidean space. A structure theory is developed in Shad and Shult [18] and Brouwer and Wilbrink [8]. Dual polar spaces were characterized by Cameron [11] as near polygons with 'classical point-quad relations'. (See also Shult [19] and Brouwer and Cohen [5].) By Yanushka's lemma ([20, Prop. 2.5]) , any quadrangle (in the collinearity graph of a near polygon) of which at least one side lies on a line with at least three points is contained in a unique geodetically closed subspace of diameter 2, necessarily a nondegenerate generalized quadrangle. Such a subspace is called a quad, and a near polygon is said to 'have quads' when any two points at distance 2 determine a quad containing them. When all lines have at least three points this is equivalent to asking that any two points at distance 2 have at least two common neighbours. In this paper we construct some infinite families of near polygons, and classify near hexagons with lines of length 3 and with quads. This paper is a compilation of the three reports Brouwer et al. [6], Brouwer and Wilbrink [7], and Brouwer [4] together with the contributions of the third author, who was referee of [6]. Our main goal is the following theorem. T H E O R E M . Let (X, L) be a near hexagon with lines of size 3 and such that any two points at distance 2 have at least two common neighbours. Then Geometriae Dedicata 49: 349-368, 1994.
doi:10.1007/bf01264034 fatcat:r2xtudd2g5hqdae6z56ags5mry