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Near polygons and Fischer spaces

A. E. Brouwer, A. M. Cohen, H. A. Wilbrink, J. J. Hall

1994
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Geometriae Dedicata
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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
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... 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