The structure of near polygons with quads

Abstract

We develop a structure theory for near polygons with quads. Main results are the existence of sub 2j-gons for 2 ~<j ~< d and the nonexistence of regular sporadic 2d-gons for d 1> 4 with s > 1 and t 2 > 1 and t 3 5 ~ t2(t 2 + 1).

The Hyperplanes of the U 4(3) Near Hexagon

“…The incidence structure E 3 is a dense near hexagon with three points on each line. The above description of the near hexagon has been taken from Brouwer & Wilbrink [5] and Brouwer et al [3]. Other descriptions of this near hexagon can be found in Aschbacher [ (3) and contains U 4 (3) as a subgroup of index 4.…”

“…Now, if Q is a quad of DH(5, 4), then Q ∩ H contains the quad Q ∩ G 3 of G 3 . It follows that H is a locally subquadrangular hyperplane of DH (5,4). By Pasini and Shpectorov [25], there exists up to isomorphism a unique locally subquadrangular hyperplane in DH(5, 4).…”

“…By Theorem 4 of Brouwer and Wilbrink [5], every two points of a dense near polygon at distance δ from each other are contained in a unique convex sub near 2δ-gon. These convex sub near 2δ-gons are called quads if δ = 2.…”

Dense Near Octagons with Four Points on Each Line, III

“…A near polygon is called dense if every line is incident with at least three points and if every two points at distance 2 have at least two common neighbours. By Lemma 19 of Brouwer and Wilbrink [2], every point of a dense near polygon S is incident with the same number of lines. We denote this number by t S + 1.…”

“…We denote this number by t S + 1. By Theorem 4 of [2], every two points of a dense near polygon at distance δ from each other are contained in a unique convex sub-2δ-gon. These convex subpolygons are called quads if δ = 2 and hexes if δ = 3.…”