Near Polygons (eBook)

Title page 4
Copyright page 5
Table of contents 6
Preface 9
Chapter 1 Introduction 12
1.1 Definition of near polygon 12
1.2 Genesis 13
1.3 Near polygons with an order 14
1.4 Parallel lines 14
1.5 Substructures 15
1.6 Product near polygons 18
1.7 Existence of quads 23
1.8 The point-quad and line-quad relations 25
1.9 Some classes of near polygons 28
1.9.1 Thin and slim near polygons 28
1.9.2 Dense near polygons 28
1.9.3 Regular near polygons 28
1.9.4 Generalized polygons 29
1.9.5 Dual polar spaces 30
1.10 Generalized quadrangles of order (2, t) 32
1.10.1 Examples 32
1.10.2 Possible orders 33
1.10.3 Generalized quadrangles of order (2, 1) 34
1.10.4 Generalized quadrangles of order 2 34
1.10.5 Generalized quadrangles of order (2, 4) 34
1.10.6 Ovoids in generalized quadrangles of order (2, t) 35
Chapter 2 Dense near polygons 37
2.1 Main results 37
2.2 The existence of convex subpolygons 38
2.3 Proof of Theorem 2.6 47
2.4 Upper bound for the diameter of Gd(x) 48
2.5 Upper bounds for t + 1 in the case of slim densenear polygons 50
2.6 Slim dense near polygons with a big convexsubpolygon 51
2.6.1 Statement of the result 51
2.6.2 Proof of Theorem 2.40 52
Chapter 3 Regular near polygons 56
3.1 Introduction 56
3.2 Some restrictions on the parameters 56
3.3 Eigenvalues of the collinearity matrix 59
Calculation of the multiplicities 61
Example 1: The case of regular near hexagons 62
Example 2: The case of regular near octagons 63
3.4 Upper bounds for t 63
3.5 Slim dense regular near hexagons 64
3.6 Slim dense regular near octagons 65
Chapter 4 Glued near polygons 66
4.1 Characterizations of product near polygons 66
4.2 Admissible d-spreads 71
4.3 Construction and elementary properties of glued near polygons 72
4.4 Basic characterization result for glued nearpolygons 77
4.5 Other characterizations of glued near polygons 80
4.5.1 Characterization of finite glued near hexagons 80
4.5.2 Characterization of general glued near polygons 82
4.5.3 Proof of Theorem 4.28 83
4.6 Subpolygons 84
4.7 Glued near polygons of type d . {0, 1} 86
4.7.1 Glued near polygons of type 0 86
4.7.2 Spreads of symmetry 86
4.7.3 Glued near polygons of type 1 89
4.7.4 Admissible triples 90
4.7.5 The sets .0(A) and .1(A) for a dense near polygon A 93
4.7.6 Extensions of spreads and automorphisms 94
4.7.7 Compatible spreads of symmetry 97
4.7.8 Compatible spreads of symmetry in product and glued nearpolygons 98
4.7.9 Near polygons of type (F1 * F2) . F3 99
Chapter 5 Valuations 102
5.1 Nice near polygons 102
5.2 Valuations of nice near polygons 103
5.3 Characterizations of classical and ovoidal valuations 105
5.4 The partial linear space Gf 107
5.5 A property of valuations 107
5.6 Some classes of valuations 108
5.6.1 Hybrid valuations 108
5.6.2 Product valuations 109
5.6.3 Diagonal valuations 110
5.6.4 Semi-diagonal valuations 110
5.6.5 Distance-j-ovoidal valuations 114
5.6.6 Extended valuations 115
5.6.7 SDPS-valuations 117
5.7 Valuations of dense near hexagons 118
5.8 Proof of Theorem 5.29 120
5.9 Proof of Theorem 5.30 124
5.10 Proof of Theorem 5.31 124
5.11 Proof of Theorem 5.32 125
Chapter 6 The known slim dense nearpolygons 130
6.1 The classical near polygons DQ(2n, 2) and DH(2n 1, 4) 130
6.2 The class Hn 136
6.3 The class Gn 138
6.3.1 Definition of Gn 138
6.3.2 Subpolygons of Gn 140
6.3.3 Lines and quads in Gn 142
6.3.4 Some properties of Gn 143
6.3.5 Determination of Aut(Gn), n = 3 144
6.3.6 Spreads in Gn 146
6.3.7 Valuations of G3 148
6.4 The class In 149
6.5 The near hexagon E1 152
6.5.1 Description of E1 in terms of the extended ternary Golaycode 153
6.5.2 Description of E1 in terms of the Coxeter cap 154
6.5.3 The valuations of E1 158
6.6 The near hexagon E2 161
6.6.1 Definition and properties of E2 161
6.6.2 The ovoids of E2 164
6.7 The near hexagon E3 168
6.8 The known slim dense near polygons 170
6.9 The elements of C3 and C4 171
6.9.1 Spreads of symmetry of Q(5, 2) 171
6.9.2 Another model for Q(5, 2) 171
6.9.3 The near polygons DH(2n-1, 4).Q(5, 2), Gn .Q(5, 2) and E1 . Q(5, 2) 173
6.9.4 Spreads of symmetry of Q(5, 2) . Q(5, 2) 174
6.9.5 Near polygons of type (Q(5, 2) . Q(5, 2)) . Q(5, 2) 174
Chapter 7 Slim dense near hexagons 176
7.1 Introduction 176
7.2 Elementary properties of slim dense near hexagons 177
7.3 Case I: S is a regular near hexagon 179
7.4 Case II: S contains grid-quads and W(2)-quads butno Q(5, 2)-quads 180
7.4.1 There exists a big W(2)-quad 180
7.4.2 No W(2)-quad is big 181
7.5 Case III: S contains grid-quads and Q(5, 2)-quads but no W(2)-quads 184
7.6 Case IV: S contains W(2)-quads and Q(5, 2)-quad sbut no grid-quads 185
7.7 Case V: S contains grid-quads, W(2)-quads and Q(5, 2)-quads 186
7.8 Appendix 190
Chapter 8 Slim dense near polygons with a nice chain of convex subpolygons 195
8.1 Overview 195
8.2 Proof of Theorem 8.1 197
8.3 Proof of Theorem 8.2 198
8.4 Proof of Theorem 8.3 201
8.5 Proof of Theorem 8.4 205
8.6 Proof of Theorem 8.5 205
8.7 Proof of Theorem 8.6 212
8.8 Proof of Theorem 8.7 212
8.9 Proof of Theorem 8.8 213
8.10 Proof of Theorem 8.9 216
Chapter 9 Slim dense near octagons 218
9.1 Some properties of slim dense near octagons 218
9.2 Existence of big hexes 219
9.3 Classification of the near octagons 226
Chapter 10 Nondense slim near hexagons 232
10.1 A few lemmas 232
10.2 Slim near hexagons with special points 233
10.2.1 Special points 233
10.2.2 Slim near hexagons of type (III) 234
10.2.3 Slim near hexagons of type (II) 234
10.2.4 Slim near hexagons of type (I) 235
10.3 Slim near hexagons without special points 235
10.3.1 Examples 235
10.3.2 Upper bounds for the number of lines through a point 237
10.4 Proof of Theorem 10.8 237
10.5 Proof of Theorem 10.9 240
10.6 Proof of Theorem 10.10 242
10.6.1 Upper bound for |G3(x*)| 242
10.6.2 Some classes of paths in G3(x*) 243
10.6.3 Some inequalities involving the values N(a, l) and Nl 246
10.6.4 The proof of Theorem 10.10 249
10.7 Slim near hexagons with an order 250
Appendix A Dense near polygons of order 254
A.1 Generalized quadrangles of order (3, t) 254
A.2 Dense near hexagons of order (3, t) 255
A.3 Dense near octagons of order (3, t) 256
A.4 Some properties of dense near 2d-gons of order (3, t) 257
A.5 Dense near polygons of order (3, t) with a nice chain of convex subpolygons 258
Bibliography 259
Index 266

"Preface (p. ix-x)

In this book, we intend to give an extensive treatment of the basic theory of general near polygons. The subject of near polygons has been around for about 25 years now. Excellent handbooks have appeared on certain important subclasses of near polygons like generalized quadrangles ([82]) and generalized polygons ([100]), but no book has ever occurred dealing with the topic of general near polygons. Although generalized polygons and especially generalized quadrangles are indispensable to the study of near polygons, we do not aim at giving a profound study of these incidence structures.

In fact, this book can be seen as complementary to the two above-mentioned books. Although generalized quadrangles and generalized polygons were intensively studied since they were introduced by Tits in his celebrated paper on triality ([96]), the terminology near polygon ?rst occurred in a paper in 1980. In [91], Shult and Yanushka showed the connection between the so-called tetrahedrally closed line-systems in Euclidean spaces and a class of point-line geometries which they called near polygons. In [91], also some very fundamental results regarding the geometric structure of near polygons were obtained, like the existence of quads, a result which was later generalized by Brouwer and Wilbrink [16] who showed that any dense near polygon has convex subpolygons of any feasible diameter.

The paper [16] gives for the ?rst time a profound study of dense near polygons. Other important papers on near polygons from the 1980s and the beginning of the 1990s deal with dual polar spaces, the classi?cation of regular near polygons in terms of their parameters and the classi?cation of the slim dense near hexagons. The subject of near polygons has regained interest in the last years. Important new contributions to the theory were the theory of glued near polygons, the theory of valuations and important breakthrough results regarding the classi?cation of dense near polygons with three and four points on every line.

These new contributions will be discussed extensively in this book. This book essentially consists of two main parts. In the ?rst part of the book, which consists of the ?rst ?ve chapters, we develop the basic theory of near polygons. In Chapters 2, 3 and 4, we study three classes of near polygons: the dense, the regular and the glued near polygons.

Our treatment of the dense and glued near polygons is rather complete. The treatment of the regular near polygons is concise and results are not always accompanied with proofs. More detailed information on regular near polygons can be found in the book Distanceregular graphs [13] by Brouwer, Cohen and Neumaier. In that book regular near polygons are considered as one of the main classes of distance-regular graphs. In Chapter 5, we discuss the notion of valuation of a near polygon which is a very important tool for classifying near polygons."