Near Polygons (Frontiers in Mathematics)

Title page......Page 4
Copyright page......Page 5
Table of contents......Page 6
Preface......Page 9
1.1 Definition of near polygon......Page 12
1.2 Genesis......Page 13
1.4 Parallel lines......Page 14
1.5 Substructures......Page 15
1.6 Product near polygons......Page 18
1.7 Existence of quads......Page 23
1.8 The point-quad and line-quad relations......Page 25
1.9.3 Regular near polygons......Page 28
1.9.4 Generalized polygons......Page 29
1.9.5 Dual polar spaces......Page 30
1.10.1 Examples......Page 32
1.10.2 Possible orders......Page 33
1.10.5 Generalized quadrangles of order (2, 4)......Page 34
1.10.6 Ovoids in generalized quadrangles of order (2, t)......Page 35
2.1 Main results......Page 37
2.2 The existence of convex subpolygons......Page 38
2.3 Proof of Theorem 2.6......Page 47
2.4 Upper bound for the diameter of Γd(x)......Page 48
2.5 Upper bounds for t + 1 in the case of slim densenear polygons......Page 50
2.6.1 Statement of the result......Page 51
2.6.2 Proof of Theorem 2.40......Page 52
3.2 Some restrictions on the parameters......Page 56
3.3 Eigenvalues of the collinearity matrix......Page 59
Calculation of the multiplicities......Page 61
Example 1: The case of regular near hexagons......Page 62
3.4 Upper bounds for t......Page 63
3.5 Slim dense regular near hexagons......Page 64
3.6 Slim dense regular near octagons......Page 65
4.1 Characterizations of product near polygons......Page 66
4.2 Admissible δ-spreads......Page 71
4.3 Construction and elementary properties of glued near polygons
......Page 72
4.4 Basic characterization result for glued nearpolygons......Page 77
4.5.1 Characterization of finite glued near hexagons......Page 80
4.5.2 Characterization of general glued near polygons......Page 82
4.5.3 Proof of Theorem 4.28......Page 83
4.6 Subpolygons......Page 84
4.7.2 Spreads of symmetry......Page 86
4.7.3 Glued near polygons of type 1......Page 89
4.7.4 Admissible triples......Page 90
4.7.5 The sets Υ0(A) and Υ1(A) for a dense near polygon A......Page 93
4.7.6 Extensions of spreads and automorphisms......Page 94
4.7.7 Compatible spreads of symmetry......Page 97
4.7.8 Compatible spreads of symmetry in product and glued nearpolygons......Page 98
4.7.9 Near polygons of type (F1 ∗ F2) ◦ F3......Page 99
5.1 Nice near polygons......Page 102
5.2 Valuations of nice near polygons......Page 103
5.3 Characterizations of classical and ovoidal valuations......Page 105
5.5 A property of valuations......Page 107
5.6.1 Hybrid valuations......Page 108
5.6.2 Product valuations......Page 109
5.6.4 Semi-diagonal valuations......Page 110
5.6.5 Distance-j-ovoidal valuations......Page 114
5.6.6 Extended valuations......Page 115
5.6.7 SDPS-valuations......Page 117
5.7 Valuations of dense near hexagons......Page 118
5.8 Proof of Theorem 5.29......Page 120
5.10 Proof of Theorem 5.31......Page 124
5.11 Proof of Theorem 5.32......Page 125
6.1 The classical near polygons DQ(2n, 2) and DH(2n − 1, 4)......Page 130
6.2 The class Hn......Page 136
6.3.1 Definition of Gn......Page 138
6.3.2 Subpolygons of Gn......Page 140
6.3.3 Lines and quads in Gn......Page 142
6.3.4 Some properties of Gn......Page 143
6.3.5 Determination of Aut(Gn), n ≥ 3......Page 144
6.3.6 Spreads in Gn......Page 146
6.3.7 Valuations of G3......Page 148
6.4 The class In......Page 149
6.5 The near hexagon E1......Page 152
6.5.1 Description of E1 in terms of the extended ternary Golaycode......Page 153
6.5.2 Description of E1 in terms of the Coxeter cap......Page 154
6.5.3 The valuations of E1......Page 158
6.6.1 Definition and properties of E2......Page 161
6.6.2 The ovoids of E2......Page 164
6.7 The near hexagon E3......Page 168
6.8 The known slim dense near polygons......Page 170
6.9.2 Another model for Q(5, 2)......Page 171
6.9.3 The near polygons DH(2n−1, 4)⊗Q(5, 2), Gn ⊗Q(5, 2) and E1 ⊗ Q(5, 2)......Page 173
6.9.5 Near polygons of type (Q(5, 2) ⊗ Q(5, 2)) ⊗ Q(5, 2)......Page 174
7.1 Introduction......Page 176
7.2 Elementary properties of slim dense near hexagons......Page 177
7.3 Case I: S is a regular near hexagon......Page 179
7.4.1 There exists a big W(2)-quad......Page 180
7.4.2 No W(2)-quad is big......Page 181
7.5 Case III: S contains grid-quads and Q(5, 2)-quads but no W(2)-quads......Page 184
7.6 Case IV: S contains W(2)-quads and Q(5, 2)-quad sbut no grid-quads......Page 185
7.7 Case V: S contains grid-quads, W(2)-quads and Q(5, 2)-quads......Page 186
7.8 Appendix......Page 190
8.1 Overview......Page 195
8.2 Proof of Theorem 8.1......Page 197
8.3 Proof of Theorem 8.2......Page 198
8.4 Proof of Theorem 8.3......Page 201
8.6 Proof of Theorem 8.5......Page 205
8.8 Proof of Theorem 8.7......Page 212
8.9 Proof of Theorem 8.8......Page 213
8.10 Proof of Theorem 8.9......Page 216
9.1 Some properties of slim dense near octagons......Page 218
9.2 Existence of big hexes......Page 219
9.3 Classification of the near octagons......Page 226
10.1 A few lemmas......Page 232
10.2.1 Special points......Page 233
10.2.3 Slim near hexagons of type (II)......Page 234
10.3.1 Examples......Page 235
10.4 Proof of Theorem 10.8......Page 237
10.5 Proof of Theorem 10.9......Page 240
10.6.1 Upper bound for |Γ3(x∗)|......Page 242
10.6.2 Some classes of paths in Γ3(x∗)......Page 243
10.6.3 Some inequalities involving the values N(a, l) and Nl......Page 246
10.6.4 The proof of Theorem 10.10......Page 249
10.7 Slim near hexagons with an order......Page 250
A.1 Generalized quadrangles of order (3, t)......Page 254
A.2 Dense near hexagons of order (3, t)......Page 255
A.3 Dense near octagons of order (3, t)......Page 256
A.4 Some properties of dense near 2d-gons of order (3, t)......Page 257
A.5 Dense near polygons of order (3, t) with a nice chain of convex subpolygons......Page 258
Bibliography......Page 259
Index......Page 266