Near Polygons (Frontiers in Mathematics)

Cover......Page 1
Near\rPolygons......Page 4
ISBN 13: 9783764375522......Page 5
Table of contents......Page 6
Preface......Page 10
1.1 Definition of near polygon......Page 14
1.2 Genesis......Page 15
1.4 Parallel lines......Page 16
1.5 Substructures......Page 17
1.6 Product near polygons......Page 20
1.7 Existence of quads......Page 25
1.8 The point-quad and line-quad relations......Page 27
1.9.3 Regular near polygons......Page 30
1.9.4 Generalized polygons......Page 31
1.9.5 Dual polar spaces......Page 32
1.10.1 Examples......Page 34
1.10.2 Possible orders......Page 35
1.10.5 Generalized quadrangles of order (2, 4)......Page 36
1.10.6 Ovoids in generalized quadrangles of order (2, t)......Page 37
2.1 Main results......Page 40
2.2 The existence of convex subpolygons......Page 41
2.3 Proof of Theorem 2.6......Page 50
2.4 Upper bound for the diameter of Γd(x)......Page 51
2.5 Upper bounds for t + 1 in the case of slim densenear polygons......Page 53
2.6.1 Statement of the result......Page 54
2.6.2 Proof of Theorem 2.40......Page 55
3.2 Some restrictions on the parameters......Page 60
3.3 Eigenvalues of the collinearity matrix......Page 63
Calculation of the multiplicities......Page 65
Example 1: The case of regular near hexagons......Page 66
3.4 Upper bounds for t......Page 67
3.5 Slim dense regular near hexagons......Page 68
3.6 Slim dense regular near octagons......Page 69
4.1 Characterizations of product near polygons......Page 70
4.2 Admissible δ-spreads......Page 75
4.3 Construction and elementary properties of glued near polygons\r......Page 76
4.4 Basic characterization result for glued nearpolygons......Page 81
4.5.1 Characterization of finite glued near hexagons......Page 84
4.5.2 Characterization of general glued near polygons......Page 86
4.5.3 Proof of Theorem 4.28......Page 87
4.6 Subpolygons......Page 88
4.7.2 Spreads of symmetry......Page 90
4.7.3 Glued near polygons of type 1......Page 93
4.7.4 Admissible triples......Page 94
4.7.5 The sets Υ0(A) and Υ1(A) for a dense near polygon A......Page 97
4.7.6 Extensions of spreads and automorphisms......Page 98
4.7.7 Compatible spreads of symmetry......Page 101
4.7.8 Compatible spreads of symmetry in product and glued nearpolygons......Page 102
4.7.9 Near polygons of type (F1 ∗ F2) ◦ F3......Page 103
5.1 Nice near polygons......Page 106
5.2 Valuations of nice near polygons......Page 107
5.3 Characterizations of classical and ovoidal valuations......Page 109
5.5 A property of valuations......Page 111
5.6.1 Hybrid valuations......Page 112
5.6.2 Product valuations......Page 113
5.6.4 Semi-diagonal valuations......Page 114
5.6.5 Distance-j-ovoidal valuations......Page 118
5.6.6 Extended valuations......Page 119
5.6.7 SDPS-valuations......Page 121
5.7 Valuations of dense near hexagons......Page 122
5.8 Proof of Theorem 5.29......Page 124
5.10 Proof of Theorem 5.31......Page 128
5.11 Proof of Theorem 5.32......Page 129
6.1 The classical near polygons DQ(2n, 2) and DH(2n − 1, 4)......Page 134
6.2 The class Hn......Page 140
6.3.1 Definition of Gn......Page 142
6.3.2 Subpolygons of Gn......Page 144
6.3.3 Lines and quads in Gn......Page 146
6.3.4 Some properties of Gn......Page 147
6.3.5 Determination of Aut(Gn), n ≥ 3......Page 148
6.3.6 Spreads in Gn......Page 150
6.3.7 Valuations of G3......Page 152
6.4 The class In......Page 153
6.5 The near hexagon E1......Page 156
6.5.1 Description of E1 in terms of the extended ternary Golaycode......Page 157
6.5.2 Description of E1 in terms of the Coxeter cap......Page 158
6.5.3 The valuations of E1......Page 162
6.6.1 Definition and properties of E2......Page 165
6.6.2 The ovoids of E2......Page 168
6.7 The near hexagon E3......Page 172
6.8 The known slim dense near polygons......Page 174
6.9.2 Another model for Q(5, 2)......Page 175
6.9.3 The near polygons DH(2n−1, 4)⊗Q(5, 2), Gn ⊗Q(5, 2) and E1 ⊗ Q(5, 2)......Page 177
6.9.5 Near polygons of type (Q(5, 2) ⊗ Q(5, 2)) ⊗ Q(5, 2)......Page 178
7.1 Introduction......Page 180
7.2 Elementary properties of slim dense near hexagons......Page 181
7.3 Case I: S is a regular near hexagon......Page 183
7.4.1 There exists a big W(2)-quad......Page 184
7.4.2 No W(2)-quad is big......Page 185
7.5 Case III: S contains grid-quads and Q(5, 2)-quads but no W(2)-quads......Page 188
7.6 Case IV: S contains W(2)-quads and Q(5, 2)-quad sbut no grid-quads......Page 189
7.7 Case V: S contains grid-quads, W(2)-quads and Q(5, 2)-quads......Page 190
7.8 Appendix......Page 194
8.1 Overview......Page 200
8.2 Proof of Theorem 8.1......Page 202
8.3 Proof of Theorem 8.2......Page 203
8.4 Proof of Theorem 8.3......Page 206
8.6 Proof of Theorem 8.5......Page 210
8.8 Proof of Theorem 8.7......Page 217
8.9 Proof of Theorem 8.8......Page 218
8.10 Proof of Theorem 8.9......Page 221
9.1 Some properties of slim dense near octagons......Page 224
9.2 Existence of big hexes......Page 225
9.3 Classification of the near octagons......Page 232
10.1 A few lemmas......Page 238
10.2.1 Special points......Page 239
10.2.3 Slim near hexagons of type (II)......Page 240
10.3.1 Examples......Page 241
10.4 Proof of Theorem 10.8......Page 243
10.5 Proof of Theorem 10.9......Page 246
10.6.1 Upper bound for |Γ3(x∗)|......Page 248
10.6.2 Some classes of paths in Γ3(x∗)......Page 249
10.6.3 Some inequalities involving the values N(a, l) and Nl......Page 252
10.6.4 The proof of Theorem 10.10......Page 255
10.7 Slim near hexagons with an order......Page 256
A.1 Generalized quadrangles of order (3, t)......Page 260
A.2 Dense near hexagons of order (3, t)......Page 261
A.3 Dense near octagons of order (3, t)......Page 262
A.4 Some properties of dense near 2d-gons of order (3, t)......Page 263
A.5 Dense near polygons of order (3, t) with a nice chain of convex subpolygons......Page 264
Bibliography......Page 266
Index......Page 274