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Translation Generalized Quadrangles

By: Joseph A Thas, Koen Thas, Hendrik Van Maldeghem

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Hardcover | 28 October 2006

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376 Pages

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Translation generalized quadrangles play a key role in the theory of generalized quadrangles, comparable to the role of translation planes in the theory of projective and affine planes. The notion of translation generalized quadrangle is a local analogue of the more global "Moufang Condition", a topic of great interest, also due to the classification of all Moufang polygons. Attention is thus paid to recent results in that direction, but also many of the most important results in the general theory of generalized quadrangles that appeared since 1984 are treated.Translation Generalized Quadrangles is essentially self-contained, as the reader is only expected to be familiar with some basic facts on finite generalized quadrangles. Proofs that are either too long or too technical are left out, or just sketched. The three standard works on generalized quadrangles are (co-)authored by the writers of this book: "Finite Generalized Quadrangles" (1984) by S E Payne and J A Thas, "Generalized Polygons" (1998) by H Van Maldeghem, and "Symmetry in Finite Generalized Quadrangles" (2004) by K Thas.

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"... the book is easily accessible by a large audience."Mathematical Reviews

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Non-Fiction Mathematics Calculus & Mathematical Analysis Combinatorics & Graph Theory Geometry Analytic Geometry Number Theory Booktopia Publisher Services World Scientific Publishing

Preface	p. xiii
Foreword	p. xv
Generalized Quadrangles	p. 1
Finite Generalized Quadrangles	p. 1
Automorphisms	p. 4
Grids and Dual Grids	p. 4
The Classical Generalized Quadrangles	p. 4
The Generalized Quadrangles T[subscript 2](O) and T[subscript 3](O) of Tits	p. 5
The Generalized Quadrangles T*[subscript 2](O)	p. 7
Orders of the Known Generalized Quadrangles	p. 8
Generalized Quadrangles with Small Parameters	p. 8
Regularity, Antiregularity and 3-Regularity	p. 11
Regularity	p. 11
Regularity and Dual Nets	p. 14
Antiregularity	p. 16
Antiregularity and Laguerre Planes	p. 18
3-Regularity	p. 22
3-Regularity and Subquadrangles	p. 23
3-Regularity, Inversive Planes, and Characterizations	p. 24
Elation and Translation Generalized Quadrangles	p. 27
Some Notions from Group Theory	p. 27
Elation Generalized Quadrangles	p. 29
Elation Generalized Quadrangles: Definition and Two Fundamental Structure Theorems	p. 29
4-Gonal Families and EGQs	p. 30
Translation Generalized Quadrangles	p. 32
Symmetry	p. 33
Translation Generalized Quadrangles: Definition	p. 34
Symmetries and Translations	p. 34
The Kernel of a Translation Generalized Quadrangle	p. 37
T(n,m,q)s and Translation Generalized Quadrangles	p. 39
Regular Pseudo-Ovals and Regular Pseudo-Ovoids	p. 42
Automorphisms of Translation Generalized Quadrangles	p. 44
Important Properties of O(n, m, q)	p. 47
Pseudo-Ovals	p. 49
Eggs	p. 52
The Stabilizer of the Base-Point of a Translation Generalized Quadrangle	p. 57
Structure of the Automorphism Group of a Translation Quadrangle	p. 59
Generalized Quadrangles and Flocks	p. 61
Flocks	p. 61
Flocks and Translation Planes	p. 62
Flocks of Ovoids and Hyperbolic Quadrics	p. 63
Flocks of Cones	p. 64
Semifield Flocks	p. 69
Known Examples of Semifield Flocks	p. 71
Generalized Quadrangles and Flocks	p. 72
Semifield Flocks and Translation Generalized Quadrangles	p. 74
Position of a Translation Line in a Flock Quadrangle	p. 74
Additive q-Clans and Translation Generalized Quadrangles	p. 74
Known Cases	p. 76
Derivation and BLT-Sets	p. 77
Constructions	p. 78
Property (G) for Generalized Quadrangles of Order (s, s[superscript 2])	p. 81
Flocks, Subquadrangles and Ovals	p. 85
Isomorphisms of Flock Quadrangles and Associated Geometries	p. 87
The Fundamental Theorem of q-Clan Geometry, and Applications	p. 87
Basic Questions on Elation Groups	p. 92
The Standard Conjectures and Questions	p. 92
Some Results by Payne and K. Thas	p. 96
Elation Generalized Quadrangles with Nonisomorphic Elation Groups	p. 98
Good Eggs	p. 101
Good Eggs and Good Translation Generalized Quadrangles	p. 101
Good Eggs and Veronese Surfaces	p. 116
Coordinatization and Applications	p. 121
Coordinatization of Eggs and Dual Eggs	p. 121
The Known Examples Revisited	p. 123
A Geometric Connection between Semifield Flocks and Good Eggs	p. 124
Further Characterizations of Good Eggs in Odd Characteristic	p. 125
Generalized Quadrangles, Nets and the Axiom of Veblen	p. 129
Generalized Quadrangles and the Axiom of Veblen	p. 129
Translation Generalized Quadrangles and the Axiom of Veblen	p. 131
Property (G) and the Axiom of Veblen	p. 132
Flock Generalized Quadrangles and the Axiom of Veblen	p. 135
Subquadrangles and the Axiom of Veblen	p. 138
Nets and Characterizations of Translation Generalized Quadrangles	p. 139
Ovoids and Subquadrangles	p. 145
Ovoids of Q(4,q)	p. 145
Subquadrangles and Ovoids	p. 146
Translation Ovoids and Semifield Flocks	p. 147
Coordinates of the Known Nonclassical Ovoids of Q(4, q)	p. 148
Subquadrangles of T(O), with O Good: the Even Case	p. 150
Subquadrangles of T(O), with O Good: the Odd Case	p. 156
Subquadrangles: Remaining Cases and Some Applications	p. 157
Translation Generalized Quadrangles with One Classical Subquadrangle	p. 160
Elation Generalized Quadrangles with a Subquadrangle	p. 163
Translation Generalized Ovals	p. 169
Translation Generalized Ovoids and Translation Generalized Ovals	p. 169
Note on the Definition of Translation Generalized Oval/Ovoid	p. 174
Characterizations of the T[subscript 2](O) of Tits	p. 175
A Characterization of Translation Generalized Ovals	p. 176
Classification of 2-Transitive Generalized Ovals in Even Characteristic	p. 177
Moufang Sets and Translation Moufang Sets	p. 181
Definition and General Results	p. 181
Finite Moufang Sets	p. 186
The Case PSL[subscript 2](q)	p. 188
The Case PSU[subscript 3](q)	p. 189
The Case Sz(q)	p. 190
The Case R(q)	p. 192
Sub Moufang Sets	p. 196
Configurations of Translation Points	p. 199
Span-Symmetric Generalized Quadrangles	p. 200
Groups Admitting a 4-Gonal Basis	p. 201
SPGQs and Moufang Sets	p. 203
Basic Structural Lemmas	p. 204
Classification of SPGQs of Order (s, t), 1 < s [less than or equal] t < s[superscript 2]	p. 208
SPGQs of Order (s, s[superscript 2])	p. 209
Generalized Quadrangles with a Line of Translation Points	p. 212
On the Classification of Translation Generalized Quadrangles	p. 216
Moufang Quadrangles with a Translation Point	p. 219
Notation	p. 219
Some General Elementary Lemmas	p. 220
The Moufang Property and Analogues	p. 225
Tits Generalized Quadrangles and Tits Systems	p. 229
Properties of Moufang Quadrangles	p. 236
Half 3-Moufang Quadrangles	p. 247
2-Moufang Quadrangles and Fong-Seitz Quadrangles	p. 252
Conclusion	p. 259
Translation Ovoids in Translation Quadrangles	p. 261
Ovoids, Elation or Translation with respect to a Flag or a Point	p. 262
Self-Polar Elation Generalized Quadrangles	p. 264
Suzuki-Tits Moufang Sets	p. 267
Polarities in the Symplectic Quadrangle W(q)	p. 267
The Suzuki-Tits Ovoids	p. 271
Subtended Elation Ovoids	p. 273
Translation Generalized Quadrangles in Projective Space	p. 279
Generalities about Lax Embeddings	p. 280
Planar Translation-Homogeneous Embeddings	p. 288
Exceptional Non-Planar Translation-Homogeneous Embeddings	p. 295
Non-Planarly Embedded Small Translation Generalized Quadrangles	p. 301
Non-Planarly Embedded Translation Generalized Quadrangles	p. 307
Open Problems	p. 317
Chapter 1	p. 317
Chapter 2	p. 318
Chapter 3	p. 318
Chapter 4	p. 320
Chapter 5	p. 320
Chapter 6	p. 321
Chapter 7	p. 321
Chapter 8	p. 322
Chapter 9	p. 322
Chapter 10	p. 322
Chapter 11	p. 323
Chapter 12	p. 323
Chapter 13	p. 324
Bibliography	p. 325
Index	p. 337
Table of Contents provided by Ingram. All Rights Reserved.

Translation Generalized Quadrangles

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