Generalized Polygons

• 1 Basic Concepts and Results.- 1.1 Introduction.- 1.2 Geometries.- 1.3 Generalized polygons.- 1.4 Generalized quadrangles.- 1.5 Projections and projectivities.- 1.6 Structure of weak generalized polygons.- 1.7 Finite and semi-finite generalized polygons.- 1.8 Subpolygons.- 1.9 Regularity.- 2 Classical Polygons.- 2.1 Introduction.- 2.2 Classical and alternative projective planes.- 2.3 Classical generalized quadrangles.- 2.4 Classical generalized hexagons.- 2.5 Classical generalized octagons.- 2.6 Table of notation for some classical polygons.- 3 Coordinatization and Further Examples.- 3.1 Introduction.- 3.2 General coordinatization theory.- 3.3 Generalized quadrangles and hexagons.- 3.4 The classical and mixed quadrangles.- 3.5 The classical hexagons.- 3.6 The Ree—Tits octagons.- 3.7 Some non-classical quadrangles.- 3.8 Other generalized polygons.- 4 Homomorphisms and Automorphism Groups.- 4.1 Introduction.- 4.2 A theorem of Pasini on epimorphisms.- 4.3 Notation and results from group theory.- 4.4 Root elations and generalized homologies.- 4.5 Collineations of classical polygons.- 4.6 Collineation groups of finite classical polygons.- 4.7 The Tits condition.- 4.8 Finite point-distance transitive and flag-transitive polygons.- 4.9 Kantor systems.- 5 The Moufang Condition.- 5.1 Introduction.- 5.2 First properties of Moufang polygons.- 5.3 Weiss’ theorem.- 5.4 Root systems.- 5.5 Commutation relations and classification.- 5.6 Another result of Weiss.- 5.7 Finite Moufang polygons.- 5.8 Simplicity of the little projective group.- 5.9 Point-minimal and line-minimal Moufang polygons.- 6 Characterizations.- 6.1 Introduction.- 6.2 Regularity in generalized quadrangles.- 6.3 Regularity in generalized hexagons.- 6.4 Regularity in generalized polygons.- 6.5 Hyperbolic andimaginary lines.- 6.6 Generalized Desargues configurations.- 6.7 Some combinatorial characterizations.- 6.8 Some algebraic characterizations.- 6.9 The perfect Ree—Tits octagons.- 7 Ovoids, Spreads and Self-Dual Polygons.- 7.1 Introduction.- 7.2 Generalities about polarities and ovoids.- 7.3 Polarities, ovoids and spreads in Moufang polygons.- 7.4 Moufang quadrangles of type (BC — CB)2.- 7.5 Polarities, conics, hyperovals and unitals in Pappian planes.- 7.6 Suzuki quadrangles and Suzuki-Tits ovoids.- 7.7 Ree hexagons and Ree-Tits ovoids.- 7.8 Amalgamations.- 8 Projectivities and Projective Embeddings.- 8.1 Introduction.- 8.2 Some more properties of the Ree-Tits octagons.- 8.3 The little projective groups of some Moufang polygons.- 8.4 Groups of projectivities of some Moufang polygons.- 8.5 Projective embeddings of generalized quadrangles.- 8.6 Ideal, weak and lax embeddings of polygons.- 8.7 Embeddings of the slim Moufang polygons.- 9 Topological Polygons.- 9.1 Introduction.- 9.2 Definition of topological polygons.- 9.3 Examples.- 9.4 General properties.- 9.5 The impact of algebraic topology.- 9.6 Transitivity properties.- 9.7 Polygons with valuation.- 9.8 Other categories.- Appendices.- A An Eigenvalue Technique.- B The Theorem of Bruck and Kleinfeld.- C Tits Diagrams for Moufang Quadrangles.- D Root Elations of Classical Polygons.- E The Ten Most Famous Open Problems.

"... If you believe that incidence geometry is out of date, this book will prove you wrong. If you are interested in geometry and want to get introduced to fascinating recent concepts, this book will be a good starting point and a trustworthy companion for most of your way. If you are already doing research in incidence geometry, this book will bring you up to date on generalized polygons and provide a very convincing notation. In addition, its bibliography and its almost complete collection of results on generalized polygons will make it an indispensable tool. If you are eager to do some research, the author offers ten open problems." -Zentralblatt Math