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 and imaginary 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.