Author | Maldeghem, Hendrik van. author |
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Title | Generalized Polygons [electronic resource] / by Hendrik van Maldeghem |
Imprint | Basel : Birkhรคuser Basel : Imprint: Birkhรคuser, 1998 |
Connect to | http://dx.doi.org/10.1007/978-3-0348-8827-1 |
Descript | XV, 502 p. 29 illus. online resource |
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 ReeโTits 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 Weissโ 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 ReeโTits 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 โ 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