Buekenhout F., Cohen A.M. Diagram Geometry: Related to Classical Groups and Buildings

Springer, 2013. — 596 p. — ISBN: 3642344526, 9783642344527.This book provides a self-contained introduction to diagram geometry. Tight connections with group theory are shown. It treats thin geometries (related to Coxeter groups) and thick buildings from a diagrammatic perspective. Projective and affine geometry are main examples. Polar geometry is motivated by polarities on diagram geometries and the complete classification of those polar geometries whose projective planes are Desarguesian is given. It differs from Tits' comprehensive treatment in that it uses Veldkamp's embeddings. The book intends to be a basic reference for those who study diagram geometry. Group theorists will find examples of the use of diagram geometry. Light on matroid theory is shed from the point of view of geometry with linear diagrams. Those interested in Coxeter groups and those interested in buildings will find brief but self-contained introductions into these topics from the diagrammatic perspective. Graph theorists will find many highly regular graphs. The text is written so graduate students will be able to follow the arguments without needing recourse to further literature. A strong point of the book is the density of examples.Geometries
The Concept of a Geometry.
Incidence Systems and Geometries.
Homomorphisms.
Subgeometries and Truncations.
Residues.
Connectedness.
Permutation Groups.
Groups and Geometries.
Diagrams.
The Digon Diagram of a Geometry.
Some Parameters for Rank Two Geometries.
Diagrams for Higher Rank Geometries.
Coxeter Diagrams.
Shadows.
Group Diagrams.
A Geometry of Type Ã_n−1.
Chamber Systems.
From a Geometry to a Chamber System.
Residues.
From Chamber Systems to Geometries.
Connectedness.
The Diagram of a Chamber System.
Groups and Chamber Systems.
Thin Geometries.
Being Thin.
Thin Geometries of Coxeter Type.
Groups Generated by Affine Reflections.
Linear Reflection Representations.
Root Systems.
Finiteness Criteria.
Finite Coxeter Groups.
Regular Polytopes Revisited.
Linear Geometries.
The Affine Space of a Vector Space.
The Projective Space of a Vector Space.
Matroids.
Matroids from Geometries.
Steiner Systems.
Geometries Related to the Golay Code.
Projective and Affine Spaces.
Perspectivities.
ProjectiveLines.
Classification of Projective Spaces.
Classification of Affine Spaces.
Apartments.
Grassmannian Geometry.
Root Filtration Spaces.
Polar Spaces.
Duality for Geometries over a Linear Diagram.
Duality and Sesquilinear Forms.
Absolutes and Reflexive Sesquilinear Forms.
Polar Spaces.
The Diagram of a Polar Space.
From Diagram to Space.
Singular Subspaces.
Other Shadow Spaces of Polar Geometries.
Root Filtration Spaces.
Polar Spaces with Thin Lines.
Projective Embeddings of Polar Spaces.
Geometric Hyperplanes and Ample Connectedness.
The Veldkamp Space.
Projective Embedding for Rank at Least Four.
Projective Embedding for Rank Three.
Automorphisms of Polar Spaces.
Embedding Polar Spaces in Absolutes.
Embedded Spaces.
Collars and Tangent Hyperplanes.
AQuasi-polarity.
Technical Results on Division Rings.
Embedding in 3-Dimensional Space.
Classical Polar Spaces.
Trace Valued Forms.
Pseudo-quadrics.
Perspective Sets.
Apartments.
Automorphisms.
Finite Classical Polar Spaces.
Buildings.
Building Axioms.
Properties of Buildings.
Tits Systems.
Shadow Spaces.
Parapolar Spaces.
Root Shadow Spaces.
Recognizing Shadow Spaces of Buildings.