CHAPTER 22

Finite Diagram Geometries Extending Buildings

Francis BUEKENHOUT

CP 216, Université Libre de Bruxelles, Boulevard du Triomphe, B-1050 Bruxelles, Belgium

Antonio PASINI

Istituto di Matematica dell'università, 15 Via del Capitano, I-53100 Siena, Italy

Contents

1. Extending buildings with restrictions

1.0. Introduction

1.1. Sporadic groups: a source and a goal

1.2. Other sources

1.3. A setting in terms of diagram, geometry, group and characteristic

1.4. A selection of allowed rank 2 residues

1.5. Classical cases

1.6. A stock of possible restrictions

1.7. Flag-transitive generalized polygons

1.8. Flag-transitive rank 2 geometries of small deficiency

1.9. The flag-transitive linear spaces

1.10. Finite or infinite universal covers?

2. Geometries over Coxeter diagrams of spherical type

2.1. Spherical diagrams and finiteness conditions

2.2. Irreducible spherical diagrams

2.3. The locally classical case

2.4. Finite thick geometries

2.5. Finite thick and geometries

3. Geometries over Coxeter diagrams of nonspherical type

3.1. Hyperbolic and other diagrams

3.2. Flag-transitive locally classical finite geometries belonging to Coxeter diagrams

3.3. Some generalizations

3.4. The tradition of thin geometries

3.5. A characterization of buildings

4. Extensions of geometries over Coxeter diagrams by linear spaces

4.1. Introduction

4.2. The diagram

4.3. The diagram

4.4. The diagram

4.5. The diagram

4.6. The diagrams and

4.7. The diagrams and

5. Extensions of geometries over Coxeter diagrams by the Petersen graph geometry

5.1. Extensions of projective geometries by the Petersen geometry

5.2. Extensions of affine geometries

5.3. Other extensions

5.4. Tilde geometries

5.5. The relationship between tilde and Petersen geometries

5.6. More examples

6. Other extensions

6.1. Back to sporadic groups

6.2. Extensions by partial geometries

6.3. All geometries for a given group

References