CHAPTER 10

Some Classes of Rank 2 Geometries

Frank DE CLERCK and Hendrik VAN MALDEGHEM

Department of Pure Mathematics and Computer Algebra, University of Ghent, Galglaan 2, B-9000 Gent, Belgium

Contents

Introduction

1. Partial geometries

1.1. Definitions

1.2. Remarks

1.3. The point graph of a partial geometry

1.4. The known models of proper partial geometries

1.5. Some characterization theorems for partial geometries

2. Semipartial geometries

2.1. Definitions

2.2. A first list of examples of proper semipartial geometries

2.3. The linear representations of semipartial geometries

2.4. Semipartial geometries and generalized quadrangles

2.5. Semipartial geometries and SPG reguli

2.6. Some characterization theorems for semipartial geometries

3. Copolar spaces

4. Near n-gons

4.1. Definitions

4.2. Classical and sporadic near n-gons

4.3. Regular near n-gons

5. Moore geometries

5.1. Moore graphs

5.2. (Generalized) Moore geometries

6. -gons

6.1. Definitions

6.2. Examples

6.3. Characterizations by automorphisms

References