CHAPTER 9

Generalized Polygons

Joseph A. THAS

Department of Pure Mathematics and Computer Algebra, University of Ghent, Krijgslaan 281, B-9000 Gent, Belgium

Contents

Introduction

1. Finite generalized polygons

1.1. Graphs

1.2. Graphs defined by finite incidence structures

1.3. Finite generalized polygons

2. Restrictions on the parameters

2.1. The theorem of Feit and Higman

2.2. The inequalities of Higman, and Haemers and Roos

3. The classical finite generalized polygons

3.1. The classical finite projective planes

3.2. The classical finite generalized quadrangles

3.3. The classical finite generalized hexagons

3.4. The classical finite generalized octagons

4. Nonclassical finite generalized quadrangles

4.1. The trivial nonclassical generalized quadrangles, grids and dual grids

4.2. The nonclassical examples of Tits

4.3. The examples of Ahrens and Szekeres, Hall, Jr., and Payne

4.4. Generalized quadrangles as group coset geometries

4.5. Isomorphisms

4.6. Open problems

5. Generalized polygons with small parameters

5.1. Generalized quadrangles with small parameters

5.2. Generalized hexagons with small parameters

5.3. Open problems

6. Ovoids, spreads, polarities and subpolygons

6.1. Ovoids and spreads

6.2. Ovoids and spreads of generalized quadrangles

6.3. Ovoids and spreads of generalized hexagons

6.4. Polarities

6.5. Subpolygons

6.6. Open problems

7. Generalized quadrangles in finite projective and affine spaces

7.1. Generalized quadrangles in finite projective spaces

7.2. Generalized quadrangles in finite affine spaces

8. Combinatorial characterizations of the finite classical generalized quadrangles and hexagons

. Introduction

8.1. Characterizations of and

8.2. Characterizations of and

8.3. Characterizations of

8.4. Theorems simultaneously characterizing several classical generalized quadrangles

8.5. Characterizations of all thick classical and dual classical generalized quadrangles

8.6. Characterizations of nonclassical generalized quadrangles

8.7. Characterizations of the finite classical generalized hexagons

8.8. A combinatorial characterization of all finite thick classical generalized n-gons, with and their duals

8.9. Open problems

9. Automorphisms of generalized polygons

9.1. Elation generalized quadrangles and translation generalized quadrangles

9.2. The sets and the generalized quadrangles

9.3. The known translation generalized quadrangles

9.4. Moufang conditions for finite generalized quadrangles

9.5. Other characterizations of finite generalized quadrangles using automorphisms

9.6. Moufang generalized n-gons with n>4

9.7. Epimorphisms

9.8. Open problems

10. Infinite generalized polygons

References