Jacques Tits Richard M. Weiss

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Moufang Polygons

Springer

Table of Contents

Part I Preliminary Results

1 Introduction 3

2 Some Definitions 7

3 Generalized Polygons 15

4 Moufang Polygons 19

5 Commutator Relations 23

6 Opposite Root Groups 27

7 A Uniqueness Lemma 31

8 A Construction 33

Part II Nine Families of Moufang Polygons

9 Alternative Division Rings, I 45

10 Indifferent and Octagonal Sets 57

11 Involutory Sets and Pseudo-Quadratic Forms 61

12 Quadratic Forms of Type Ee, E7 and E$, I 71

13 Quadratic Forms of Type Ee, E7 and E&, II 91

14 Quadratic Forms of Type F4 125

15 Hexagonal Systems, I 133

16 An Inventory of Moufang Polygons 163

17 Main Results 175

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viii Table of Contents

Part III The Classification of Moufang Polygons

18 A Bound o n n 179

19 Triangles 185

20 Alternative Division Rings, II 191

21 Quadrangles 203

22 Quadrangles of Involution Type 215

23 Quadrangles of Quadratic Form Type 229

24 Quadrangles of Indifferent Type 239

25 Quadrangles of Pseudo-Quadratic Form Type, I 243

26 Quadrangles of Pseudo-Quadratic Form Type, II 251

27 Quadrangles of Type E6, E7 and Es 275

28 Quadrangles of Type FA 285

29 Hexagons 301

30 Hexagonal Systems, II 319

31 Octagons 339

32 Existence 353

Part IV More Results on Moufang Polygons

33 BN-Pairs 367

34 Finite Moufang Polygons 375

35 Isotopes 381

36 Isomorphic Hexagonal Systems 391

37 Automorphisms 397

38 Isomorphic Quadrangles 419

Table of Contents ix

Part V Moufang Polygons and Spherical Buildings

39 Chamber Systems 427

40 Spherical Buildings 447

41 Classical, Algebraic and Mixed Buildings 477

42 Appendix 489

Bibliography 521

Index of Notation 527

Index 529

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