Contents
Introduction ix
Acknowledgements xv
Chapter 1. The objects: Kac-Moody groups, root data and Tits buildings 1
1.1. Kac-Moody groups and Tits functors 1
1.2. Root data 3
1.3. Tits buildings 6
1.4. Twin root data and twin buildings: a short dictionary 8
Chapter 2. Basic tools from geometric group theory 11
2.1. CAT(0) geometry 11
2.2. Rigidity of algebraic-group-actions on trees 14
Chapter 3. Kac-Moody groups and algebraic groups 17
3.1. Bounded subgroups 17
3.2. Adjoint representation of Tits functors 19
3.3. A few facts from the theory of algebraic groups 23
Chapter 4. Isomorphisms of Kac-Moody groups: an overview 27
4.1. The isomorphism theorem 27
4.2. Diagonalizable subgroups and their centralizers 30
4.3. Completely reducible subgroups and their centralizers 35
4.4. Basic recognition of the ground field 39
4.5. Detecting rank one subgroups of Kac-Moody groups 40
4.6. Images of diagonalizable subgroups under Kac-Moody group
isomorphisms 43
4.7. A technical auxiliary to the isomorphism theorem 43
Chapter 5. Isomorphisms of Kac-Moody groups in characteristic zero 45
5.1. Rigidity of SL2(Q)-actions on CAT(0) polyhedral complexes 45
5.2. Homomorphisms of Chevalley groups over Q to Kac-Moody groups 48
5.3. Regularity of diagonalizable subgroups 51
5.4. Proof of the isomorphism theorem 52
Chapter 6. Isomorphisms of Kac-Moody groups in positive characteristic 53
6.1. On bounded subgroups of Kac-Moody groups 53
6.2. Homomorphisms of certain algebraic groups to Kac-Moody groups 54
6.3. Images of certain small subgroups under Kac-Moody group
isomorphisms 59
6.4. Proof of the isomorphism theorem 62
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