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-actionsontrees ................ 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
Chapter 7. Homomorphisms of Kac-Moody groups to algebraic
groups .............................................. 63
7.1. The non-linearity theorem ................................. 63
7.2. A combinatorial characterization of affine Coxeter
groups .................................................... 63
7.3. On infinite root Systems .................................. 66
7.4. Proof of the non-linearity theorem ........................ 70
Chapter 8. Unitary forms of Kac-Moody groups ................... 73
8.1. Introduction .............................................. 73
8.2. Definitions ............................................... 73
8.3. Isomorphisms of unitary forms ............................. 75
8.4. Non-linearity ............................................. 78
Bibliography ................................................... 79
Index .......................................................... 83
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