# "Abstract" Homomorphisms of Split Kac-Moody Groups

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*Pierre-Emmanuel Caprace*

This work is devoted to the isomorphism
problem for split Kac-Moody groups over arbitrary fields. This
problem turns out to be a special case of a more general problem,
which consists in determining homomorphisms of isotropic semisimple
algebraic groups to Kac-Moody groups, whose image is
bounded. Since Kac-Moody groups possess natural actions on twin
buildings, and since their bounded subgroups can be characterized by
fixed point properties for these actions, the latter is actually a
rigidity problem for algebraic group actions on twin buildings. The
author establishes some partial rigidity results, which we use to
prove an isomorphism theorem for Kac-Moody groups over
arbitrary fields of cardinality at least \(4\). In particular, he
obtains a detailed description of automorphisms of Kac-Moody
groups. This provides a complete understanding of the structure of the
automorphism group of Kac-Moody groups over ground fields of
characteristic \(0\).

The same arguments allow to treat unitary forms of complex Kac-Moody
groups. In particular, the author shows that the Hausdorff topology that these
groups carry is an invariant of the abstract group structure.

Finally, the author proves the non-existence of cocentral
homomorphisms of Kac-Moody groups of indefinite type over
infinite fields with finite-dimensional target. This provides a
partial solution to the linearity problem for Kac-Moody
groups.

#### Table of Contents

# Table of Contents

## "Abstract" Homomorphisms of Split Kac-Moody Groups

- Contents v6 free
- Introduction ix10 free
- Acknowledgements xv16 free
- Chapter 1. The objects: Kac-Moody groups, root data and Tits buildings 118 free
- Chapter 2. Basic tools from geometric group theory 1128
- Chapter 3. Kac-Moody groups and algebraic groups 1734
- Chapter 4. Isomorphisms of Kac-Moody groups: an overview 2744
- 4.1. The isomorphism theorem 2744
- 4.2. Diagonalizable subgroups and their centralizers 3047
- 4.3. Completely reducible subgroups and their centralizers 3552
- 4.4. Basic recognition of the ground field 3956
- 4.5. Detecting rank one subgroups of Kac-Moody groups 4057
- 4.6. Images of diagonalizable subgroups under Kac-Moody group isomorphisms 4360
- 4.7. A technical auxiliary to the isomorphism theorem 4360

- Chapter 5. Isomorphisms of Kac-Moody groups in characteristic zero 4562
- Chapter 6. Isomorphisms of Kac-Moody groups in positive characteristic 5370
- Chapter 7. Homomorphisms of Kac-Moody groups to algebraic groups 6380
- Chapter 8. Unitary forms of Kac-Moody groups 7390
- Bibliography 7996
- Index 83100