INTRODUCTION xiii

on the cardinality of the

field3;

however, non-linearity results for Kac-Moody groups

over finite fields seem to be much harder to prove. In particular, the techniques

developed by B. R´ emy to tackle this problem are considerably more elaborated

than the ones we use to prove Theorem E. Actually, according to B. R´ emy’s work,

the algebraic group point of view on Kac-Moody groups should be replaced by

a discrete group point in the case of finite ground fields4; this allows to combine

classical arguments from the theory of algebraic groups with tools from dynamics

and ergodic theory (see [R´ em03] for a survey).

Although the Kac-Moody groups considered in Theorem A are split, it is prob-

able that some of the ideas developed in this context can also be used to study the

isomorphism problem in the non-split case (see [R´ em02b] for the relative theory

of Kac-Moody groups). As an illustration of this possibility, we include the solu-

tion of the isomorphism problem for unitary forms of complex Kac-Moody groups

(see Theorem 8.2). These unitary forms were defined and studied by V. Kac and

D. Peterson (see [KP87] and references therein). In the finite-dimensional case,

they coincide with the compact semisimple Lie groups. In the aﬃne case, they co-

incide (up to a central extension by a copy of

S1)

with the so-called ‘algebraic’ loop

groups of the compact semisimple Lie groups. In the indefinite type case, no such

convenient description is known. However, in all cases, unitary forms of complex

Kac-Moody groups carry a natural structure of connected Hausdorff topological

group, and it follows from our result that any epimorphism with central kernel

between two such unitary forms is continuous. This is of course well known in the

finite-dimensional case.

We now come to the organization of the text. The first chapter collects stan-

dard prerequisites on Kac-Moody groups, their root data and twin buildings. The

second chapter is devoted to CAT(0) geometry. After reviewing some standard def-

initions, we recall Bruhat-Tits fixed point theorem and mention some consequences

for groups with bounded generation. The third chapter reviews B. R´emy’s construc-

tion of the adjoint representation of Kac-Moody groups, as well as the relationship

between the adjoint action and the action on the twin building. These first three

chapters are essentially preliminary and contain nothing new (although the Jor-

dan decomposition of bounded elements of Kac-Moody groups over field of positive

characteristic does not seem to appear in the literature). The next three chapters

are devoted to the isomorphism problem for Kac-Moody groups; they constitute

the heart of this work. Chapter 4 contains a statement of the isomorphism theorem,

and proceeds next to a detailed study of diagonalizable and completely reducible

subgroups of Kac-Moody groups. The rest of the proof of the isomorphism theorem

is divided up among Chapter 5 and Chapter 6, which correspond respectively to

the case of characteristic 0 and positive characteristic. Chapter 5 also contains the

3More

precisely, one expects that Kac-Moody groups of type A over finite fields are non-linear

as soon as the Coxeter diagram M (A) associated with A is non-spherical and non-aﬃne. There

are however generalized Cartan matrices A of indefinite type such that M (A) is a Coxeter diagram

of type

˜

A 1. Over finite fields, the corresponding Kac-Moody groups should be considered with

special care, see §4.1.2.

4A

Kac-Moody group over a finite field is finitely generated and embeds as a lattice in the

product of the automorphism groups of its two buildings; this automorphism group is canonically

endowed with a locally compact totally disconnected topology. These facts fail for Kac-Moody

groups over infinite fields.