INTRODUCTION xi

A. This adjoint representation, constructed by B. R´ emy [R´ em02b, Chapter 9], is

functorial and should be compared to the adjoint representation of a group scheme

on its algebra of distributions.

A striking feature of these two actions is that they are strongly related. The

main relationship to keep in mind is the following: The adjoint action of a subgroup

of G(K) is locally finite if and only if this subgroup has fixed points in both halves B+,

B− of the twin building B. A subgroup satisfying one of these equivalent conditions

is called a bounded subgroup. The adjoint representation can be used to endow

certain bounded subgroups with a structure of algebraic groups and serves in this

way as a substitute for an algebro-geometric structure for G(K).

A key observation made in [CM05b] and inspired by [KW92], is that the

conclusions of Theorem A would follow for a given Kac-Moody group isomorphism

whenever one shows that this isomorphism maps bounded subgroups to bounded

subgroups. This observation relies on the understanding of the structure of maximal

bounded subgroups, which allows to reduce the Kac-Moody group isomorphism

problem to the finite-dimensional case, for which a complete solution is available

in [BT73]. In this way, the isomorphism problem for Kac-Moody groups becomes

a special case of the following.

Problem. Let K be a Kac-Moody group, G be a connected reductive F-isotropic

F-group and ϕ : G(F) → K be a homomorphism. Find conditions under which ϕ

has bounded image.

In view of the above description of bounded subgroups of Kac-Moody groups,

this problem may be viewed as a rigidity problem for reductive group actions on twin

buildings. It turns out that for split Kac-Moody groups acting on one-dimensional

buildings, this problem can be completely solved by means of the following result.

Theorem. (J. Tits [Tit77]) Let G(F) act on a tree T , where G is a semisim-

ple algebraic F-group of positive F-rank. Then one of the following holds, where

G†(F) denotes the subgroup of G(F) generated by the F-points of the unipotent

radicals of the Borel subgroups of G defined over F.

(i)

G†(F)

has a global fixed point.

(ii)

G†(F)

has no global fixed point but a unique fixed end.

(iii) F-rank(G) = 1 and the root datum of G(F) has a valuation such that the

corresponding Bruhat-Tits tree has G†(F)-equivariant embedding in T .

Specializing this result to G(F)-actions on one-dimensional twin buildings, one

obtains the following.

Corollary B. Let K be a split Kac-Moody group whose twin building is one-

dimensional and G be a connected reductive F-group of positive F-rank. Then every

homomorphism of

G†(F)

to K has bounded image.

This motivates the search for rigidity results analogous to Tits’ theorem but

for higher dimensional buildings. An example of such an analogue is provided by

the following.

Theorem C. Let Γ := SL2(Q) act by cellular isometries on a CAT(0) polyhe-

dral complex X. Then one of the following holds:

(i) Every finitely generated subgroup of Γ has a fixed point in X.