(ii) There exists finitely many primes p1, . . . , pn such that for each i = 1, . . . , n,
there exists a Γpi -equivariant embedding of the vertices of the Bruhat-Tits
pi-adic tree Ti in X, where Γpi = SL2(Z[
]). Moreover, for each integer
m prime to all pi’s, the group SL2(Z[
]) has fixed points in X.
The basic ingredient of the proof of this theorem is a result of M. Brid-
son [Bri99] which describes arbitrary abelian group actions on CAT(0) polyhedral
complexes (see Proposition 2.8 below). One also needs the fact that the group
]) has bounded generation [Mor05].
Since no assumption on the local compactness of X is made in Theorem C,
this result applies in particular to all buildings of finite rank (see [Dav98]). In the
special case of Kac-Moody buildings, one obtains the following.
Corollary D. Let K be a Kac-Moody group and G be a Q-split semisimple
algebraic Q-group. Then every homomorphism of G(Q) to K has bounded image.
This corollary is the key ingredient of the proof of Theorem A over fields of
characteristic 0. Though similar in spirit, the proof of Theorem A in positive
characteristic follows a slightly different strategy. In the latter, one considers ho-
momorphisms of a F-isotropic reductive F group to a split Kac-Moody group whose
restriction to the center of the reductive group is injective. The main idea, which
was at the heart of [CM05a], is to study the action of the semisimple part on the
fixed point set of the abelian part in the twin building. Combining the aforemen-
tioned result of M. Bridson, a fixed point theorem for automorphism groups of twin
buildings by B. M¨uhlherr [M¨ uh94] and Borel-Tits’ description of centralizers of
tori in reductive groups [BT65], one shows essentially that the image of the center
centralizes a subgroup of G(K) which is of Kac-Moody type but not necessarily split.
If the dimension of the center is large enough, the twin building of this Kac-Moody
group becomes one-dimensional, which makes Tits’ theorem available again. How-
ever, the presence of a possibly non-trivial anisotropic kernel creates some technical
diﬃculties coming from the existence of anisotropic elements in algebraic groups
(see Theorem 6.6 below for a precise statement).
It is rather natural to consider the ‘dual’ of the problem addressed above and
study homomorphisms of Kac-Moody groups to reductive groups. The question of
the existence of injective such homomorphisms is known as the linearity problem
for Kac-Moody groups, to which the following result gives a partial answer.
Theorem E. Let A be a generalized Cartan matrix, G be a Tits functor of type
A and K be an infinite field. Let F be a field, n be an integer and ϕ : G(K) → GLn(F)
be a homomorphism with central kernel. Then every indecomposable component of
the generalized Cartan matrix A is of finite or aﬃne type.
This shows in particular that there does not exist any cocentral homomorphism
of an indefinite type Kac-Moody group over an infinite field to a reductive group.
Note that modulo the conjectural simplicity of indefinite type Kac-Moody groups,
this shows the nonexistence of any nontrivial homomorphism of these groups with
The linearity problem has been considered for Kac-Moody groups over finite
fields by B. R´ emy, who proved that Kac-Moody groups of certain hyperbolic types
over suﬃciently big finite fields are non-linear (see [R´ em02a] and [R´ em04]). It is
expected that the conclusions of Theorem E actually hold without any restriction