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.
has a global fixed point.
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
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
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.