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. We establish
some partial rigidity results, which we use to prove an isomorphism theorem for
Kac-Moody groups over arbitrarily fields of cardinality at least 4. In particular,
we obtain 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, we show that the Hausdorff topology that these groups
carry is an invariant of the abstract group structure.
Finally, we prove 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.
Received by the editor February 3, 2006.
2000 Mathematics Subject Classification. Primary 17B40, 20E36, 20E42, 20G15, 22E65,
Key words and phrases. Kac-Moody group, algebraic group, building, non-positively