Linear images of Kac-Moody groups and super-rigidity
by
Bertrand Remy
Université de Grenoble I (Joseph Fourier)

We are interested in finite-dimensional linear representations of Kac-Moody groups. The ground fields of the latter groups are supposed to be finite, but no assumption is made for the representations (they are abstract group homomorphisms from discrete finitely generated groups to any general linear group). We show the following dichotomy. Let \Lambda be a Kac-Moody group over a finite field of characteristic p. Then: either \Lambda admits faithful linear representation - and this provides a topological embedding of the corresponding topological Kac-Moody group to a non-Archimedean Lie group of characteristic p, or all linear images of \Lambda are virtually solvable groups. This enables to exhibit finitely generated Kac-Moody groups all of whose linear images are finite. The proofs mixes arguments from Tits system theory, dynamics in algebraic groups over local fields and super-rigidity of commensurators.

Date received: July 8, 2003