A lattice in more than two Kac--Moody groups is arithmetic

Details A lattice in more than two Kac--Moody groups is arithmetic A lattice in more than two Kac--Moody groups is arithmetic

2008-12-07

arxiv.org/abs/0812.1383v2

Mathematics

Group Theory

Subsection 2.B was modified and an example was added there

Scientific paper

Let $\Gamma$ be an irreducible lattice in a product of n infinite irreducible complete Kac-Moody groups of simply laced type over finite fields. We show that if n is at least 3, then each Kac-Moody groups is in fact a simple algebraic group over a local field and $\Gamma$ is an arithmetic lattice. This relies on the following alternative which is satisfied by any irreducible lattice provided n is at least 2: either $\Gamma$ is an S-arithmetic (hence linear) group, or it is not residually finite. In that case, it is even virtually simple when the ground field is large enough. More general CAT(0) groups are also considered throughout.

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