Let Γ 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≥3, then each Kac–Moody groups is in fact a simple algebraic group over a local field and Γ is an arithmetic lattice. This relies on the following alternative which is satisfied by any irreducible lattice provided n≥2:
either Γ is an S-arithmetic (hence linear) group, or Γ 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.