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

Caprace, Pierre-Emmanuel;Monod, Nicolas
(2012) Israel Journal of Mathematics — Vol. 190, p. 413-444 (2012)

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Abstract
Let Γ < G 1 × … × G n be an irreducible lattice in a product of infinite irreducible complete Kac-Moody groups of simply laced type over finite fields. We show that if n ≥ 3, then each G i is a simple algebraic group over a local field and Γ is an S-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.
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Caprace, P.-E., & Monod, N. (2012). A lattice in more than two Kac-Moody groups is arithmetic. Israel Journal of Mathematics, 190, 413-444. https://doi.org/10.1007/s11856-012-0006-3 (Original work published 2012)