Israel Journal of Mathematics

, Volume 190, Issue 1, pp 413–444 | Cite as

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

  • Pierre-Emmanuel Caprace
  • Nicolas Monod


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.


Algebraic Group Weyl Group Coxeter Group Open Subgroup Index Subgroup 
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Copyright information

© Hebrew University Magnes Press 2012

Authors and Affiliations

  1. 1.UCLouvainLouvain-la-NeuveBelgium
  2. 2.EPFLLausanneSwitzerland

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