Inventiones mathematicae

, Volume 176, Issue 1, pp 169–221 | Cite as

Simplicity and superrigidity of twin building lattices

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Abstract

Kac–Moody groups over finite fields are finitely generated groups. Most of them can naturally be viewed as irreducible lattices in products of two closed automorphism groups of non-positively curved twinned buildings: those are the most important (but not the only) examples of twin building lattices. We prove that these lattices are simple if the corresponding buildings are irreducible and not of affine type (i.e. they are not Bruhat–Tits buildings). Many of them are finitely presented and enjoy property (T). Our arguments explain geometrically why simplicity fails to hold only for affine Kac–Moody groups. Moreover we prove that a nontrivial continuous homomorphism from a completed Kac–Moody group is always proper. We also show that Kac–Moody lattices fulfill conditions implying strong superrigidity properties for isometric actions on non-positively curved metric spaces. Most results apply to the general class of twin building lattices.

Keywords

Normal Subgroup Weyl Group Coxeter Group Index Subgroup Coxeter System 
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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Département de MathématiquesUniversité catholique de LouvainLouvain-la-NeuveBelgium
  2. 2.Université Lyon 1, CNRS UMR 5208 – Institut Camille Jordan, Bâtiment du doyen Jean BraconnierUniversité de LyonVilleurbanne CedexFrance

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