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Simplicity and superrigidity of twin building lattices

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Caprace, Pierre-Emmanuel [UCL] Remy, Bertrand

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.

Document type Article de périodique (Journal article) – Article de recherche
Access type Accès restreint
Publication date 2009
Language Anglais
Journal information "Inventiones Mathematicae" - Vol. 176, no. 1, p. 169-221 (2009)
Peer reviewed yes
Publisher Springer (New York)
issn 0020-9910
e-issn 1432-1297
Publication status Publié
Affiliation UCL
Links
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Bibliographic reference Caprace, Pierre-Emmanuel ; Remy, Bertrand. Simplicity and superrigidity of twin building lattices. In: Inventiones Mathematicae, Vol. 176, no. 1, p. 169-221 (2009)
Permanent URL http://hdl.handle.net/2078.1/35725