Comptes Rendus
no. 3
Group Theory
Superrigidity for irreducible lattices and geometric splitting
[La super-rigidité des réseaux irréductibles et un théorème de décomposition]

Présenté par : Étienne Ghys

Nicolas Monod1
1 Department of Mathematics, University of Chicago, 5734, South University Avenue, Chicago, IL 60637, USA
Comptes Rendus. Mathématique, Volume 340 (2005) no. 3, pp. 185-190.

Nous exposons des résultats de super-rigidité pour les actions de réseaux irréductibles en géométrie de Hadamard, singulière ou non. Une de nos motivations est de présenter une preuve élémentaire du théorème de super-rigidité de Margulis pour les réseaux uniformes dans les groupes algébriques semi-simples (non simples) ; nos méthodes s'appliquent cependant aux réseaux dans des produits de groupes complètement généraux. Notre preuve repose notamment sur un théorème de décomposition qui généralise le théorème de Lawson–Yau/Gromoll–Wolf aux dimensions infinies, ou plus précisément aux espaces CAT(0) complets généraux.

We propose general superrigidity results for actions of irreducible lattices on CAT(0) spaces. In particular, we obtain a new and self-contained proof of Margulis' superrigidity theorem for uniform irreducible lattices in non-simple groups. However, the statements hold for lattices in products of arbitrary groups; likewise, the geometric representations need not be linear. The proof uses notably a new splitting theorem which can be viewed as an infinite-dimensional and singular generalization of the Lawson–Yau/Gromoll–Wolf theorem.

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DOI : 10.1016/j.crma.2004.12.023
Nicolas Monod 1

1 Department of Mathematics, University of Chicago, 5734, South University Avenue, Chicago, IL 60637, USA 
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Nicolas Monod. Superrigidity for irreducible lattices and geometric splitting. Comptes Rendus. Mathématique, Volume 340 (2005) no. 3, pp. 185-190. doi : 10.1016/j.crma.2004.12.023. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.12.023/

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[2] D. Gromoll; J.A. Wolf Some relations between the metric structure and the algebraic structure of the fundamental group in manifolds of nonpositive curvature, Bull. Amer. Math. Soc., Volume 77 (1971), pp. 545-552

[3] J. Jost; S.-T. Yau Harmonic maps and rigidity theorems for spaces of nonpositive curvature, Commun. Anal. Geom., Volume 7 (1999) no. 4, pp. 681-694

[4] H.B. Lawson; S.-T. Yau Compact manifolds of nonpositive curvature, J. Differential Geom., Volume 7 (1972), pp. 211-228

[5] G.A. Margulis Arithmeticity of the irreducible lattices in the semisimple groups of rank greater than 1, Invent. Math., Volume 76 (1984) no. 1, pp. 93-120

[6] G.A. Margulis Discrete Subgroups of Semisimple Lie Groups, Springer-Verlag, Berlin, 1991

[7] N. Monod, Superrigidity for irreducible lattices and geometric splitting, Preprint, 2003

[8] N. Monod, Arithmeticity vs. non-linearity for irreducible lattices, Preprint, 2004

[9] V. Schroeder A splitting theorem for spaces of nonpositive curvature, Invent. Math., Volume 79 (1985) no. 2, pp. 323-327

[10] T.N. Venkataramana On superrigidity and arithmeticity of lattices in semisimple groups over local fields of arbitrary characteristic, Invent. Math., Volume 92 (1988) no. 2, pp. 255-306

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