Comptes Rendus
no. 3
Group Theory
Superrigidity for irreducible lattices and geometric splitting
Étienne Ghys
Nicolas Monod1
1Department of Mathematics, University of Chicago, 5734, South University Avenue, Chicago, IL 60637, USA
Comptes Rendus. Mathématique, Volume 340 (2005) no. 3, pp. 185-190

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.

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.

Received:
Accepted:
Published online:
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 
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
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[1] M.R. Bridson; A. Haefliger Metric Spaces of Non-Positive Curvature, Grundlehren Math. Wiss., vol. 319, Springer, Berlin, 1999

[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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