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Superrigidity for irreducible lattices and geometric splitting

Author: Nicolas Monod
Journal: J. Amer. Math. Soc. 19 (2006), 781-814
MSC (2000): Primary 22Exx; Secondary 53Cxx, 20F65
Published electronically: March 21, 2006
MathSciNet review: 2219304
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Abstract: We prove general superrigidity results for actions of irreducible lattices on CAT$(0)$ spaces, first in terms of the ideal boundary, and then for the intrinsic geometry (also for infinite-dimensional spaces). In particular, one obtains a new and self-contained proof of Margulis’ superrigidity theorem for uniform irreducible lattices in non-simple groups. The proofs rely on simple geometric arguments, including a splitting theorem which can be viewed as an infinite-dimensional (and singular) generalization of the Lawson-Yau/Gromoll-Wolf theorem. Appendix A gives a very elementary proof of commensurator superrigidity; Appendix B proves that all our results also hold for certain non-uniform lattices.

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Additional Information

Nicolas Monod
Affiliation: Department of Mathematics, The University of Chicago, 5734 University Avenue, Chicago, Illinois 60637
Address at time of publication: Université de Genève, 2-4, rue du Lièvre, CP 64, CH-1211 Genève 4, Switzerland
MR Author ID: 648787

Keywords: Superrigidity, splitting, lattices, Hadamard spaces.
Received by editor(s): December 13, 2004
Published electronically: March 21, 2006
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.