Superrigidity for irreducible lattices and geometric splitting

Monod, Nicolas

doi:10.1090/S0894-0347-06-00525-X

Superrigidity for irreducible lattices and geometric splitting
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by Nicolas Monod;

J. Amer. Math. Soc. 19 (2006), 781-814

DOI: https://doi.org/10.1090/S0894-0347-06-00525-X

Published electronically: March 21, 2006

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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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