Negative curvature from a cohomological viewpoint and cocycle superrigidity

Nicolas Monod; Yehuda Shalom

doi:10.1016/j.crma.2003.10.002

Group Theory

Negative curvature from a cohomological viewpoint and cocycle superrigidity

Presented by: Étienne Ghys

Nicolas Monod ¹; Yehuda Shalom ²

¹ Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, USA
² School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel

Comptes Rendus. Mathématique, Volume 337 (2003) no. 10, pp. 635-638.

Abstract
Résumé

In the framework of general negatively curved spaces, we present new superrigidity results and introduce new techniques based on bounded cohomology. This applies to irreducible lattices, and more generally to cocycles, of products of arbitrary locally compact groups. Together with a new vanishing result for higher rank groups, this also generalizes and unifies all previously known results in that direction. The non-vanishing results provide a large class of examples for our results on orbit equivalence rigidity (Monod and Shalom, Ann. of Math., in press). We prove the ‘toy-case’ of actions on trees.

Nous proposons de nouvelles méthodes cohomologiques pour établir des énoncés de superrigidité dans le cadre général des espaces métriques à courbure strictement négative. Nos résultats s'appliquent aux réseaux irréductibles, ou plus généralement aux cocycles, pour des produits de groupes localement compacts généraux. Avec le concours d'un nouveau théorème d'annulation, on subsume et généralise de la sorte tous les résultats qui allaient dans ce sens ; en outre, les énoncés de non annulation fournissent une vaste classe d'exemples pour nos résultats en équivalence orbitale (Monod et Shalom, Ann. of Math., in press). Nous donnons une preuve dans le cas particulièrement simple des arbres.

Received: 2003-08-04
Accepted: 2003-10-05
Published online: 2003-11-05

DOI: 10.1016/j.crma.2003.10.002

Author's affiliations:

Nicolas Monod ¹; Yehuda Shalom ²

¹ Department of Mathematics, University of Chicago, 5734 S. University Avenue, Chicago, IL 60637, USA
² School of Mathematical Sciences, Tel-Aviv University, Tel-Aviv 69978, Israel

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Nicolas Monod; Yehuda Shalom. Negative curvature from a cohomological viewpoint and cocycle superrigidity. Comptes Rendus. Mathématique, Volume 337 (2003) no. 10, pp. 635-638. doi : 10.1016/j.crma.2003.10.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.10.002/

References
Cited by

[1] M. Burger; N. Monod Continuous bounded cohomology and applications to rigidity theory, Geom. Funct. Anal., Volume 12 (2002) no. 2, pp. 219-280

[2] V.A. Kaimanovich, Double ergodicity of the Poisson boundary and applications to bounded cohomology, Geom. Funct. Anal., in press

[3] I. Mineyev, N. Monod, Y. Shalom, Ideal bicombing for hyperbolic groups and applications, Topology, in press

[4] N. Monod Continuous Bounded Cohomology of Locally Compact Groups, Lecture Notes in Math., 1758, Springer, Berlin, 2001

[5] N. Monod, Y. Shalom, Cocycle superrigidity and bounded cohomology for negatively curved spaces, Preprint

[6] N. Monod, Y. Shalom, Orbit equivalence rigidity and bounded cohomology, Ann. of Math., in press

[7] Y. Shalom Rigidity of commensurators and irreducible lattices, Invent. Math., Volume 141 (2000) no. 1, pp. 1-54

[8] R.J. Zimmer Ergodic theory and semisimple groups, Birkhäuser, Basel, 1984

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