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Article Dans Une Revue Annales Scientifiques de l'École Normale Supérieure Année : 2020

Hyperbolic rigidity of higher rank lattices

Résumé

We prove that any action of a higher rank lattice on a Gromov-hyperbolic space is elementary. More precisely, it is either elliptic or parabolic. This is a large generalization of the fact that any action of a higher rank lattice on a tree has a fixed point. A consequence is that any quasi-action of a higher rank lattice on a tree is elliptic, i.e. it has Manning's property (QFA). Moreover, we obtain a new proof of the theorem of Farb-Kaimanovich-Masur that any morphism from a higher rank lattice to a mapping class group has finite image, without relying on the Margulis normal subgroup theorem nor on bounded cohomology. More generally, we prove that any morphism from a higher rank lattice to a hierarchically hyperbolic group has finite image. In the Appendix, Vincent Guirardel and Camille Horbez deduce rigidity results for morphisms from a higher rank lattice to various outer automorphism groups.

Dates et versions

hal-01814136 , version 1 (12-06-2018)

Identifiants

Citer

Thomas Haettel, Vincent Guirardel, Camille Horbez. Hyperbolic rigidity of higher rank lattices. Annales Scientifiques de l'École Normale Supérieure, 2020, 53 (2), pp.439-468. ⟨10.24033/asens.2425⟩. ⟨hal-01814136⟩
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