Hyperbolic rigidity of higher rank lattices

TL;DR

The paper demonstrates that higher rank lattices have highly constrained actions on hyperbolic spaces, leading to rigidity results and finite image properties for morphisms into various groups, broadening understanding of their geometric and algebraic behavior.

Contribution

It generalizes fixed point results for higher rank lattices acting on trees to hyperbolic spaces and provides new proofs for morphism finiteness without relying on traditional theorems.

Findings

Actions on Gromov-hyperbolic spaces are elementary (elliptic or parabolic).

Morphisms from higher rank lattices to hierarchically hyperbolic groups have finite image.

New proof of finiteness of morphisms to mapping class groups without Margulis theorem.

Abstract

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.