Hyperbolic actions of higher-rank lattices come from rank-one factors

TL;DR

This paper demonstrates that higher-rank lattice actions on hyperbolic spaces are essentially derived from their rank-one components, establishing a clear connection between lattice structure and hyperbolic actions.

Contribution

It shows that all non-elementary actions of higher-rank lattices on hyperbolic spaces originate from rank-one factors, clarifying the structure of such actions.

Findings

All higher-rank lattice actions on hyperbolic spaces come from rank-one factors.

Non-elementary actions on unbounded hyperbolic spaces are characterized by this origin.

The results unify the understanding of lattice actions in hyperbolic geometry.

Abstract

We study actions of higher rank lattices $\Gamma<G$ on hyperbolic spaces, and we show that all such actions satisfying mild properties come from the rank-one factors of $G$. In particular, all non-elementary actions on an unbounded hyperbolic space are of this type.