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

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.

26 pages, to appear in Ergodic theory and Dynamical Systems. In v1 there was a mistake due to non-separability of certain spaces. The main theorem still applies to Lie groups but it is now weaker than in v1, and we introduce the new notion of coarse metric ergodicity to prove it