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

Uri Bader, Pierre-Emmanuel Caprace, Alex Furman, Alessandro Sisto

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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. Our results also apply to lattices in products of trees, so that for example Burger--Mozes groups have exactly two non-elementary actions on a hyperbolic space, up to a natural equivalence.
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