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

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

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.
Rigidity of group actions on hyperbolic spaces
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12 days ago
anna ▴ 30

Can theorem 1.1 be extended to the case where we replace the standard rank one groups with locally compact hyperbolic groups which act nicely on some model space? For example, the automorphism group of a hyperbolic building.

spaces hyperbolic rigidity • 16 views