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.
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.