Virtual boundaries of Hadamard spaces with admissible actions of higher rank

Abstract

Any discrete action of a group on a locally compact Hadamard space extends to a topological action on the virtual boundary. Croke and Kleiner introduced a class of so-called admissible actions and associated geometric data which determine the topological conjugacy class of the boundary action. They also posed the question whether their results hold for a wider class of actions. We show that, for the natural generalization, their question has to be answered in the negative: There is an admissible action of higher rank on a pair of Hadamard spaces with equivalent geometric data and an equivariant quasi-isometry which does not extend continuously to the virtual boundaries.