Virtual boundaries of Hadamard spaces with admissible actions of higher rank (Q329952)

A Hadamard space is a complete and simply connected length space of nonpositive curvature in the sense of Alexandrov. The metric of a Hadamard space is convex. That is why it is uniquely geodesic by the Cartan-Hadamard theorem. \textit{C. B. Croke} and \textit{B. Kleiner} [Geom. Funct. Anal. 12, No. 3, 479--545 (2002; Zbl 1035.53116)] have introduced the class of admissible actions and associated geometric data on the virtual boundary of a locally compact Hadamard space. They also have posed a problem of extending of their results. The author presents the following result that is a negative answer to a question of Croke and Kleiner. Theorem. There are admissible actions of rank 3 on a pair of Hadamard spaces with equivalent generalized geometric data and an equivariant quasi-isometry which does not extend continuously to the virtual boundary.