Virtual boundaries of Hadamard spaces with admissible actions of higher rank

TL;DR

This paper investigates the extension of group actions on Hadamard spaces to their boundaries, showing that certain generalized admissible actions of higher rank do not extend continuously, unlike previously understood cases.

Contribution

It demonstrates that for a broader class of admissible actions of higher rank, the boundary extension property fails, answering a question posed by Croke and Kleiner in the negative.

Findings

Not all admissible higher rank actions extend to boundary homeomorphisms.

Equivalent geometric data do not guarantee boundary extension.

Counterexamples exist with equivariant quasi-isometries not extending continuously.

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.