Boundaries Of Nonpositively Curved Groups Of The Form G x Z^n
P. Bowers, K. Ruane
We prove that whenever G is negatively curved and \Gamma=G x Z^n acts geometrically on two CAT(0)-spaces X and X', their visual boundaries are \Gamma-equivariantly homeomorphic, answering affirmatively a question of Gromov in this special case. However, we give a simple example that shows that the natural \Gamma-equivariant quasi-isometry from X to X' does not continuously extend to a homeomorphism of visual boundaries, a situation that does occur if \Gamma is negatively curved. We show further that the \Gamma-rational endpoints of X form a dense subset of the visual boundary, generalizing the analogous result that holds in the context of negative curvature.