Nonplanar graphs in boundaries of CAT(0) groups

Abstract

Croke and Kleiner constructed two homeomorphic locally CAT(0) complexes whose universal covers have visual boundaries that are not homeomorphic. We construct two homeomorphic locally CAT(0) complexes so that the visual boundary of one universal cover contains a nonplanar graph, while the visual boundary of the other does not. In contrast, we prove for any two locally CAT(0) metrics on the Croke–Kleiner complex, if a finite graph embeds in the visual boundary of one universal cover, then the graph embeds in the visual boundary of the other.