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.

中文翻译：

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CAT（0）组边界中的非平面图

Croke和Kleiner构造了两个同胚局部CAT（0）复合体，它们的通用封面具有非同胚视觉边界。我们构造了两个同胚局部CAT（0）配合物，因此一个通用封面的视觉边界包含一个非平面图，而另一个则没有。相比之下，我们证明了对于Croke-Kleiner复数上的任何两个局部CAT（0）度量，如果将有限图嵌入一个通用封面的视觉边界中，则该图也将嵌入另一个通用覆盖物的视觉边界中。