Nonplanar graphs in boundaries of CAT(0) groups

TL;DR

This paper explores the visual boundaries of universal covers of locally CAT(0) complexes, demonstrating the existence of nonplanar graphs in some boundaries and establishing conditions for graph embeddings across different metrics.

Contribution

It constructs examples of locally CAT(0) complexes with differing boundary properties and proves embedding invariance for finite graphs across metrics on the same complex.

Findings

Existence of nonplanar graphs in the visual boundary of some universal covers.

Embedding of finite graphs in boundaries is invariant under different metrics on the same complex.

Constructed examples contrast previous results on boundary homeomorphism types.

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.