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Graphs and groups with tree-like properties. (English) Zbl 0696.05027
The author investigates conditions under which the Cayley graph of a finitely generated, infinite group looks like a tree. For a locally finite, infinite graph, three different notions of “metric” type are introduced to capture “tree-like”. The graph may have (1) a “uniformly spanning tree”, (2) a certain triangulation property, or (3) all its ends of finite diameter. After studying the interrelations among these properties for arbitrary graphs, the author proves that all three are equivalent for vertex-transitive graphs. A Cayley graph has one of these properties if and only if it arises from a finite extension of a free group.
Reviewer: St.Althoen

MSC:
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
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