Graphs and groups with tree-like properties

AbstractWhen does the Cayley graph of a finitely generated, infinite group look similar to a tree? For a locally finite, infinite graph, one can introduce different notions of “metric” type saying that it is tree-like: the graph may have (1) a “uniformly spanning tree,” (2) a certain triangulation property, or (3) all its ends may have finite diameters. After studying the interrelations between these properties for arbitrary graphs, it is proved that they 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