Vertex-Transitive Graphs and Accessibility

We call an infinite graph G accessible if there exists a natural number k such that any two ends of G can be separated by k edges. C. T. C. Wall′s accessibility conjecture for finitely generated groups has a simple and attractive graph version: Every locally finite Cayley graph is accessible. Wall′s conjecture has recently been disproved by M.J. Dunwoody. In this paper we show that all locally finite, 2-transitive graphs and all 1-transitive graphs of prime degree are accessible. We prove that every locally finite, vertex-transitive graph with at least one thick end has a thick end with a 2-way infinite geodesic, while no thin end has a 2-way infinite geodesic. Furthermore, those ends in a locally finite, accessible vertex-transitive graph which have a 2-way infinite geodesic are precisely the thick ends. In addition, there are only finitely many non-isomorphic thick ends. We obtain these and other results from a precise description of the end structure of every locally finite, accessible, vertex-transitive graph. We also investigate the ends in inaccessible vertex-transitive graphs.