Groups acting on locally finite graphs - a survey of the infinitely ended case  pp. 426-456

Groups acting on locally finite graphs - a survey of the infinitely ended case

By R G Möller

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The study of infinite graphs has many aspects and various connections with other fields. There are the classical graph theoretic problems in infinite settings (see the survey by Thomassen [49]); there are special graph theoretical questions which have no direct analogues for finite graphs, such as questions about ends (see [7], [44] and the monograph [6]); Ramsey graph theory with its connections to set theory; the study of spectra of infinite graphs and random walks on infinite graphs (see the surveys [32] and [58]); the study of group actions on infinite graphs.

This survey is on the last subject, or rather on a small corner of the last subject. As is usual one concentrates on the case where the automorphism group acts transitively on the graph. The study of group actions can then be spilt up into three cases according to whether the graph under investigation has one, two or infinitely many ends. A graph has one end if there is always just one infinite component when finitely many vertices are removed from the graph. (“Component” will always mean a connected component in the graph theoretical sense.) The case of graphs with only one end is the hardest one, but in the special case of graphs with polynomial growth there are some very nice results (see [23]). The two ended case is the easiest one: roughly speaking these graphs all look like fat lines and one can say that they are very well understood (see [29] and [22]). Then there is the infinitely ended case, which is the one that this paper is all about.