Vertex-Transitive Graphs and Accessibility

Analogues of Cayley graphs for topological groups

“…Structure trees. The fundamental results behind the theory of structure trees are from the book by Dicks and Dunwoody [6,Chapter II], but the connections and uses of this theory to study infinite graphs and group actions on such graphs are developed in [34] and [53]. The survey paper [36] gives an overview of this technique and the main results on group actions on graphs with infinitely many ends.…”

“…This construction is first described in [8, Theorem 2.1]. It is also treated in [6, Section II.1], [28], [36] and [53]. The reader is referred to those references for more details and proofs.…”

“…We use this result to prove that the stabilizer of a vertex in a structure tree (as in Theorem 11) is compactly generated. To do so we use the following construction from [53,Section 7]. Let X be a graph and − → T a structure tree of X with respect to some tree set consisting of tight cuts (i.e.…”

“…Suppose G is a group acting transitively on X. (Note that in [53,Section 7] the group action considered is the action of the full automorphism group of X, but all the arguments hold true for any transitive group action on X.) Given a vertex α in − → T we want to produce a connected subgraph X α of X such that G α acts with finitely many orbits on X α .…”

“…In Section 3.3 we focus on the concept of accessibility. The natural translation of the definition of an accessible group into terms appropriate for our totally disconnected locally compact groups is shown to be equivalent to the graph X being accessible in the sense of [53]. In Section 3.5 we investigate the action of group elements on a rough Cayley graph and how this action can be used to divide G into three disjoint classes (ellpitic, parabolic, hyperbolic) resembling the classification of isometries in hyperbolic geometry.…”