Groups acting on graphs

Abstract

In the first part of this thesis we investigate the automorphism groups of regular trees. In the second part we look at the action of the automorphism group of a locally finite graph on the ends of the graph. The two part are not directly related but trees play a fundamental role in both parts. Let Tn be the regular tree of valency n. Put G := Aut(Tn) and let G0 be the subgroup of G that is generated by the stabilisers of points. The main results of the first part are : Theorem 4.1 Suppose 3 ≤ n &lt; N0 and α ϵ Tn. Then Gα (the stabiliser of α in G) contains 22N0 subgroups of index less than 22N0. Theorem 4.2 Suppose 3 ≤ n &lt; N0 and H ≤ G with G : H |&lt; 2N0. Then H = G or H = G0 or H fixes a point or H stabilises an edge. Theorem 4.3 Let n = N0 and H ≤ G with | G : H |&lt; 2N0. Then H = G or H = G0 or there is a finite subtree ϕ of Tn such that G(ϕ) ≤ H ≤ G{ϕ}. These are proved by finding a concrete description of the stabilisers of points in G, using wreath products, and also by making use of methods and results of Dixon, Neumann and Thomas [Bull. Lond. Math. Soc. 18, 580-586]. It is also shown how one is able to get short proofs of three earlier results about the automorphism groups of regular trees by using the methods used to prove these theorems. In their book Groups acting on graphs, Warren Dicks and M. J. Dunwoody [Cambridge University Press, 1989] developed a powerful technique to construct trees from graphs. An end of a graph is an equivalence class of half-lines in the graph, with two half-lines, L1 and L2, being equivalent if and only if we can find the third half-line that contains infinitely many vertices of both L1 and L2. In the second part we point out how one can, by using this technique, reduce questions about ends of graphs to questions about trees. This allows us both to prove several new results and also to give simple proofs of some known results concerning fixed points of group actions on the ends of a locally finite graph (see Chapter 10). An example of a new result is the classification of locally finite graphs with infinitely many ends, whose automorphism group acts transitively on the set of ends (Theorem 11.1).</p