On Infinite, Cubic, Vertex-Transitive Graphs with Applications to Totally Disconnected, Locally Compact Groups

We study groups acting vertex-transitively and non-discretely on connected, cubic graphs (regular graphs of degree 3). Using ideas from Tutte's fundamental papers in 1947 and 1959, it is shown that if the action is edge-transitive, then the graph has to be a tree. When the action is not edge-transitive Tutte's ideas are still useful and can, amongst other things, be used to fully classify the possible two-ended graphs. Results about cubic graphs are then applied to Willis' scale function from the theory of totally disconnected, locally compact groups. Some of the results in this paper have most likely been known to experts but most of them are not stated explicitly with proofs in the literature.