On the Structure of Vertex Stabilizers in Vertex Transitive Graphsnatisni
Abstract: In these lectures I would love to give some insight into the structure of vertex stabilisers of groups of automorphisms of (not necessarily finite) connected vertex-transitive graphs, that is, graphs admitting a group of automorphisms acting transitively on the vertices of the graph. This area of research has rich history that goes at least as far back as the 1949 pioneering paper of Tutte, and the astonishing work of (for instance) Goldschmidt and Weiss in the late 1960s. Lectures will mainly focus on two classes of graphs: graphs of small valency (3 and 4), and graphs having a group of automorphisms acting vigorously (the stabiliser of a vertex acts primitively on the neighbourhood of a vertex). Recently, this theory has been used to obtain a complete census of cubic vertex-transitive graphs on up to 1280 vertices. As an application of the theory of vertex stabilisers, we describe in some detail how to obtain this census and hopefully how to improve it in the near future.