On the Structure of Vertex Stabilizers in Vertex Transitive Graphs

Lecturer: Pablo Spiga, University of Milano-Bicocca, Italy

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.