Lifting a prescribed group of automorphisms of graphs

Original · EN

In this paper we are interested in lifting a prescribed group of automorphisms of a finite graph via regular covering projections. Here we describe with an example the problems we address and refer to the introductory section for the correct statements of our results. Let P be the Petersen graph, say, and let:P→ P be a regular covering projection. With the current covering machinery, it is straightforward to find with the property that every subgroup of (P) lifts via. However, for constructing peculiar examples and in applications, this is usually not enough. Sometimes it is important, given a subgroup G of (P), to find along which G lifts but no further automorphism of P does. For instance, in this concrete example, it is interesting to find a covering of the Petersen graph lifting the alternating group A₅ but not the whole symmetric group S₅. (Recall that (P) S₅.) Some other time it is important, given a subgroup G of (P), to find with the property that (P) is the lift of G. Typically, it is desirable to find satisfying both conditions. In a very broad sense, this might remind wallpaper patterns on surfaces: the group of symmetries of the dodecahedron is S₅, and there is a nice colouring of the dodecahedron (found also by Escher) whose group of symmetries is just A₅. In this paper, we address this problem in full generality.