Arc-transitive abelian regular covers of cubic graphs

Quite a lot of attention has been paid recently to the construction of edge- or arc-transitive covers of symmetric graphs. In most cases, the approach has involved voltage graph techniques, which are excellent for finding regular covers in which the group of covering transformations is either cyclic or elementary abelian, or more generally, homocyclic, but are not so easy to use when the covering group has other forms — even when it is abelian but not homocyclic. In this paper, a different approach is introduced that can be used more widely. This new approach takes a universal group for the action of the automorphism group of the base graph, and uses Reidemeister–Schreier theory to obtain a presentation for a ‘universal covering group’, and some representation theory and other methods for determining suitable quotients. This approach is then used to find all arc-transitive abelian regular covers of K4, K3,3, the cube Q3, and the Petersen graph. A sequel will do the same for the Heawood graph.