H.Servatius Connectivity and Planarity of Cayley graphs

Abstract. The question of which groups admit planar Cayley graphs goes back over 100 years, having been settled for finite groups by Maschke in 1896. Since that time, various authors have studied infinite planar Cayley graphs which satisfy additional special conditions. We consider the question of which groups possess any planar Cayley graphs at all by categorizing such graphs according to their connectivity. Like planarity, connectivity is a fundamental concept in graph theory. We show that having a Cayley graph which is less than three–connected has strong implications for the structure of the group. In the planar case, the decomposition imposed by low connectivity allows us to reduce the problem to the case where the Cayley graph is three–connected, where geometric techniques can be employed. 1. Introduction. Given a discrete group of isometries of the sphere, Euclidean plane or hyperbolic plane, one can construct a fundamental region whose translates tessellate the space. The dual of such a configuration is in fact a (modified) Cayley graph for the group. Figure 1 illustrates thi