Parabolic and Hyperbolic Packings

In this chapter we discuss countably infinite connected simple graphs that are locally finite, that is, the vertex degrees are finite. In a similar fashion to the previous chapter, an infinite planar graph is a connected infinite graph such that there exists a drawing of it in the plane. We recall that a drawing is a correspondence sending vertices to points of ℝ² and edges to continuous curves between the corresponding vertices such that no two edges cross. An infinite planar map is an infinite planar graph equipped with a set of cyclic permutations {σ_v: v ∈ V } of the neighbors of each vertex v, such that there exists a drawing of the graph which respects these permutations, that is, the clockwise order of edges emanating from a vertex v coincides with σ_v.