Discrete Riemann Maps and the Parabolicity of Tilings

The classical Riemann Mapping Theorem has many discrete analogues. One of these, the Finite Riemann Mapping Theorem of Cannon, Floyd, Parry, and others, describes finite tilings of quadrilaterals and annuli. It relates to several combinatorial moduli, similar in nature to the classical modulus. The first chapter surveys some of these discrete analogues. The next chapter considers appropriate extensions to infinite tilings of half-open quadrilaterals and annuli. In this chapter we prove some results about combinatorial moduli for such tilings. The final chapter considers triangulations of open topological disks. It has been shown that one can classify such triangulations as either parabolic or hyperbolic, depending on whether an associated combinatorial modulus is infinite or finite. We obtain a criterion for parabolicity in terms of the degrees of vertices that lie within a specified distance of a given base vertex.