Discrete Riemann Maps and the Parabolicity of Tilings
Abstract
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.
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- Doctoral Dissertations [13025]