Type of Document Dissertation Author Repp, Andrew S. Author's Email Address arepp@math.vt.edu URN etd-41398-14113 Title Discrete Riemann Maps and the Parabolicity of Tilings Degree PhD Department Mathematics Advisory Committee

Advisor Name Title Floyd, William J. Committee Chair Haskell, Peter E. Committee Member Linnell, Peter A. Committee Member McCoy, Robert A. Committee Member Thomson, James E. Committee Member Keywords

- Tilings
- Parabolic
- Modulus
- Riemann Map
Date of Defense 1998-05-04 Availability unrestricted AbstractThe classical Riemann Mapping Theorem has many discreteanalogues. 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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