Title page for ETD etd-41398-14113

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
  • Tilings
  • Parabolic
  • Modulus
  • Riemann Map
Date of Defense 1998-05-04
Availability unrestricted
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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