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
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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