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Title page for ETD etd-03272006-143429

Type of Document Dissertation
Author Mirani, Mozhgan
Author's Email Address mozhgan.mirani@vanderbilt.edu
URN etd-03272006-143429
Title Classical Trees and Ultrametric Spaces
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Bruce Hughes Committee Chair
Guoliang Yu Committee Member
John G. Ratcliffe Committee Member
Michael L. Mihalik Committee Member
Thomas W. Kephart Committee Member
  • ultrametric spaces
  • quasi-conformal homeomorphisms
  • Topology
  • category
Date of Defense 2006-03-01
Availability unrestricted
In this paper it is established that there is a faithful functor E from the category T whose objects are locally finite classical trees of minimal vertex degree three and whose morphisms are classes of quasi-isometries to the category U whose objects are perfect compact ultrametric spaces and whose morphisms are bi-Hölder homeomorphisms.

The image of morphisms under E are also quasi-conformal. If two quasi-conformal homeomorphisms are images of a morphisms under the functor E, their composition is also a quasi-conformal homeomorphism.

It is not known in more general cases

exactly when quasi-conformal homeomorphisms are closed under composition. Quasi-conformal homeomorphisms are studied in great depth and numerous examples of quasi-conformal homeomorphisms are given. Examples are also provided that show that compositions of quasi-conformal homeomorphisms need not be quasi-conformal.

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