Classical Trees and Ultrametric Spaces

Abstract

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.