Dimension zero at all scales
by
Atish Mitra
University Of Tennessee - Knoxville
Coauthors: N. Brodskiy, J. Dydak, J. Highes

We consider the notion of dimension in two categories: the category of separable metric spaces and Lipschitz maps, and the category of separable metric spaces and uniform maps. A uniform treatment is given to the large scale dimension and the small scale dimension. We show that in both categories a space has dimension zero if and only if it is equivalent to an ultrametric space. There is a universal zero-dimensional space in both categories. Spaces of dimension zero are characterized by means of extensions of maps to the unit 0-sphere and by means of retractions to subspaces. Any countable group of asymptotic dimension zero is coarsely equivalent to a direct sum of cyclic groups. We construct uncountably many examples of coarsely inequivalent ultrametric spaces.

Date received: February 28, 2006