Coarse structure of ultrametric spaces with applications

Abstract

We show how to decompose all separable ultrametric spaces into a "Lego" combinations of scaled versions of full simplices. To do this we introduce metric resolutions of large scale metric spaces, which describe how a space can be broken up into roughly independent pieces. We use these metric resolutions to define the coarse disjoint union of large scale metric spaces, which provides a way of attaching large scale metric spaces to each other in a "coarsely independent way". We use these notions to construct universal spaces in the categories of separable and proper metric spaces of asymptotic dimension $0$, respectively. In doing so we generalize a similar result of Dranishnikov and Zarichnyi as well as Nag\'orko and Bell. However, the new application is a universal space for proper metric spaces of asymptotic dimension $0$, something that eluded those authors. We finish with a description of some countable groups that can serve as such universal spaces.