Coarse structure of ultrametric spaces with applications

TL;DR

This paper introduces a novel decomposition method for ultrametric spaces using metric resolutions, enabling the construction of universal spaces for certain classes of large scale metric spaces, with applications to countable groups.

Contribution

It develops the concept of metric resolutions and coarse disjoint unions to construct universal spaces for separable and proper ultrametric spaces of asymptotic dimension zero, extending previous results.

Findings

Decomposition of ultrametric spaces into scaled simplices

Construction of universal spaces for asymptotic dimension zero

Application to certain countable groups

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.