Coarse structure of ultrametric spaces with applications

Yuankui Ma, Jeremy Siegert, Jerzy Dydak

Research output: Contribution to journalArticlepeer-review

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 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 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 Nagórko 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.

Original languageEnglish
Article number5
JournalEuropean Journal of Mathematics
Volume9
Issue number1
DOIs
StatePublished - 1 Mar 2023

Keywords

  • Asymptotic dimension
  • Coarse geometry
  • Ultrametric spaces
  • Universal spaces

ASJC Scopus subject areas

  • General Mathematics

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