2022/03/10 by Yuankui Ma, Ma, Yuankui, Jeremy Siegert +3 · 1 citation
Mathematics · #Advanced Topology and Set Theory #FOS: Mathematics #Geometric Topology (math.GT) #Mathematical Dynamics and Fractals #Metric Geometry (math.MG) #Primary 54D35 #Secondary 20F69 #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2203.05329
openalex publication_date 2022/03/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 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.