Zig-zag chains and metric equivalences between ultrametric spaces

2009/09/01 by Álvaro Martínez-Pérez, Martínez-Pérez, Álvaro
Mathematics · #18B30 (Primary) #37F20 #54E35 (Secondary) #Advanced Topics in Algebra #Advanced Topology and Set Theory #Combinatorics (math.CO) #FOS: Mathematics #Fixed Point Theorems Analysis #Geometric Topology (math.GT) #math.CO #math.GT #msc:18B30 #msc:37F20 #msc:54E35

paper · pdf · doi:10.48550/arxiv.0909.0153

16 pages

arxiv created 2009/09/01 · openalex publication_date 2009/09/01 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/03

Abstract

We study the classification of ultrametric spaces based on their small scale geometry (uniform homeomorphism), large scale geometry (coarse equivalence) and both (all scale uniform equivalences). We prove that these equivalences can be characterized with parallel constructions using a combinatoric tool called common zig-zag chain.

Citations

• The coarse classification of homogeneous ultra-metric spaces
• Inverse sequences, rooted trees and their end spaces

Related

• An introduction to the geometry of ultrametric spaces
• Coarse structure of ultrametric spaces with applications
• Ultrametrics and complete multipartite graphs
• Combinatorial properties of ultrametrics and generalized ultrametrics
• Forbidden Four Cycle, Star Graphs and Isometric Embeddings