Zig-zag chains and metric equivalences between ultrametric spaces

TL;DR

This paper introduces a combinatorial tool called common zig-zag chain to classify ultrametric spaces based on their small, large, and all-scale geometric equivalences, providing a unified framework for understanding their metric properties.

Contribution

It presents a novel combinatorial approach using zig-zag chains to characterize various types of metric equivalences between ultrametric spaces.

Findings

Zig-zag chains effectively classify ultrametric spaces by their geometric equivalences.

Unified framework for small, large, and all-scale geometric classifications.

Provides new insights into the structure of ultrametric spaces.

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.