Assouad-Nagata dimension of minor-closed metrics

TL;DR

This paper determines the Assouad-Nagata dimension for all minor-closed metric spaces, unifying results from graph theory and Riemannian geometry to deepen understanding of metric space complexity.

Contribution

It provides a comprehensive characterization of the Assouad-Nagata dimension for minor-closed metrics, extending previous results in graph minors and Riemannian surfaces.

Findings

Unified theorems for Assouad-Nagata dimension of minor-closed metrics

Generalized known results for $H$-minor free graphs

Extended understanding to Riemannian surfaces with finite Euler genus

Abstract

Assouad-Nagata dimension addresses both large and small scale behaviors of metric spaces and is a refinement of Gromov's asymptotic dimension. A metric space $M$ is a minor-closed metric if there exists an (edge-)weighted graph $G$ satisfying a fixed minor-closed property such that the underlying space of $M$ is the vertex-set of $G$, and the metric of $M$ is the distance function in $G$. Minor-closed metrics naturally arise when removing redundant edges of the underlying graphs by using edge-deletion and edge-contraction. In this paper, we determine the Assouad-Nagata dimension of every minor-closed metric. Our main theorem simultaneously generalizes known results about the asymptotic dimension of $H$-minor free unweighted graphs and about the Assouad-Nagata dimension of complete Riemannian surfaces with finite Euler genus.