Assouad-Nagata dimension of minor-closed metrics

TL;DR

This work settles the Assouad-Nagata dimension for minor-closed metric families generated by weighted graphs, revealing sharp dichotomies tied to minor-exclusion: the AN-dimension is infinite only if all finite graphs appear as minors, and it drops to 2, 1, or 0 depending on the absence of particular finite graphs. The authors develop a constructive framework that reduces AN-dimension questions to weak-diameter colorings via r-th powers and to tree-decompositions with gadgets, near embeddings, and hierarchical condensations. They prove that for any fixed finite graph H, the class of $H$-minor-free weighted graphs has AN-dimension at most 2, and planar-H cases attain AN-dimension at most 1; these results extend known asymptotic-dimension bounds and recover planar-genus cases with a unified approach. The findings yield algorithmic implications, offering polynomial-time procedures for computing the witness colorings and establishing weak-sparse partition schemes, with consequences for groups, continuous spaces, and graph classes embeddable on surfaces. Overall, the paper provides a comprehensive, constructive resolution of the AN-dimension for minor-closed metric families and connects discrete and continuous coarse-geometric phenomena through a common structural framework.

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.

Figures (2)

• Figure 1: An example of the $(8,\frac{1}{4},3,1)$-heirachy $(H,\phi_H,I,B)$ of ${\mathcal{P}}=({\mathcal{P}}_i: i \in {\mathbb N})$, where $X=[6]$, ${\mathcal{P}}_0=\{\{x\}: x \in [6]\}$, ${\mathcal{P}}_1={\mathcal{P}}_2 = \{\{1\},\{2,3\},\{4\},\{5\},\{6\}\}$, ${\mathcal{P}}_i = \{\{1,4,5\},\{2,3,6\}\}$ for every $i \geq 3$. Note that $I=\{0,1,3\}$, $H$ is the graph in the picture, and $B$ is the set of the six vertices at the bottom.
• Figure 2: An example of the $(U_E,U_E',\ell,\theta,\mu)$-condensation of $(G,\phi,T,{\mathcal{X}})$, where $U_E=\{e_1,e_2\}$ and $U_E'=\{e_1\}$.

Theorems & Definitions (44)

• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Corollary 1.6
• Theorem 1.7
• Theorem 1.8
• Lemma 1.9: bbeglps_merged
• Lemma 3.1
• Lemma 3.2