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 $G$ in a fixed minor-closed family 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. It is a common generalization of known results for the asymptotic dimension of $H$-minor free unweighted graphs and the Assouad-Nagata dimension of some 2-dimensional continuous spaces (e.g.\ complete Rienmannian surfaces with finite Euler genus) and their corollaries.

翻译：Assouad-Nagata维数同时度量度量空间的大尺度与小尺度行为，是Gromov渐近维数的精细化概念。若存在某个固定子式封闭族中的（边）加权图$G$，使得度量空间$M$的底层集合为$G$的顶点集，且$M$的度量由$G$中的距离函数定义，则称$M$为子式封闭度量空间。子式封闭度量空间自然产生于通过边删除与边收缩剔除底层图冗余边的过程中。本文确定了所有子式封闭度量空间的Assouad-Nagata维数。该结果统一推广了关于不含$H$子式的无加权图的渐近维数、若干二维连续空间（如具有有限欧拉亏格的完备黎曼曲面）的Assouad-Nagata维数及其推论等已知结论。