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.

翻译：Assouad-Nagata维数同时处理度量空间的大尺度与小尺度行为，是Gromov渐近维数的一种精细化。若存在满足某固定小闭包性质的（边）赋权图$G$，使得度量空间$M$的底空间是$G$的顶点集，且$M$的度量是$G$中的距离函数，则称$M$为小闭包度量。通过边删除与边收缩操作去除底图中冗余边时，自然会产生小闭包度量。本文确定了所有小闭包度量的Assouad-Nagata维数。我们的主要定理同时推广了关于$H$-minor自由无权图渐近维数的已知结果，以及关于有限欧拉亏格完备黎曼曲面Assouad-Nagata维数的已有结论。