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. It is a common generalization of 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 and their corollaries.

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概念。度量空间$M$称为子闭度量，若存在满足固定子闭性质的（边）赋权图$G$，使得$M$的底空间为$G$的顶点集，且$M$的度量为$G$中的距离函数。子闭度量自然地出现在通过边删除和边收缩移除底图冗余边的过程中。本文确定了所有子闭度量的Assouad-Nagata维数。该结果统一推广了已知结论：关于不含$H-子式的无赋权图的渐近维数，以及关于有限欧拉亏格的完备黎曼曲面及其推论的Assouad-Nagata维数。