Assouad-Nagata dimension of minor-closed metrics

08/23/2023

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by   Chun-Hung Liu, et al.

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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 Riemannian surfaces with finite Euler genus) and their corollaries.