Assouad–Nagata dimension of minor‐closed metrics

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 is a minor‐closed metric if there exists an (edge‐)weighted graph satisfying a fixed minor‐closed property such that the underlying space of is the vertex‐set of , and the metric of is the distance function in . 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 ‐minor free unweighted graphs and about the Assouad–Nagata dimension of complete Riemannian surfaces with finite Euler genus (Bonamy et al., Asymptotic dimension of minor‐closed families and Assouad–Nagata dimension of surfaces, JEMS (2024)).