Work

Assouad–Nagata dimension of minor‐closed metrics

Year: 2025

Type: article

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)).

Source: Proceedings of the London Mathematical Society

DOI: 10.1112/plms.70032

Author: Chun‐Hung Liu

Institution: Texas A&M University

Language: English

Topic: Geometric Analysis and Curvature Flows

Subfield: Applied Mathematics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $4,700

Funder: National Science Foundation