Assouad-Nagata dimension of minor-closed metrics
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866915203835232256 |
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| author | Liu, Chun-Hung |
| author_facet | Liu, Chun-Hung |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2308_12273 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Assouad-Nagata dimension of minor-closed metrics Liu, Chun-Hung Combinatorics Discrete Mathematics Group Theory Geometric Topology Metric Geometry 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. |
| title | Assouad-Nagata dimension of minor-closed metrics |
| topic | Combinatorics Discrete Mathematics Group Theory Geometric Topology Metric Geometry |
| url | https://arxiv.org/abs/2308.12273 |