Coarse geometry of quasi-transitive graphs beyond planarity
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909654801448960 |
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| author | Esperet, Louis Giocanti, Ugo |
| author_facet | Esperet, Louis Giocanti, Ugo |
| contents | We study geometric and topological properties of infinite graphs that are quasi-isometric to a planar graph of bounded degree. We prove that every locally finite quasi-transitive graph excluding a minor is quasi-isometric to a planar graph of bounded degree. We use the result to give a simple proof of the result that finitely generated minor-excluded groups have Assouad-Nagata dimension at most 2 (this is known to hold in greater generality, but all known proofs use significantly deeper tools). We also prove that every locally finite quasi-transitive graph that is quasi-isometric to a planar graph is $k$-planar for some $k$ (i.e. it has a planar drawing with at most $k$ crossings per edge), and discuss a possible approach to prove the converse statement. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2312_08902 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Coarse geometry of quasi-transitive graphs beyond planarity Esperet, Louis Giocanti, Ugo Combinatorics Metric Geometry We study geometric and topological properties of infinite graphs that are quasi-isometric to a planar graph of bounded degree. We prove that every locally finite quasi-transitive graph excluding a minor is quasi-isometric to a planar graph of bounded degree. We use the result to give a simple proof of the result that finitely generated minor-excluded groups have Assouad-Nagata dimension at most 2 (this is known to hold in greater generality, but all known proofs use significantly deeper tools). We also prove that every locally finite quasi-transitive graph that is quasi-isometric to a planar graph is $k$-planar for some $k$ (i.e. it has a planar drawing with at most $k$ crossings per edge), and discuss a possible approach to prove the converse statement. |
| title | Coarse geometry of quasi-transitive graphs beyond planarity |
| topic | Combinatorics Metric Geometry |
| url | https://arxiv.org/abs/2312.08902 |