Accessibility, planar graphs, and quasi-isometries
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866918497963999232 |
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| author | MacManus, Joseph Paul |
| author_facet | MacManus, Joseph Paul |
| contents | We prove that a connected, locally finite, quasi-transitive graph which is quasi-isometric to a planar graph is necessarily accessible. This leads to a complete classification of the finitely generated groups which are quasi-isometric to planar graphs. In particular, such a group is virtually a free product of free and surface groups, and thus virtually admits a planar Cayley graph. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2310_15242 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Accessibility, planar graphs, and quasi-isometries MacManus, Joseph Paul Group Theory Combinatorics Metric Geometry 20F65 (Primary) 05C10, 20E08, 51F30 (Secondary) We prove that a connected, locally finite, quasi-transitive graph which is quasi-isometric to a planar graph is necessarily accessible. This leads to a complete classification of the finitely generated groups which are quasi-isometric to planar graphs. In particular, such a group is virtually a free product of free and surface groups, and thus virtually admits a planar Cayley graph. |
| title | Accessibility, planar graphs, and quasi-isometries |
| topic | Group Theory Combinatorics Metric Geometry 20F65 (Primary) 05C10, 20E08, 51F30 (Secondary) |
| url | https://arxiv.org/abs/2310.15242 |