Graphs and groups with unique geodesics
Fuente:
arXiv
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| Autori principali: | , , , , |
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866909427591806976 |
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| author | Elder, Murray Gardam, Giles Piggott, Adam Spriano, Davide Townsend, Kane |
| author_facet | Elder, Murray Gardam, Giles Piggott, Adam Spriano, Davide Townsend, Kane |
| contents | A connected graph is called \emph{geodetic} if there is a unique geodesic between each pair of vertices. In this paper we prove that if a finitely generated group admits a Cayley graph which is geodetic, then the group must be virtually free. Before now, it was open whether finitely generated and geodetic implied hyperbolic. In fact we prove something more general: if a quasi-transitive locally finite connected undirected graph is geodetic then it is quasi-isometric to a tree. Our main tool is to define a \emph{boundary} of a graph and understand how the local behaviour influences it when the graph is geodetic. Our results unify, and represent significant progress on, research initiated by Ore, Shapiro, and Madlener and Otto. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2311_03730 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Graphs and groups with unique geodesics Elder, Murray Gardam, Giles Piggott, Adam Spriano, Davide Townsend, Kane Group Theory Combinatorics 05C75, 05C12, 20F65, 20F67, 68Q42 A connected graph is called \emph{geodetic} if there is a unique geodesic between each pair of vertices. In this paper we prove that if a finitely generated group admits a Cayley graph which is geodetic, then the group must be virtually free. Before now, it was open whether finitely generated and geodetic implied hyperbolic. In fact we prove something more general: if a quasi-transitive locally finite connected undirected graph is geodetic then it is quasi-isometric to a tree. Our main tool is to define a \emph{boundary} of a graph and understand how the local behaviour influences it when the graph is geodetic. Our results unify, and represent significant progress on, research initiated by Ore, Shapiro, and Madlener and Otto. |
| title | Graphs and groups with unique geodesics |
| topic | Group Theory Combinatorics 05C75, 05C12, 20F65, 20F67, 68Q42 |
| url | https://arxiv.org/abs/2311.03730 |