A Cayley graph for $F_{2}\times F_{2}$ which is not minimally almost convex
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arXiv
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| Format: | Preprint |
| Publié: |
2016
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| _version_ | 1866910473291563008 |
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| author | Price, Andrew Elvey |
| author_facet | Price, Andrew Elvey |
| contents | We give an example of a Cayley graph $Γ$ for the group $F_{2}\times F_{2}$ which is not minimally almost convex (MAC). On the other hand, the standard Cayley graph for $F_{2}\times F_{2}$ does satisfy the falsification by fellow traveler property (FFTP), which is strictly stronger. As a result, any Cayley graph property $K$ lying between FFTP and MAC (i.e., $\text{FFTP}\Rightarrow K\Rightarrow\text{MAC}$) is dependent on the generating set. This includes the well known properties FFTP and almost convexity, which were already known to depend on the generating set as well as Poénaru's condition $P(2)$ and the basepoint loop shortening property for which dependence on the generating set was previously unknown. We also show that the Cayley graph $Γ$ does not have the loop shortening property, so this property also depends on the generating set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1611_00101 |
| institution | arXiv |
| publishDate | 2016 |
| record_format | arxiv |
| spellingShingle | A Cayley graph for $F_{2}\times F_{2}$ which is not minimally almost convex Price, Andrew Elvey Group Theory We give an example of a Cayley graph $Γ$ for the group $F_{2}\times F_{2}$ which is not minimally almost convex (MAC). On the other hand, the standard Cayley graph for $F_{2}\times F_{2}$ does satisfy the falsification by fellow traveler property (FFTP), which is strictly stronger. As a result, any Cayley graph property $K$ lying between FFTP and MAC (i.e., $\text{FFTP}\Rightarrow K\Rightarrow\text{MAC}$) is dependent on the generating set. This includes the well known properties FFTP and almost convexity, which were already known to depend on the generating set as well as Poénaru's condition $P(2)$ and the basepoint loop shortening property for which dependence on the generating set was previously unknown. We also show that the Cayley graph $Γ$ does not have the loop shortening property, so this property also depends on the generating set. |
| title | A Cayley graph for $F_{2}\times F_{2}$ which is not minimally almost convex |
| topic | Group Theory |
| url | https://arxiv.org/abs/1611.00101 |