Gespeichert in:
| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2411.18465 |
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Inhaltsangabe:
- We construct a unimodular random rooted graph with maximal degree $d\geq 3$ and upper growth rate $d-1$, which does not have a growth rate. Abért, Fraczyk and Hayes showed that for a unimodular random tree, if the upper growth rate is at least $\sqrt{d-1}$, then the growth rate exists, and asked with some scepticism if this may hold for more general graphs. Our construction shows that the answer is negative. We also provide a non-hyperfinite example of a unimodular random graph with no growth rate. This may be of interest in light of a conjecture of Abért that unimodular Riemannian surfaces of bounded negative curvature always have growth.