The asymptotic dimension of the grand arc graph is infinite
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arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866915432807530496 |
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| author | Kopreski, Michael C. |
| author_facet | Kopreski, Michael C. |
| contents | Let $Σ$ be a compact, orientable surface of genus $g$, and let $Γ$ be a relation on $π_0(\partial Σ)$ such that the prescribed arc graph $\mathcal{A}(Σ,Γ)$ is Gromov-hyperbolic and non-trivial. We show that $\operatorname{asdim} \mathcal{A}(Σ,Γ) \geq -χ(Σ) - 1$, from which we prove that the asymptotic dimension of the grand arc graph is infinite. More generally, an arc and curve model on $Σ$ is a graph of simple arc and curves on $Σ$, on which $\operatorname{PMap}(Σ)$ acts by permuting vertices. We prove that any connected, Gromov-hyperbolic cocompact arc and curve model $\mathcal{M}$ has $\operatorname{asdim} \mathcal{M} \geq g - \lceil\frac{1}{2} χ(Σ)\rceil$, and that a broad class of arc and curve models on infinite-type surfaces has infinite asymptotic dimension. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_03603 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The asymptotic dimension of the grand arc graph is infinite Kopreski, Michael C. Geometric Topology Let $Σ$ be a compact, orientable surface of genus $g$, and let $Γ$ be a relation on $π_0(\partial Σ)$ such that the prescribed arc graph $\mathcal{A}(Σ,Γ)$ is Gromov-hyperbolic and non-trivial. We show that $\operatorname{asdim} \mathcal{A}(Σ,Γ) \geq -χ(Σ) - 1$, from which we prove that the asymptotic dimension of the grand arc graph is infinite. More generally, an arc and curve model on $Σ$ is a graph of simple arc and curves on $Σ$, on which $\operatorname{PMap}(Σ)$ acts by permuting vertices. We prove that any connected, Gromov-hyperbolic cocompact arc and curve model $\mathcal{M}$ has $\operatorname{asdim} \mathcal{M} \geq g - \lceil\frac{1}{2} χ(Σ)\rceil$, and that a broad class of arc and curve models on infinite-type surfaces has infinite asymptotic dimension. |
| title | The asymptotic dimension of the grand arc graph is infinite |
| topic | Geometric Topology |
| url | https://arxiv.org/abs/2402.03603 |