Right-angled Artin groups and the cohomology basis graph
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| Main Authors: | , , , |
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| Format: | Preprint |
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2023
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| _version_ | 1866911937036550144 |
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| author | Flores, Ramón Kahrobaei, Delaram Koberda, Thomas Coz, Corentin Le |
| author_facet | Flores, Ramón Kahrobaei, Delaram Koberda, Thomas Coz, Corentin Le |
| contents | Let $Γ$ be a finite graph and let $A(Γ)$ be the corresponding right-angled Artin group. From an arbitrary basis $\mathcal B$ of $H^1(A(Γ),\mathbb F)$ over an arbitrary field, we construct a natural graph $Γ_{\mathcal B}$ from the cup product, called the \emph{cohomology basis graph}. We show that $Γ_{\mathcal B}$ always contains $Γ$ as a subgraph. This provides an effective way to reconstruct the defining graph $Γ$ from the cohomology of $A(Γ)$, to characterize the planarity of the defining graph from the algebra of $A(Γ)$, and to recover many other natural graph-theoretic invariants. We also investigate the behavior of the cohomology basis graph under passage to elementary subminors, and show that it is not well-behaved under edge contraction. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_05495 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Right-angled Artin groups and the cohomology basis graph Flores, Ramón Kahrobaei, Delaram Koberda, Thomas Coz, Corentin Le Group Theory Combinatorics Geometric Topology Let $Γ$ be a finite graph and let $A(Γ)$ be the corresponding right-angled Artin group. From an arbitrary basis $\mathcal B$ of $H^1(A(Γ),\mathbb F)$ over an arbitrary field, we construct a natural graph $Γ_{\mathcal B}$ from the cup product, called the \emph{cohomology basis graph}. We show that $Γ_{\mathcal B}$ always contains $Γ$ as a subgraph. This provides an effective way to reconstruct the defining graph $Γ$ from the cohomology of $A(Γ)$, to characterize the planarity of the defining graph from the algebra of $A(Γ)$, and to recover many other natural graph-theoretic invariants. We also investigate the behavior of the cohomology basis graph under passage to elementary subminors, and show that it is not well-behaved under edge contraction. |
| title | Right-angled Artin groups and the cohomology basis graph |
| topic | Group Theory Combinatorics Geometric Topology |
| url | https://arxiv.org/abs/2309.05495 |