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Right-angled Artin groups and the cohomology basis graph
- Part of
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- Journal:
- Proceedings of the Edinburgh Mathematical Society , First View
- Published online by Cambridge University Press:
- 03 February 2025, pp. 1-21
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- Article
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Let Γ be a finite graph and let
$A(\Gamma)$ be the corresponding right-angled Artin group. From an arbitrary basis 
$\mathcal B$ of 
$H^1(A(\Gamma),\mathbb F)$ over an arbitrary field, we construct a natural graph 
$\Gamma_{\mathcal B}$ from the cup product, called the cohomology basis graph. We show that 
$\Gamma_{\mathcal B}$ always contains Γ as a subgraph. This provides an effective way to reconstruct the defining graph Γ from the cohomology of 
$A(\Gamma)$, to characterize the planarity of the defining graph from the algebra of 
$A(\Gamma)$ and to recover many other natural graph-theoretic invariants. We also investigate the behaviour of the cohomology basis graph under passage to elementary subminors and show that it is not well-behaved under edge contraction.
