Right-angled Artin groups and the cohomology basis graph

2023/09/11 by Flores, Ramón, Kahrobaei, Delaram, Koberda, Thomas +1
#Combinatorics (math.CO) #FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)

paper · doi:10.48550/arxiv.2309.05495

Abstract

Let Γ be a finite graph and let A(Γ) be the corresponding right-angled Artin group. From an arbitrary basis \mathcal B of H¹(A(Γ),\mathbb F) over an arbitrary field, we construct a natural graph Γ_{\mathcal B} from the cup product, called the 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.

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