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Right-angled Artin groups and the cohomology basis graph

Part of: Graph theory Special aspects of infinite or finite groups Basic linear algebra

Published online by Cambridge University Press:  03 February 2025

Ramón Flores ,
Delaram Kahrobaei ,
Thomas Koberda Open the ORCID record for Thomas Koberda [Opens in a new window]  and
Corentin Le Coz
Ramón Flores*
Affiliation:
Department of Geometry and Topology, University of Seville, Seville, Spain
Delaram Kahrobaei
Affiliation:
Departments of Mathematics and Computer Science, Queens College, CUNY, New York, USA Department of Computer Science, University of York, York, UK Department of Computer Science and Engineering, New York University, Tandon School of Engineering, New York, USA The Initiative for the Theoretical Sciences, CUNY Graduate Center, New York, USA
Thomas Koberda
Affiliation:
Department of Mathematics, University of Virginia, Charlottesville, VA, USA
Corentin Le Coz
Affiliation:
Department of Mathematics: Algebra and Geometry (WE01), Universiteit Gent, Ghent, Belgium
*
Corresponding author: Ramon Flores, email: ramonjflores@us.es
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Abstract

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.

Keywords

determinantcohomologyminorright-angled Artin groupplanar graphouterplanar graph

MSC classification

Primary: 15A15: Determinants, permanents, other special matrix functions
Secondary: 20F36: Braid groups; Artin groups 05C10: Planar graphs; geometric and topological aspects of graph theory 05C83: Graph minors
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , First View , pp. 1 - 21
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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