Right-angled Artin groups and the cohomology basis graph

TL;DR

This paper introduces the cohomology basis graph for right-angled Artin groups, enabling the reconstruction of the defining graph from cohomology and revealing connections between algebraic and graph-theoretic properties.

Contribution

It constructs the cohomology basis graph from an arbitrary basis of first cohomology, linking algebraic structures to the original graph and its invariants.

Findings

The cohomology basis graph always contains the original graph as a subgraph.

It provides a method to reconstruct the defining graph from cohomology.

The cohomology basis graph is not well-behaved under edge contraction.

Abstract

Let $\Gamma$ 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 \emph{cohomology basis graph}. We show that $\Gamma_{\mathcal B}$ always contains $\Gamma$ as a subgraph. This provides an effective way to reconstruct the defining graph $\Gamma$ 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 behavior of the cohomology basis graph under passage to elementary subminors, and show that it is not well-behaved under edge contraction.