Constructing presentations of subgroups of right-angled Artin groups

Let $G$ be the right-angled Artin group associated to the flag complex $\Sigma$ and let $\pi:G\to\Z$ be its canonical height function. We investigate the presentation theory of the groups $\Gamma_n=\pi^{-1}(n\Z)$ and construct an algorithm that, given $n$ and $\Sigma$, outputs a presentation of optimal deficiency on a minimal generating set, provided $\Sigma$ is triangle-free; the deficiency tends to infinity as $n\to\infty$ if and only if the corresponding Bestvina-Brady kernel $\bigcap_n\Gamma_n$ is not finitely presented, and the algorithm detects whether this is the case. We explain why there cannot exist an algorithm that constructs finite presentations with these properties in the absence of the triangle-free hypothesis. We explore what is possible in the general case, describing how to use the configuration of 2-simplices in $\Sigma$ to simplify presentations and giving conditions on $\Sigma$ that ensure that the deficiency goes to infinity with $n$. We also prove, for general $\Sigma$, that the abelianized deficiency of $\Gamma_n$ tends to infinity if and only if $\Sigma$ is 1-acyclic, and discuss connections with the relation gap problem.