One-relator quotients of Partially Commutative Groups

We generalise a key result of one-relator group theory, namely Magnus's Freiheitssatz, to partially commutative groups, under sufficiently strong conditions on the relator. The main theorem shows that under our conditions, on an element $r$ of a partially commutative group $\mathbb{G}$, certain Magnus subgroups embed in the quotient $G=\mathbb{G}/N(r)$; that if $r=s^n$ has root $s$ in $\mathbb{G}$ then the order of $s$ in $G$ is $n$, and under slightly stronger conditions that the word problem of $G$ is decidable. We also give conditions under which the question of which Magnus subgroups of $\mathbb{G}$ embed in $G$ reduces to the same question in the minimal parabolic subgroup of $\mathbb{G}$ containing $r$. In many cases this allows us to characterise Magnus subgroups which embed in $G$, via a condition on $r$ and the commutation graph of $\mathbb{G}$, and to find further examples of quotients $G$ where the word and conjugacy problems are decidable. We give evidence that situations in which our main theorem applies are not uncommon, by proving that for cycle graphs with a chord $\Gamma$, almost all cyclically reduced elements of the partially commutative group $\mathbb{G}(\Gamma)$ satisfy the conditions of the theorem.