Polyhedral products, and relations in the commutator subgroup of a right-angled Coxeter group

We give a criterion of the existence of a presentation with a single relation for the commutator subgroup $RC_{\mathscr{K}}'$ of a right-angled Coxeter group $RC_{\mathscr{K}}$. Namely, we prove that $RC_{\mathscr{K}}'$ is a one-relator group if and only if $\mathscr{K}$ is either a $p$-cycle with $p\geqslant4$ or has the form ($p$-cycle)$*\Delta^q$ with $p\geqslant4$ and $g\geqslant0$, where $\Delta^q$ is a $q$-simplex.