One-relator groups and algebras related to polyhedral products

We link distinct concepts of geometric group theory and homotopy theory through underlying combinatorics. For a flag simplicial complex $K$, we specify a necessary and sufficient combinatorial condition for the commutator subgroup $R{C}_K^{\mathrm{\prime }}$ of a right-angled Coxeter group, viewed as the fundamental group of the real moment-angle complex ${\mathcal{R}}_K$, to be a one-relator group; and for the Pontryagin algebra $H_{\ast }(\mathrm{\Omega }{\mathcal{Z}}_K)$ of the moment-angle complex to be a one-relator algebra. We also give a homological characterisation of these properties. For $R{C}_K^{\mathrm{\prime }}$, it is given by a condition on the homology group $H_2({\mathcal{R}}_K)$, whereas for $H_{\ast }(\mathrm{\Omega }{\mathcal{Z}}_K)$ it is stated in terms of the bigrading of the homology groups of ${\mathcal{Z}}_K$.