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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 
$RC_K'$ 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 
$H_{*}(\Omega \mathcal {Z}_K)$ of the moment-angle complex to be a one-relator algebra. We also give a homological characterization of these properties. For 
$RC_K'$, it is given by a condition on the homology group 
$H_2(\mathcal {R}_K)$, whereas for 
$H_{*}(\Omega \mathcal {Z}_K)$ it is stated in terms of the bigrading of the homology groups of 
$\mathcal {Z}_K$.
