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 $\mathcal {Z}_K$.

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与多面体产品相关的单相关组和代数

我们通过基础组合学将几何群论和同伦论的不同概念联系起来。对于标志单纯复形$K$，我们为交换子群指定了一个充分必要的组合条件$RC_K'$一个直角 Coxeter 群，被视为实矩角复形的基本群$\数学{R}_K$，成为一个单一的关系群体；对于 Pontryagin 代数$H_{*}(\Omega \mathcal {Z}_K)$矩角复数是一个单相关代数。我们还给出了这些属性的同调表征。为了$RC_K'$, 它由同调群上的一个条件给出$H_2(\mathcal {R}_K)$, 而对于$H_{*}(\Omega \mathcal {Z}_K)$它是根据同源群的二分类来表述的$\数学{Z}_K$.