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One-relator groups and algebras related to polyhedral products
- Part of
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- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 152 / Issue 1 / February 2022
- Published online by Cambridge University Press:
- 15 January 2021, pp. 128-147
- Print publication:
- February 2022
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- Article
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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 
$\mathcal {Z}_K$.
