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arXiv · 2002.11476

One-relator groups and algebras related to polyhedral products

Jelena Grbić·Marina Ilyasova·Taras Panov·George Simmons
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Abstract

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

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Jelena Grbić, Marina Ilyasova, Taras Panov, George Simmons. 2022-03-17. One-relator groups and algebras related to polyhedral products. https://doi.org/10.1017/prm.2020.101

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