One-relator groups and algebras related to polyhedral products

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

16 pages, published version

Cited by in corpus (3)

• Pontryagin algebras and the LS-category of moment-angle complexes in the flag case
• Cartesian subgroups in graph products of groups
• Polyhedral products, graph products and p-central series