One-relator loop homology algebras of moment-angle complexes

For a flag complex $K$, we give a necessary and sufficient combinatorial condition for the loop homology algebra of the moment-angle complex $\mathcal Z_K$ to be a one-relator algebra. The condition we specify also characterises when the commutator subgroup of the right-angled Coxeter group corresponding to $K$ is a one-relator group, providing a combinatorial link between distinct concepts of geometric group theory and homotopy theory. Returning to moment-angle complexes, we give other equivalent algebraic and homotopy-theoretic formulations of our condition. In the non-flag case, many of these equivalences no longer hold, and we will present some of the key differences.