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
$\mathcal {Z}_K$
.
A First Approximation to Homotopy Theory
