Dual cube complex of a wallspace
= Dual cube complex of a wallspace
{title2=$C(S,\mathcal W)$}
The vertices of the dual cube complex are consistent choices of one halfspace of every wall that differ from a principal choice on only finitely many walls. Two vertices are joined when their choices differ on one wall, and higher cubes fill the resulting hypercubes. The result is a <CAT(0) cube complex>.