Fix $x\in S$ and let $\sigma_x$ be its <principal vertex of a dual cube complex>. The combinatorial distance in the dual complex is the <wall metric>:
$$
d_C(\sigma_x,g\sigma_x)
=d_{\mathcal W}(x,gx)
=\#\{W\in\mathcal W:W\text{ separates }x\text{ from }gx\}.
$$
Consequently the <wall-metric properness criterion> says that the action on $C$ is metrically proper provided this number tends to infinity as $g$ leaves every finite subset of $G$. Equivalently, for every $R$, only finitely many $g$ separate $x$ from $gx$ by at most $R$ walls.

Solution

Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 134 / 2 / b / Solution

A point $x\in S$ determines a principal vertex by choosing, for every wall, the halfspace containing $x$. The combinatorial distance between the principal vertices of $x$ and $y$ is $d_{\mathcal W}(x,y)$.


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 Incoming links: Principal vertex of a dual cube complex