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

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: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.