Translation lengths in crossing and contact graphs of (quasi-)median graphs

Given a quasi-median graph $X$, the crossing graph $\mathrm{\Delta }X$ and the contact graph $\mathrm{\Gamma }X$ are natural hyperbolic models of $X$. In this article, we show that the asymptotic translation length in $\mathrm{\Delta }X$ or $\mathrm{\Gamma }X$ of an isometry of $X$ is always rational. Moreover, if $X$ is hyperbolic, these rational numbers can be written with denominators bounded above uniformly; this is not true in full generality. Finally, we show that, if the quasi-median graph $X$ is constructible in some sense, then there exists an algorithm computing the translation length of every computable isometry. Our results encompass contact graphs in CAT(0) cube complexes and extension graphs of right-angled Artin groups.