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

TL;DR

This paper investigates the translation lengths of isometries in hyperbolic models of quasi-median graphs, showing they are always rational and providing algorithms for their computation in certain cases.

Contribution

It proves the rationality of asymptotic translation lengths in crossing and contact graphs of quasi-median graphs and offers an algorithm for computing these lengths in constructible cases.

Findings

Translation lengths in crossing and contact graphs are always rational.

In hyperbolic cases, these lengths have bounded denominators.

An algorithm exists for computing translation lengths in constructible quasi-median graphs.

Abstract

Given a quasi-median graph $X$, the crossing graph $\Delta X$ and the contact graph $\Gamma X$ are natural hyperbolic models of $X$. In this article, we show that the asymptotic translation length in $\Delta X$ or $\Gamma X$ 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.