Geodesic
Translation lengths in crossing and contact graphs of quasi-median graphs
Anthony Genevois1 orcid
1University of Montpellier, Montpellier, France
Groups, geometry, and dynamics, Tome 19 (2025) no. 1, pp. 343-391

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DOI

Given a quasi-median graph X, the crossing graph ΔX and the contact graph ΓX are natural hyperbolic models of X. In this article, we show that the asymptotic translation length in ΔX or ΓX 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.
DOI : 10.4171/ggd/822
Classification : 20F65, 20F10, 20F67
Mots-clés : quasi-median graphs, crossing graph, contact graph, translation length, graph products of groups

Anthony Genevois  1

1 University of Montpellier, Montpellier, France 
Anthony Genevois. Translation lengths in crossing and contact graphs of quasi-median graphs. Groups, geometry, and dynamics, Tome 19 (2025) no. 1, pp. 343-391. doi: 10.4171/ggd/822
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