August 26, 2025Open Access

Quasi-isometric classification of right-angled Artin groups II: Several infinite out cases

Puntos clave

Any RAAG in identified classes is quasi-isometric to another if and only if they are commensurable, revealing a deep connection between these concepts.
The existence of an algorithm to determine quasi-isometry based on defining graphs enhances understanding of these groups' structural properties.
The study emphasizes the challenge of straightening quasi-isometries compared to finite outer cases, necessitating innovative approaches for infinite scenarios.
Deformation arguments and cubulation techniques are introduced to address complexities in the classification process for these groups.

Resumen

We are motivated by the question that for which class of right-angled Artin groups (RAAGs), the quasi-isometric classification coincides with commensurability classification. This is previously known to hold for RAAGs with finite outer automorphism groups. In this paper, we identify two classes of RAAGs, where their outer automorphism groups are allowed to contain adjacent transvections and partial conjugations, hence are infinite. If G belongs to one of these classes, then any other RAAG G' is quasi-isometric to G if and only if G' is commensurable with G . We also show that in such cases, there exists an algorithm to determine whether two RAAGs are quasi-isometric by looking at their defining graphs. Compared to the finite out case, the main issue we need to deal with here is that one may not be able to straighten the quasi-isometries in a canonical way. We introduce a deformation argument, as well as techniques from cubulation to deal with this issue.

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