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

Abstract

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 $G^{\prime }$ is quasi-isometric to $G$ if and only if $G^{\prime }$ 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.