Quasi-isometric classification of right-angled Artin groups, I: The finite out case

Let [math] and [math] be two right-angled Artin groups. We show they are quasi-isometric if and only if they are isomorphic, under the assumption that the outer automorphism groups [math] and [math] are finite. If we only assume [math] is finite, then [math] is quasi-isometric to [math] if and only if [math] is isomorphic to a subgroup of finite index in [math] . In this case, we give an algorithm to determine whether [math] and [math] are quasi-isometric by looking at their defining graphs.