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

Abstract

Let $G$ and $G^{\prime }$ 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 $Out\left(G\right)$ and $Out\left(G^{\prime }\right)$ are finite. If we only assume $Out\left(G\right)$ is finite, then $G^{\prime }$ is quasi-isometric to $G$ if and only if $G^{\prime }$ is isomorphic to a subgroup of finite index in $G$. In this case, we give an algorithm to determine whether $G$ and $G^{\prime }$ are quasi-isometric by looking at their defining graphs.