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

Let $G$ and $G^{\mathrm{\prime }}$ be two right-angled Artin groups (RAAG). We show they are quasi-isometric iff they are isomorphic, under the assumption that $Out(G)$ and $Out(G^{\mathrm{\prime }})$ are finite. If only $Out(G)$ is finite, then $G^{\mathrm{\prime }}$ is quasi-isometric $G$ iff $G^{\mathrm{\prime }}$ is isomorphic to a finite index subgroup of $G$. In this case, we give an algorithm to determine whether $G$ and $G^{\mathrm{\prime }}$ are quasi-isometric by looking at their defining graphs.