We study atomic right-angled Artin groups – those whose defining graph has no cycles of length , and no separating vertices, separating edges, or separating vertex stars. We show that these groups are not quasi-isometrically rigid, but that an intermediate form of rigidity does hold. We deduce from this that two atomic groups are quasi-isometric iff they are isomorphic.
"The asymptotic geometry of right-angled Artin groups, I." Geom. Topol. 12 (3) 1653 - 1699, 2008. https://doi.org/10.2140/gt.2008.12.1653