The asymptotic geometry of right-angled Artin groups, I

Abstract

We study atomic right-angled Artin groups – those whose defining graph has no cycles of length $\le 4$, 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.