  The main goal of this note is to suggest an algebraic approach to the
quasi-isometric classification of partially commutative groups (alias
right-angled Artin groups). More precisely, we conjecture that if the partially
commutative groups $\mathbb{G}(\Delta)$ and $\mathbb{G}(\Gamma)$ are
quasi-isometric, then $\mathbb{G}(\Delta)$ is a (nice) subgroup of
$\mathbb{G}(\Gamma)$ and vice-versa. We show that the conjecture holds for all
known cases of quasi-isometric classification of partially commutative groups,
namely for the classes of $n$-tress and atomic graphs.
  As in the classical Mostow rigidity theory for irreducible lattices, we
relate the quasi-isometric rigidity of the class of atomic partially
commutative groups with the algebraic rigidity, that is with the co-Hopfian
property of their $\mathbb{Q}$-completions.

Group Theory