  The action dimension of a discrete group $\Gamma$ is the smallest dimension
of a contractible manifold which admits a proper action of $\Gamma$. Associated
to any flag complex $L$ there is a right-angled Artin group, $A_L$. We compute
the action dimension of $A_L$ for many $L$. Our calculations come close to
confirming the conjecture that if an $\ell^2$-Betti number of $A_L$ in degree
$l$ is nonzero, then the action dimension of $A_L$ is $\ge 2l$.
