  Consider an ergodic non-singular action $\Gamma \cc B$ of a countable group
on a probability space. The type of this action codes the asymptotic range of
the Radon-Nikodym derivative, also called the {\em ratio set}. If $\Gamma \cc
X$ is a pmp (probability-measure-preserving) action, then the ratio set of the
product action $\Gamma \cc B\times X$ is contained in the ratio set of $\Gamma
\cc B$. So we define the {\em stable ratio set} of $\Gamma \cc B$ to be the
intersection over all pmp actions $\Gamma \cc X$ of the ratio sets of $\Gamma
\cc B\times X$. By analogy, there is a notion of {\em stable type} which codes
the stable ratio set of $\Gamma \cc B$. This concept is crucially important for
the identification of the limit in pointwise ergodic theorems established by
the author and Amos Nevo.
  Here, we establish a general criteria for a nonsingular action of a countable
group on a probability space to have stable type $III_\lambda$ for some
$\lambda >0$. This is applied to show that the action of a non-elementary
Gromov hyperbolic group on its boundary with respect to a quasi-conformal
measure is not type $III_0$ and, if it is weakly mixing, then it is not stable
type $III_0$.
