The type and stable type of the boundary of a Gromov hyperbolic group

Abstract

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 $III_0$.