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

Abstract

Consider an ergodic non-singular action $Γ\cc B$ 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 $Γ\cc X$ is a pmp (probability-measure-preserving) action, then the ratio set of the product action $Γ\cc B\times X$ is contained in the ratio set of $Γ\cc B$. So we define the {\em stable ratio set} of $Γ\cc B$ to be the intersection over all pmp actions $Γ\cc X$ of the ratio sets of $Γ\cc B\times X$. By analogy, there is a notion of {\em stable type} which codes the stable ratio set of $Γ\cc B$. 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 $III_λ$ for some $λ>0$. 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 $III_0$.

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