On interpreting Patterson--Sullivan measures of geometrically finite groups as Hausdorff and packing measures

Abstract

We provide a new proof of a theorem whose proof was sketched by Sullivan ('82), namely that if the Poincaré exponent of a geometrically finite Kleinian group $G$ is strictly between its minimal and maximal cusp ranks, then the Patterson--Sullivan measure of $G$ is not proportional to the Hausdorff or packing measure of any gauge function. This disproves a conjecture of Stratmann ('97, '06).