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

TL;DR

This paper offers a new proof that for certain geometrically finite Kleinian groups, the Patterson--Sullivan measure cannot be expressed as a Hausdorff or packing measure, disproving a longstanding conjecture.

Contribution

It provides a novel proof of a theorem relating Patterson--Sullivan measures to Hausdorff and packing measures, resolving a conjecture from Stratmann.

Findings

Patterson--Sullivan measure is not proportional to Hausdorff or packing measure under specified conditions

Disproves Stratmann's conjecture from 1997 and 2006

Clarifies the relationship between geometric finiteness and measure equivalence

Abstract

We provide a new proof of a theorem whose proof was sketched by Sullivan ('82), namely that if the Poincar\'e 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).