Abstract
We provide a new proof of a theorem whose proof was sketched by Sullivan [Disjoint spheres, approximation by imaginary quadratic numbers, and the logarithm law for geodesics.Acta Math.149(3–4) (1982), 215–237], 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 [Multiple fractal aspects of conformal measures; a survey.Workshop on Fractals and Dynamics (Mathematica Gottingensis, 5). Eds. M. Denker, S.-M. Heinemann and B. Stratmann. Springer, Berlin, 1997, pp. 65–71; Fractal geometry on hyperbolic manifolds.Non-Euclidean Geometries (Mathematical Applications (N.Y.), 581). Springer, New York, 2006, pp. 227–247].
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