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

Published online by Cambridge University Press:  21 July 2015

DAVID SIMMONS
DAVID SIMMONS*
Affiliation:
Ohio State University, Department of Mathematics, 231 W. 18th Avenue, Columbus, OH 43210-1174, USA email simmons.465@osu.edu
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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].

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 8 , December 2016 , pp. 2675 - 2686
Copyright
© Cambridge University Press, 2015 

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