The Patterson Measure for Geometrically Finite Groups with Parabolic Elements, New and Old

B O Stratmann, S L Velani

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We study the Patterson measure on the limit set of a geometrically finite Kleinian group with parabolic elements. Using the conformality of the Patterson measure in combination with basic hyperbolic geometry a Global Shadow Lemma is obtained. This macroscopic measure estimate is exploited to derive a Khintchine-type theorem, which is then used for a finer analysis of the density of the limit set in terms of Hausdorff measure.