Mesures de Hausdorff de l'ensemble limite de groupes kleiniens g\'eom\'etriquement finis

TL;DR

This paper proves that for geometrically finite Kleinian groups acting on hyperbolic 3-space, the Poincaré exponent equals the Hausdorff dimension of the limit set, and compares associated natural measures.

Contribution

It establishes the equality between the Poincaré exponent and the Hausdorff dimension for these groups and analyzes the relationships between different natural measures on the limit set.

Findings

Poincaré exponent equals Hausdorff dimension of the limit set.

Comparison of Patterson measure with Hausdorff and packing measures.

Insights into measure-theoretic properties of limit sets.

Abstract

We prove here that the Poincar\'e exponent of a geometrically finite group od isometries of the 3-dimensionnal hyperbolic space coincides with the Hausdorff dimension of its limit set. We also compare the natural measures supported by this set: the Patterson measure and the Hausdorff and packing measures corresponding to the standart jauge.