The Poincaré Exponent and the Dimensions of Kleinian Limit Sets

Abstract

Kleinian groups are discrete groups of isometries of hyperbolic space. Their actions give rise to intricate fractal limit sets on the boundary at infinity and there is great interest in estimating the “dimension” of these limit sets. As an invitation to this fascinating area, we provide a proof of the (well-known) result that the Poincaré exponent of a nonelementary Kleinian group is a lower bound for the upper box dimension of the limit set. Our proof uses only elementary hyperbolic and fractal geometry.