The Poincar\'e exponent and the dimensions of Kleinian limit sets

TL;DR

This paper proves that the Poincaré exponent of a non-elementary Kleinian group bounds the upper box dimension of its limit set, using elementary hyperbolic and fractal geometric methods.

Contribution

It offers a new, elementary proof of a well-known relationship between the Poincaré exponent and the limit set's dimension.

Findings

Poincaré exponent bounds the upper box dimension of the limit set.

Elementary geometric methods suffice for the proof.

Reinforces the fundamental link between hyperbolic geometry and fractal dimensions.

Abstract

We provide a proof of the (well-known) result that the Poincar\'e exponent of a non-elementary Kleinian group is a lower bound for the upper box dimension of the limit set. Our proof only uses elementary hyperbolic and fractal geometry.