Homological Dimension and Critical Exponent of Kleinian Groups

Abstract

We prove an inequality between the relative homological dimension of a Kleinian group \(\Gamma \subset {\rm Isom}({\mathbb{H}}^n)\) and its critical exponent. As an application of this result we show that for a geometrically finite Kleinian group Γ, if the topological dimension of the limit set of Γ equals its Hausdorff dimension, then the limit set is a round sphere.