Homological Dimension and Critical Exponent of Kleinian Groups

We prove an inequality between the relative homological dimension of a Kleinian group 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. Received: March 2007, Revision: October 2007, Accepted: October 2007