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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. [ABSTRACT FROM AUTHOR]