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.
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Received: March 2007, Revision: October 2007, Accepted: October 2007
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Kapovich, M. Homological Dimension and Critical Exponent of Kleinian Groups. GAFA Geom. funct. anal. 18, 2017–2054 (2009). https://doi.org/10.1007/s00039-009-0705-z
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DOI: https://doi.org/10.1007/s00039-009-0705-z