Geometric and Functional Analysis

, Volume 18, Issue 6, pp 2017–2054 | Cite as

Homological Dimension and Critical Exponent of Kleinian Groups

  • Michael Kapovich
Open Access


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.

Keywords and phrases:

Kleinian groups homological dimension Hausdorff dimension critical exponent 

AMS Mathematics Subject Classification:

30F40 20F67 

Copyright information

© Birkhäuser Verlag, Basel 2009

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaDavisUSA

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