Homological Dimension and Critical Exponent of Kleinian Groups

2009/02/10 by Michael Kapovich · 2 citations
Mathematics · #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Point processes and geometric inequalities

paper · pdf · doi:10.1007/s00039-009-0705-z

openalex publication_date 2009/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/31

Abstract

We prove an inequality between the relative homological dimension of a Kleinian group Γ ⊂ \rm Isom(ℍⁿ) 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.

Citations

• Tameness of hyperbolic 3-manifolds

Cited by

• Dimension rigidity in conformal structures
• Michael Kapovich [wikipedia]

Related

• Homological dimension and critical exponent of Kleinian groups
• Hausdorff dimension and Kleinian groups
• Critical exponents and dimension for generalised limit sets
• The Poincaré exponent and the dimensions of Kleinian limit sets
• The critical exponent, the Hausdorff dimension of the limit set and the convex core entropy of a Kleinian group