# Kleinian Groups which Are Limits of Geometrically Finite Groups

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*Ken’ichi Ohshika*

Ahlfors conjectured in 1964 that the limit set of every finitely generated Kleinian group either has Lebesgue measure \(0\) or is the entire \(S^2\). We prove that this conjecture is true for purely loxodromic Kleinian groups which are algebraic limits of geometrically finite groups. What we directly prove is that if a purely loxodromic Kleinian group \(\Gamma\) is an algebraic limit of geometrically finite groups and the limit set \(\Lambda_\Gamma\) is not the entire \(S^2_\infty\), then \(\Gamma\) is topologically (and geometrically) tame, that is, there is a compact 3-manifold whose interior is homeomorphic to \({\mathbf H}^3/\Gamma\). The proof uses techniques of hyperbolic geometry considerably and is based on works of Maskit, Thurston, Bonahon, Otal, and Canary.

#### Table of Contents

# Table of Contents

## Kleinian Groups which Are Limits of Geometrically Finite Groups

- Contents v6 free
- Abstract vii8 free
- Introduction ix10 free
- Chapter 1. Preliminaries 114 free
- Chapter 2. Statements of theorems 1326
- Chapter 3. Characteristic compression bodies 1528
- Chapter 4. The Masur domain and Ahlfors' conjecture 1932
- Chapter 5. Branched covers and geometric limit 5366
- Chapter 6. Non-realizable measured laminations 5972
- Chapter 7. Strong convergence of function groups 8396
- Chapter 8. Proof of the main theorem 87100
- Bibliography 111124
- Index 115128