# Kleinian Groups Which are Limits of Geometrically Finite Groups (Memoirs of the American Mathematical Society illustrated Edition)

By: Ken'ichi Ohshika (author)Paperback

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£64.95

### Description

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 ${\mathbf H}^3[LAMBDA]Gamma$. The proof uses techniques of hyperbolic geometry considerably and is based on works of Maskit, Thurston, Bonahon, Otal, and Canary.

### Contents

Preliminaries Statements of theorems Characteristic compression bodies The Masur domain and Ahlfors' conjecture Branched covers and geometric limit Non-realizable measured laminations Strong convergence of function groups Proof of the main theorem Bibliography Index.

### Product Details

• ISBN13: 9780821837726
• Format: Paperback
• Number Of Pages: 116
• ID: 9780821837726
• ISBN10: 0821837729
• edition: illustrated Edition

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