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Kleinian Groups which Are Limits of Geometrically Finite Groups
Ken'ichi Ohshika, Osaka University, Japan
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Memoirs of the American Mathematical Society
2005; 116 pp; softcover
Volume: 177
ISBN-10: 0-8218-3772-9
ISBN-13: 978-0-8218-3772-6
List Price: US$64
Individual Members: US$38.40
Institutional Members: US$51.20
Order Code: MEMO/177/834
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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

  • 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
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