Kleinian Groups Which are Limits of Geometrically Finite Groups (Memoirs of the American Mathematical Society No. 177 Illustrated edition)

Kleinian Groups Which are Limits of Geometrically Finite Groups (Memoirs of the American Mathematical Society No. 177 Illustrated edition)

By: Ken'ichi Ohshika (author)Paperback

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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.

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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

  • publication date: 15/08/2005
  • ISBN13: 9780821837726
  • Format: Paperback
  • Number Of Pages: 116
  • ID: 9780821837726
  • weight: 261
  • ISBN10: 0821837729
  • edition: Illustrated edition

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