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Kleinian Groups which Are Limits of Geometrically Finite Groups
 
Ken’ichi Ohshika Osaka University, Osaka, Japan
Kleinian Groups which Are Limits of Geometrically Finite Groups
eBook ISBN:  978-1-4704-0435-2
Product Code:  MEMO/177/834.E
List Price: $68.00
MAA Member Price: $61.20
AMS Member Price: $40.80
Kleinian Groups which Are Limits of Geometrically Finite Groups
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Kleinian Groups which Are Limits of Geometrically Finite Groups
Ken’ichi Ohshika Osaka University, Osaka, Japan
eBook ISBN:  978-1-4704-0435-2
Product Code:  MEMO/177/834.E
List Price: $68.00
MAA Member Price: $61.20
AMS Member Price: $40.80
  • Book Details
     
     
    Memoirs of the American Mathematical Society
    Volume: 1772005; 116 pp
    MSC: Primary 57; 30;

    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
     
     
    • Chapters
    • 1. Preliminaries
    • 2. Statements of theorems
    • 3. Characteristic compression bodies
    • 4. The Masur domain and Ahlfors’ conjecture
    • 5. Branched covers and geometric limit
    • 6. Non-realizable measured laminations
    • 7. Strong convergence of function groups
    • 8. Proof of the main theorem
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 1772005; 116 pp
MSC: Primary 57; 30;

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.

  • Chapters
  • 1. Preliminaries
  • 2. Statements of theorems
  • 3. Characteristic compression bodies
  • 4. The Masur domain and Ahlfors’ conjecture
  • 5. Branched covers and geometric limit
  • 6. Non-realizable measured laminations
  • 7. Strong convergence of function groups
  • 8. Proof of the main theorem
Review Copy – for publishers of book reviews
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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