
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 |

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 |
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Book DetailsMemoirs of the American Mathematical SocietyVolume: 177; 2005; 116 ppMSC: 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.
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Table of Contents
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Chapters
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1. Preliminaries
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2. Statements of theorems
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3. Characteristic compression bodies
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4. The Masur domain and Ahlfors’ conjecture
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5. Branched covers and geometric limit
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6. Non-realizable measured laminations
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7. Strong convergence of function groups
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8. Proof of the main theorem
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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.
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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
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8. Proof of the main theorem