Kleinian groups which are limits of geometrically finile groups / Ken© ichi Ohshika.

Format:

Book

Author/Creator:

Ōshika, Ken'ichi, 1961- author.

Series:

Memoirs of the American Mathematical Society ; no. 834.

Memoirs of the American Mathematical Society, 0065-9266 ; number 834

Language:

English

Subjects (All):

Kleinian groups.

Low-dimensional topology.

Geometry, Hyperbolic.

Physical Description:

1 online resource (136 p.)

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2005.

Language Note:

English

Summary:

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:

""Contents""; ""Abstract""; ""Introduction""; ""Chapter 1. Preliminaries""; ""1.A. Generalities""; ""1.B. Compact cores, ends of hyperbolic 3-manifolds""; ""1.C. Geodesic and measured laminations""; ""1.D. Masur domain""; ""1.E. Pleated surfaces""; ""1.F. Train tracks""; ""1.G. Algebraic and geometric convergence""; ""Chapter 2. Statements of theorems""; ""Chapter 3. Characteristic compression bodies""; ""Chapter 4. The Masur domain and Ahlfors' conjecture""; ""4.A. The main result in this chapter""

""4.B. Realization by pleated surfaces for measured laminations on the exterior boundaries of compression bodies""""4.C. Approximation by train tracks""; ""4.D. Realization by pleated surfaces""; ""4.E. A product neighbourhood of the end""; ""Chapter 5. Branched covers and geometric limit""; ""Chapter 6. Non-realizable measured laminations""; ""Chapter 7. Strong convergence of function groups""; ""Chapter 8. Proof of the main theorem""; ""8.A. A special case""; ""8.B. The existence of a homeomorphism""; ""8.C. Lemmata for the proof of Lemma 8.2""

""8.D. Proof of Lemma 8.2 and Proposition 8.1""""8.E. Concluding the proof of Theorem 2.1""; ""Bibliography""; ""Index""; ""A""; ""B""; ""C""; ""E""; ""G""; ""I""; ""K""; ""L""; ""M""; ""P""; ""Q""; ""R""; ""S""; ""T""; ""U""; ""V""; ""W""

Notes:

"September 2005, volume 177, number 834 (second of 4 numbers)."

Includes bibliographical references (pages 111-113) and index.

Description based on print version record.

ISBN:

1-4704-0435-4