Kleinian Groups Which are Limits of Geometrically Finite Groups

Ken'ichi Ohshika (Autor)

Buch | Softcover

116 Seiten

2005 | illustrated Edition
American Mathematical Society (Verlag)
978-0-8218-3772-6 (ISBN)

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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 $S^2$. This title intends to prove that this conjecture is true for purely loxodromic Kleinian groups which are algebraic limits of geometrically finite groups.

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.

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.

Erscheint lt. Verlag	1.9.2005
Reihe/Serie	Memoirs of the American Mathematical Society
Zusatzinfo	illustrations
Verlagsort	Providence
Sprache	englisch
Gewicht	263 g
Themenwelt	Mathematik / Informatik ► Mathematik
ISBN-10	0-8218-3772-9 / 0821837729
ISBN-13	978-0-8218-3772-6 / 9780821837726
Zustand	Neuware