Ahlfors' theorem of finiteness

In mathematics , Ahlfors' finiteness theorem describes the geometrically finite ends of hyperbolic 3-manifolds .

Ahlfors' theorem of finiteness

Let be a finitely generated Klein group and its region of discontinuity . ${\displaystyle \Gamma }$ ${\displaystyle \Omega \subset \partial _{\infty }\mathbb {H} ^{3}}$

Then it has finitely many connected components and each of these connected components is a compact Riemann surface with a finite number of dots . ${\displaystyle \Gamma \backslash \Omega }$

Quantitative version

The following two inequalities are due to Bers.

Let be a non-elementary Klein group with generators, then is ${\displaystyle \Gamma }$ ${\displaystyle N}$

${\displaystyle Area(\Gamma \backslash \Omega )\leq 4\pi (N-1)}$

with equality only for Schottky groups .

For every -invariant connected component holds ${\displaystyle \Gamma }$ ${\displaystyle \Omega _{1}}$

${\displaystyle Area(\Gamma \backslash \Omega )\leq 2Area(\Gamma \backslash \Omega _{1})}$

with equality only for Fuchsian groups of the first kind .

Higher dimension

For finitely generated, discrete subgroups of , no finiteness theorem holds in general. Counterexamples were given by Kapovich and Potyagailo in 1991 . ${\displaystyle Isom(\mathbb {H} ^{n}),n\geq 4}$

history

The theorem was proved by Ahlfors in 1964 and the proof was completed by Greenberg in 1967 . According to Ahlfors, Bers had previously proven the analogous theorem for Fuchsian groups . Dennis Sullivan later gave a simpler proof , using analogies to the iteration of rational functions.

Individual evidence

1. ↑ L. Bers: Inequalities for finitely generated Kleinian groups , Journal d'Analyse Mathématique 18, 23-41, 1967.
2. ↑ M. Kapovich, L. Potyagailo: On absence of Ahlfors' finiteness theorem for Kleinian groups in dimension 3 , Topology and its Applications 40, 83-91, 1991.
3. ↑ M. Kapovich, L. Potyagailo: On absence of Ahlfors' and Sullivan's finiteness theorems for Kleinian groups in higher dimensions , Siberian Math. Journ. 32, 61-73, 1991.
4. ^ L. Ahlfors: Finitely generated Kleinian groups , American Journal of Mathematics 86, 413-429, 1964.
5. ^ L. Greenberg: On a theorem of Ahlfors and conjugate subgroups of Kleinian groups , American Journal of Mathematics 89, 56-68, 1967.