Jørgensen's inequality

In hyperbolic geometry , a branch of mathematics , the Jørgensen inequality is a necessary condition for the discreteness of groups of isometries in 3-dimensional hyperbolic space . It goes back to Troels Jørgensen . ${\displaystyle \mathbb {H} ^{3}}$

Inequality

Assuming a non-elementary Klein group generated by two matrices , then the inequality holds ${\displaystyle \Gamma }$ ${\displaystyle f,g\in PSL(2,\mathbb {C} )}$

${\displaystyle \vert Sp(f)^{2}-4\vert +\vert Sp(\left[f,g\right])-2\vert \geq 1}$,

where denotes the trace of a matrix and the commutator of two matrices. ${\displaystyle Sp(.)}$ ${\displaystyle \left[.,.\right]}$

The condition clearly says that two elements which create a non-elementary discrete group cannot be too close to the identity .

equality

The only hyperbolic 3-manifold whose fundamental group of 2 elements with is generated, is the complement of the aft node . ${\displaystyle f,g}$ ${\displaystyle \vert Sp(f)^{2}-4\vert +\vert Sp(\left[f,g\right])-2\vert =1}$

The only discrete subgroups of , which are generated by 2 elements with , are the hyperbolic triangle groups of the signature with . ${\displaystyle PSL(2,\mathbb {R} )}$ ${\displaystyle f,g}$ ${\displaystyle \vert Sp(f)^{2}-4\vert +\vert Sp(\left[f,g\right])-2\vert =1}$ ${\displaystyle (2,3,q)}$ ${\displaystyle 7\leq q\leq \infty }$

Furthermore, for every discrete group with . ${\displaystyle r\geq 1}$ ${\displaystyle \Gamma \subset PSL(2,\mathbb {C} )}$ ${\displaystyle inf_{f,g\colon \langle f,g\rangle =\Gamma }\left\{\vert Sp(f)^{2}-4\vert +\vert Sp(\left[f,g\right])-2\vert \right\}=r}$

Applications

• The Jørgensen inequality is used in numerous proofs of convergence in Klein's group theory.
• Jørgensen's original application was the proof of the following convergence theorem: Let be a non-elementary Klein group and a sequence of isomorphisms of that converges to a homomorphism , then is a Klein group and is an isomorphism. ${\displaystyle \Gamma }$ ${\displaystyle (\phi _{n})_{n}}$ ${\displaystyle \Gamma }$ ${\displaystyle \phi :\Gamma \rightarrow PSL(2,\mathbb {C} )}$ ${\displaystyle \phi (\Gamma )}$ ${\displaystyle \phi }$
• If is parabolic, one gets the classic result about the existence of precisely invariant horospheres . ${\displaystyle f}$
• There are numerous generalizations of the Jørgensen inequality to discrete groups of isometries of other metric spaces.

literature

• Jørgensen, Troels: On discrete groups of Möbius transformations. Amer. J. Math. 98 (1976) no. 3, 739-749. pdf
• Matsuzaki, Katsuhiko; Taniguchi, Masahiko: Hyperbolic manifolds and Kleinian groups. Oxford Mathematical Monographs. Oxford Science Publications. The Clarendon Press, Oxford University Press, New York, 1998. ISBN 0-19-850062-9 (Chapter 2.2)

Individual evidence

1. ↑ Callahan: Jørgensen number and arithmeticity
2. ^ T. Jørgensen, M. Kiikka: Some extreme discrete groups. Ann. Acad. Sci. Fenn. Ser. AI Math., 1 (2): 245-248, 1975.
3. ↑ Y. Yamashita, R. Yamazaki: The realization problem for Jørgensen numbers . ArXiv