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Jørgensen number and arithmeticity


Author: Jason Callahan
Journal: Conform. Geom. Dyn. 13 (2009), 160-186
MSC (2000): Primary 30F40; Secondary 57M05, 57M25, 57M50
DOI: https://doi.org/10.1090/S1088-4173-09-00196-9
Published electronically: July 23, 2009
MathSciNet review: 2525101
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Abstract: The Jørgensen number of a rank-two non-elementary Kleinian group $ \Gamma$ is

$\displaystyle J(\Gamma) = \inf\{\vert\mathrm{tr}^2 X - 4\vert + \vert\mathrm{tr} [X, Y] - 2\vert : \langle X, Y \rangle = \Gamma \}. $

Jørgensen's Inequality guarantees $ J(\Gamma) \geq 1$, and $ \Gamma$ is a Jørgensen group if $ J(\Gamma) = 1$. This paper shows that the only torsion-free Jørgensen group is the figure-eight knot group, identifies all non-cocompact arithmetic Jørgensen groups, and establishes a characterization of cocompact arithmetic Jørgensen groups. The paper concludes with computations of $ J(\Gamma)$ for several non-cocompact Kleinian groups including some two-bridge knot and link groups.


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Additional Information

Jason Callahan
Affiliation: Department of Mathematics, The University of Texas at Austin, 1 University Station C1200, Austin, Texas 78712 and Department of Mathematics, St. Edward’s University, 3001 South Congress Avenue, Austin, Texas 78704
Email: callahan@math.utexas.edu; jasonc@stedwards.edu

DOI: https://doi.org/10.1090/S1088-4173-09-00196-9
Received by editor(s): May 14, 2009
Published electronically: July 23, 2009
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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