Conformal Geometry and Dynamics

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ISSN 1088-4173

Jørgensen number and arithmeticity

Author(s): Jason Callahan
Journal: Conform. Geom. Dyn. 13 (2009), 160-186.
MSC (2000): Primary 30F40; Secondary 57M05, 57M25, 57M50
Posted: July 23, 2009
MathSciNet review: 2525101
Retrieve article in: PDF

Abstract | References | Similar articles | Additional information

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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