Jørgensen number and arithmeticity

Callahan, Jason

doi:10.1090/S1088-4173-09-00196-9

Jørgensen number and arithmeticity
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by Jason Callahan

Conform. Geom. Dyn. 13 (2009), 160-186

DOI: https://doi.org/10.1090/S1088-4173-09-00196-9

Published electronically: July 23, 2009

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Abstract:

The Jørgensen number of a rank-two non-elementary Kleinian group $\Gamma$ is \[ J(\Gamma ) = \inf \{|\mathrm {tr}^2 X - 4| + |\mathrm {tr} [X, Y] - 2| : \langle X, Y \rangle = \Gamma \}. \] 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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