Jrgensen Number and Arithmeticity
Abstract
A Jrgensen group is a non-elementary Kleinian group that can be generated by two elements for which equality holds in Jrgensen's Inequality. This paper shows that the only torsion-free Jrgensen group is the figure-eight knot group, identifies all non-cocompact arithmetic Jrgensen groups, and establishes a characterization of cocompact arithmetic Jrgensen groups. The paper also defines and computes the Jrgensen number of several non-cocompact Kleinian groups including some two-bridge knot and link groups.
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