Jørgensen number and arithmeticity

Abstract

The Jørgensen number of a rank-two non-elementary Kleinian group Γ \Gamma is \[ J ( Γ ) = inf { | t r 2 X − 4 | + | t r [ X , Y ] − 2 | : ⟨ X , Y ⟩ = Γ } . J(\Gamma ) = \inf \{|\mathrm {tr}^2 X - 4| + |\mathrm {tr} [X, Y] - 2| : \langle X, Y \rangle = \Gamma \}. \] Jørgensen’s Inequality guarantees J ( Γ ) ≥ 1 J(\Gamma ) \geq 1 , and Γ \Gamma is a Jørgensen group if J ( Γ ) = 1 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 ( Γ ) J(\Gamma ) for several non-cocompact Kleinian groups including some two-bridge knot and link groups.