Jørgensen number and arithmeticity

Abstract

Abstract. The Jørgensen number of a rank-two non-elementary Kleinian group Γ isJørgensen's Inequality guarantees J(Γ) ≥ 1, and Γ is a Jørgensen group if J(Γ) = 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) for several noncocompact Kleinian groups including some two-bridge knot and link groups.

The realization problem for Jørgensen numbers

“…Remark 4.4. Li-Oichi-Sato [11] [10] [12] considered the case µ = ik, k ∈ R. They conjectured that, for any Jørgensen group G of parabolic type, there exists a marked group G σ,ik = M, N σ,ik , σ ∈ C \ {0}, k ∈ R, such that G σ,ik is conjugate to G. Later, Callahan [3] found counter examples for this conjecture. Now, we compute some examples.…”

“…Jørgensen's inequality is sharp, and if J(G) = 1, G is called a Jørgensen group, There are many results on Jørgensen groups in the literature [5,8,10,11,12,15,18,19,23]. Also, calculating Jørgensen numbers can be difficult and interesting challenge [3,6,20]. Oichi-Sato [16] asked the following natural question: Question 1.1 (The realization problem).…”

“…When is there a non-elementary Kleinian group whose Jorgensen number is equal to r? Oichi-Sato [16] claimed that if r = 1, 2, 3 or r ≥ 4, then there is a non-elementary Kleinian group whose Jørgensen number is equal to r. (See also [3,23,24].) Though Jørgensen's inequality is sharp, constructing a Kleinian group with small Jørgensen Figure 1.…”

“…In this paper, we show that, for any r ≥ 1, there exists a non-elementary Kleinian group whose Jørgensen number is equal to r. This answers a question posed by Oichi and Sato. We also present our computer generated picture which estimates Jørgensen numbers from above in the diagonal slice of Schottky space.Oichi-Sato [16] claimed that if r = 1, 2, 3 or r ≥ 4, then there is a non-elementary Kleinian group whose Jørgensen number is equal to r. (See also [3,23,24].) Though Jørgensen's inequality is sharp, constructing a Kleinian group with small Jørgensen 2010 Mathematics Subject Classification.…”

On Gehring–martin–tan Groups With an Elliptic Generator

“…A description of the set of all two-generated discrete nonelementary groups of isometries H 3 for which the equality holds in the Jørgensen inequality is an open problem of special interest. For many elegant results concerning this problem, see [4,7,17] and references therein.…”