Jørgensen’s inequality for classical Schottky groups of real type. (English) Zbl 0928.30026

Let \(G=\langle A_1,A_2\rangle\) be a non-elementary discrete (Möbius transformation) group. Define \(J(G)=| tr^2(A_1)- 4|+| tn (A_1A_2A_1^{-1} A_2^{-1})-2|\). Jorgensen’s amazing inequality states that \(J(G)\geq 1\). This paper gives (best possible) lower bounds for certain classes of classical real Schottky groups. The classes are defined by geometric properties of the axes of \(A_1\) and \(A_2\). There is a fine bibliography of the considerable amount of related work.


30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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