A geometric approach to Jørgensen's inequality (Q755930)

Let \(<A,B>\) be a non-elementary purely hyperbolic subgroup of PSL(2,\({\mathbb{R}})\) generated by A and B. It is shown that \(| tr[A,B]- 2| >16\) if the asees of A and B are disjoint and \(| tr[A,B]- 2| >4\) if the asees of A and B intersect and these inequalities are sharp. This result is a trivial consequence and reformulation of a result in [\textit{G. Rosenberger}, Math. Ann. 199, 213-227 (1972; Zbl 0244.20057)] or in [\textit{N. Purzitsky}, Math. Z. 126, 209-223 (1972; Zbl 0232.20104)].