Kleinian groups and iteration of quadratic maps (Q1123262)

If \(X,Y\in SL(2,{\mathbb{C}})\) generate a discrete non elementary group then, according to \textit{T. Jørgensen} [Am. J. Math. 98, 739-749 (1976; Zbl 0336.30007)], the inequality \(| tr^ 2(X)-4| +| tr(XYX^{- 1}Y^{-1})| \geq 1\) holds, where tr denotes the trace. Two similar but stronger estimates are shown to be true under the additional assumption \(3<| tr^ 2(X)| <5\), which essentially means that X, as a Möbius transformation, is not strictly loxodromic. The idea of proof is due to \textit{C. L. Siegel} [Math. Ann. 133, 127-138 (1957; Zbl 0079.039)]: an iteration process in SL(2,\({\mathbb{C}})\) is translated into the iteration of a quadratic function of one complex variable, and one has to study the basin of attraction of a fixed point of that function. By direct computation certain discs are shown to belong to that basin.