Kleinian groups of small Hausdorff dimension are classical Schottky groups. I (Q1035313)

The author proves that there exists a universal positive number \(\lambda\) such that any 2-generated nonelementary Kleinian group whose limit set has Hausdorff dimension less than \(\lambda\) is a classical Schottky group. This result can be viewed as a converse to a result of \textit{P. G. Doyle} in [Acta Math. 160, No. 3--4, 249--284 (1988; Zbl 0649.30036)], who showed that there exists a universal upper bound on the Hausdorff dimension of the limit sets of finitely generated classical Schottky groups and of \textit{R. S. Phillips} and \textit{P. Sarnak} in [Acta Math. 155, 173--241 (1985; Zbl 0611.30037)], which proved that there exists a universal upper bound on the Hausdorff dimension of the limit sets of classical Schottky groups of dimension greater than 3.