All finitely generated Kleinian groups of small Hausdorff dimension are classical Schottky groups

Abstract

This is the second part of the works on Hausdorff dimensions of Schottky groups. It has been conjectured that the Hausdorff dimensions of nonclassical Schottky groups are strictly bounded from below. In this second part of our works we provide a resolution of this conjecture, we prove that there exists a universal positive number $\lambda>0$, such that any finitely-generated non-elementary Kleinian groups with limit set of Hausdorff dimension $<\lambda$ are classical Schottky groups. We will generalize our previous technologies given in \cite{HS} to prove our general result. Our result can be consider as a converse to \cite{Doyle}.