Higher dimensional Kleinian groups of small Hausdorff dimension are classical Schottky groups II

This is the higher dimensional continuation of our works on Hausdorff dimension of Kleinian groups {Hou2010}, {Hou2020}. In this paper we prove that for all r≥ 2 there exists λᵣ>0, which is independent of the dimension N of the hyperbolic space, such that any finitely generated non-elementary torsion-free Kleinian group of (HN) generated by r elements with limit set of Hausdorff dimension <λᵣ is a classical Schottky group. For two generators the bound can be taken to be $1/3$. The proof relies on our previous works in {Hou2010}, {Hou2020} and on a metric carrier graph reduction. Our result can also be considered as the complete converse to the well-known theorem of Phillips-Sarnak {PS} for spaces of any dimension N.