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

TL;DR

This paper proves that all finitely generated Kleinian groups with sufficiently small Hausdorff dimension are classical Schottky groups, confirming a conjecture about the lower bound of Hausdorff dimensions for nonclassical Schottky groups.

Contribution

It establishes a universal positive lower bound for Hausdorff dimensions below which all such groups are classical Schottky groups, extending previous methods.

Findings

Existence of a universal positive number λ such that groups with Hausdorff dimension < λ are classical Schottky groups

Resolution of the conjecture that nonclassical Schottky groups have Hausdorff dimensions bounded away from zero

Generalization of previous techniques to prove the main result

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}.