Kleinian groups of small Hausdorff dimension are classical Schottky groups. I

Abstract

It has been conjectured that the Hausdorff dimensions of nonclassical Schottky groups are strictly bounded from below. In this first part of our work on this conjecture, we prove that there exists a universal positive number $\lambda$ greater than $0$ such that any $2$–generated nonelementary Kleinian group with limit set of Hausdorff dimension less than $\lambda$ is a classical Schottky group.