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

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Yong Hou, Zanty Electronics
Fine Hall 314

By the inspirational works of Peter Sarnak and Ralph Phillips (ACTA 1985), we know that all classical Schottky groups (dim n >3) must have Hausdorff dimension strictly bounded away from dim n-1. Later Peter Doyle (ACTA 1988) showed that it is also true for n=3. But the natural question of nonclassical Schottky groups should have Hausdorff dimension strictly bounded from below away from 0 remains open. In this second part of our works on geometric structure of Kleinian groups of small Hausdorff dimensions we provide positive solution to this question. In particular 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 \$