Kleinian groups with small limit sets (Q1325168)

The authors prove that if \(\Gamma\) is a finitely generated Kleinian group with limit set \(L(G)\) such that \(\dim_ H(L(\Gamma))< 1\) then \(\Gamma\) is indeed geometrically finite and is almost quasi-conformally equivalent to a Fuchsian group of the second kind. This result is complemented by similar slightly different results for \(\dim_ H(L(\Gamma))=1\). Under the much stronger assumption that \(\Gamma\) is convex cocompact similar results were known and were due to Bowen, Braam and Sullivan. The idea behind the proof of this remarkable result is surprisingly simple. First of all by Selberg's Lemma one may assume that \(\Gamma\) is torsion-free. Using results of Chuckrow and Maskit (and a result of Ahlfors-Beurling showing that as the 1-measure of \(L(\Gamma)\) is zero every univalent analytic function on \(\mathbb{C}_ \infty-L(\Gamma)\) is Möbius), the authors deduce either \(\Gamma\) contains a group for which one knows that \(\dim_ H(L(\Gamma))\geq 1\) or it is quasi-conformal to a Fuchsian group. This suffices in this case. When \(\dim_ H(L(\Gamma))=1\) the argument is a little more complex but runs along the same lines. The authors also give a number of variants.