Circle packings and co-compact extensions of Kleinian groups (Q1066310)

In this paper we prove: Theorem: Let \(\Gamma\) be a geometrically finite Kleinian group without cusps. Then there exist arbitrarily small quasi- conformal deformations \(\Gamma_{\varepsilon}\) of \(\Gamma\), and groups \(\Gamma_{\varepsilon}^{**}\) containing \(\Gamma\), such that \(H^ 3/\Gamma_{\varepsilon}^{**}\) is compact. When \(\Gamma\) contains cusps, we obtain \(\Gamma_{\varepsilon}^{**}\) containing \(\Gamma_{\varepsilon}\) so that \(H^ 3/\Gamma_{\varepsilon}^{**}\) has finite volume. We also obtain results on circle packings of Riemann surfaces. The proof makes use of continued fraction coordinates for circle packings, introduced in the author's study of classical Schottky groups.