DOMAINS OF DISCONTINUITY FOR ALMOST-FUCHSIAN GROUPS

Abstract. An almost-Fuchsian group Γ < Isom+(H3) is a quasi-Fuchsi-an group such that the quotient hyperbolic manifold H3/Γ contains a closed incompressible minimal surface with principal curvatures con-tained in (−1, 1).We show that the domain of discontinuity of an almost-Fuchsian group contains many balls of a fixed spherical radius R> 0 in C ∪ {∞} = ∂∞(H3). This yields a necessary condition for a quasi-Fuchsian group to be almost-Fuchsian which involves only conformal ge-ometry. As an application, we prove that there are no doubly-degenerate geometric limits of almost-Fuchsian groups. 1