Domains of discontinuity for almost-Fuchsian Groups

TL;DR

This paper investigates the geometric properties of almost-Fuchsian groups, showing their domains of discontinuity contain many balls, and establishes conditions distinguishing them from other quasi-Fuchsian groups, with implications for their limits.

Contribution

It provides a new geometric criterion for almost-Fuchsian groups based on their domains of discontinuity and rules out certain degenerations.

Findings

Domains of discontinuity contain many fixed-radius balls.

Necessary conformal geometric condition for almost-Fuchsian groups.

No doubly-degenerate limits of almost-Fuchsian groups.

Abstract

An almost-Fuchsian group is a quasi-Fuchsian group such that the quotient hyperbolic manifold contains a closed incompressible minimal surface with principal curvatures contained in (-1,1). We show that the domain of discontinuity of an almost-Fuchsian group contains many balls of a fixed spherical radius in the visual boundary of hyperbolic 3-space. This yields a necessary condition for a quasi-Fuchsian group to be almost-Fuchsian which involves only conformal geometry. As an application, we prove that there are no doubly-degenerate geometric limits of almost-Fuchsian groups.