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Domains of discontinuity for almost-Fuchsian groups

Author: Andrew Sanders
Journal: Trans. Amer. Math. Soc. 369 (2017), 1291-1308
MSC (2010): Primary 53A10, 30F40; Secondary 37F30
Published electronically: August 18, 2016
MathSciNet review: 3572274
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Abstract: An almost-Fuchsian group $ \Gamma <\mathrm {Isom}^{+}(\mathbb{H}^3)$ is a quasi-Fuchsian group such that the quotient hyperbolic manifold $ \mathbb{H}^3/\Gamma $ 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 $ R>0$ in $ \mathbb{C}\cup \{\infty \} =\partial _{\infty }(\mathbb{H}^3).$ 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.

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Additional Information

Andrew Sanders
Affiliation: Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607

Keywords: Minimal surfaces, quasi-Fuchsian groups, hyperbolic 3-manifolds, hyperbolic Gauss map, quasi-conformal maps
Received by editor(s): October 23, 2013
Received by editor(s) in revised form: June 3, 2015
Published electronically: August 18, 2016
Additional Notes: The author gratefully acknowledges partial support from the National Science Foundation Postdoctoral Research Fellowship
Article copyright: © Copyright 2016 American Mathematical Society

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