## Domains of discontinuity for almost-Fuchsian groups

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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.## References

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## Additional Information

**Andrew Sanders**- Affiliation: Department of Mathematics, Statistics and Computer Science, University of Illinois at Chicago, Chicago, Illinois 60607
- Email: andysan@uic.edu
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**369**(2017), 1291-1308 - MSC (2010): Primary 53A10, 30F40; Secondary 37F30
- DOI: https://doi.org/10.1090/tran/6789
- MathSciNet review: 3572274