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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Domains of discontinuity for almost-Fuchsian groups
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by Andrew Sanders PDF
Trans. Amer. Math. Soc. 369 (2017), 1291-1308 Request permission

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
  • 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