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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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A family of non-injective skinning maps with critical points
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by Jonah Gaster PDF
Trans. Amer. Math. Soc. 368 (2016), 1911-1940 Request permission

Abstract:

Certain classes of 3-manifolds, following Thurston, give rise to a ‘skinning map’, a self-map of the Teichmüller space of the boundary. This paper examines the skinning map of a 3-manifold $M$, a genus-2 handlebody with two rank-1 cusps. We exploit an orientation-reversing isometry of $M$ to conclude that the skinning map associated to $M$ sends a specified path to itself and use estimates on extremal length functions to show non-monotonicity and the existence of a critical point. A family of finite covers of $M$ produces examples of non-immersion skinning maps on the Teichmüller spaces of surfaces in each even genus, and with either $4$ or $6$ punctures.
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Additional Information
  • Jonah Gaster
  • Affiliation: Department of Mathematics, Statistics, and Computer Science, University of Illinois - Chicago, 322 Science and Engineering Offices (M/C 249), 851 S. Morgan Street, Chicago, Illinois 60607-7045
  • Address at time of publication: Department of Mathematics, Boston College, Chestnut Hill, Massachusetts 02467
  • Email: gaster@math.uic.edu, gaster@bc.edu
  • Received by editor(s): January 10, 2014
  • Published electronically: June 15, 2015
  • Additional Notes: The author gratefully acknowledges support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367, “RNMS: GEometric structures And Representation varieties” (the GEAR Network).
  • © Copyright 2015 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 1911-1940
  • MSC (2010): Primary 51-XX, 54-XX
  • DOI: https://doi.org/10.1090/tran/6400
  • MathSciNet review: 3449228