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Transactions of the American Mathematical Society

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A family of non-injective skinning maps with critical points


Author: Jonah Gaster
Journal: Trans. Amer. Math. Soc. 368 (2016), 1911-1940
MSC (2010): Primary 51-XX, 54-XX
DOI: https://doi.org/10.1090/tran/6400
Published electronically: June 15, 2015
MathSciNet review: 3449228
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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

DOI: https://doi.org/10.1090/tran/6400
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).
Article copyright: © Copyright 2015 American Mathematical Society