A family of non-injective skinning maps with critical points

Gaster, Jonah

doi:10.1090/tran/6400

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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