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Acta Mathematica

, Volume 215, Issue 1, pp 55–126 | Cite as

Skinning maps are finite-to-one

  • David Dumas
Article
  • 311 Downloads

Abstract

We show that Thurston’s skinning maps of Teichmüller space have finite fibers. The proof centers around a study of two subvarieties of the \({{\rm SL}_2(\mathbb{C})}\) character variety of a surface—one associated with complex projective structures, and the other associated with a 3-manifold. Using the Morgan–Shalen compactification of the character variety and author’s results on holonomy limits of complex projective structures, we show that these subvarieties have only a discrete set of intersections.

Along the way, we introduce a natural stratified Kähler metric on the space of holomorphic quadratic differentials on a Riemann surface and show that it is symplectomorphic to the space of measured foliations. Mirzakhani has used this symplectomorphism to show that the Hubbard–Masur function is constant; we include a proof of this result. We also generalize Floyd’s theorem on the space of boundary curves of incompressible, boundary-incompressible surfaces to a statement about extending group actions on \({\Lambda}\) -trees.

Keywords

Length Function Isometric Embedding Compacti Cation Saddle Connection Measured Foliation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Institut Mittag-Leffler 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at ChicagoChicagoU.S.A

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