Skinning maps are finite-to-one

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.

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Correspondence to David Dumas.

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Dumas, D. Skinning maps are finite-to-one. Acta Math 215, 55–126 (2015). https://doi.org/10.1007/s11511-015-0129-6

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Keywords

  • Length Function
  • Isometric Embedding
  • Compacti Cation
  • Saddle Connection
  • Measured Foliation