Skinning maps are finite-to-one

David Dumas Department of Mathematics, Statistics, and Computer Science, University of Illinois at Chicago

Complex Variables and Complex Analysis mathscidoc:1701.08004

Acta Mathematica, 215, (1), 55-126, 2012.6
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.
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  title={Skinning maps are finite-to-one},
  author={David Dumas},
  booktitle={Acta Mathematica},
David Dumas. Skinning maps are finite-to-one. 2012. Vol. 215. In Acta Mathematica. pp.55-126.
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