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 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 -trees.
Citation
David Dumas. "Skinning maps are finite-to-one." Acta Math. 215 (1) 55 - 126, 2015. https://doi.org/10.1007/s11511-015-0129-6
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