Skinning maps are finite-to-one

• 成果类型: Article
• 署名作者: Dumas, David
• 署名单位: University of Illinois System; University of Illinois Chicago; University of Illinois Chicago Hospital
• 刊物名称: ACTA MATHEMATICA
• ISSN/ISSBN: 0001-5962
• DOI: 10.1007/s11511-015-0129-6
• 发表日期: 2015
• 页码: 55-126
• 关键词: hyperbolic structures; trees; SPACES; REPRESENTATIONS; degenerations; VALUATIONS; foliations; CURVES

摘要

We show that Thurston's skinning maps of Teichmuller 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 Kahler 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.