Skinning maps are finite-to-one

TL;DR

This paper proves that Thurston's skinning maps have finite fibers by analyzing character varieties and introduces a stratified Kahler metric, connecting complex structures, measured foliations, and group actions.

Contribution

It demonstrates the finiteness of fibers of skinning maps and introduces a stratified Kahler metric linked to measured foliations, extending previous results.

Findings

Thurston's skinning maps have finite fibers.

A natural stratified Kahler metric is introduced and shown to be symplectomorphic to measured foliations.

The Hubbard-Masur function is proven to be constant.

Abstract

We show that Thurston's skinning maps of Teichmuller space have finite fibers. The proof centers around a study of two subvarieties of the SL_2(C) character variety of a surface, one associated to complex projective structures and the other associated to a 3-manifold. Using the Morgan-Shalen compactification of the character variety and the results of [arXiv:1105.5102] 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 Lambda-trees.