A key construction in Thurston's proof of the existence of hyperbolic structures on Haken manifolds is the so-called "skinning map" associated to a 3-manifold $M$ with boundary whose interior admits a hyperbolic metric. The map
$$\sigma\_M:\mathrm{Teich}(\partial M) \to \mathrm{Teich}(\overline{\partial M})$$ takes a conformal structure on the boundary and produces a new one. Conformal structures on $\partial M$ are in bijection with geometrically finite hyperbolic metrics on $M$. See [here](https://people.math.harvard.edu/%7Ectm/papers/home/text/papers/rs/rs.pdf) for a quick reference.


The problem is, I'm having trouble understanding what exactly this map is. The most I've gathered is that one can lift a hyperbolic structure on $M$ to the covering space $\tilde M$ of the interior corresponding to the image of $\pi\_1(\partial M)\to\pi\_1(M)$, and that this induces a new conformal structure, but how? In particular, how can the covering space $\tilde M$ be described as a manifold? Is it just $\partial M\times \mathbb{R}$?



Definition of Thurston's skinning map
Let’s simplify to the case where $M$ has exactly one boundary component, say $\partial M = S$. So the hyperbolic structures on $M$ are parametrised by the conformal structures on $S$. Fix one such conformal structure $\rho$ and lift the resulting hyperbolic structure on $M$ to the cover corresponding to $\pi\_1(S)$. As you suggest, this cover is homeomorphic to $S \times \mathbb{R}$. So it has two ends, each giving a conformal structure on $S$. One of these is $\rho$, the conformal structure we started with. The other is the (orientation reverse of) $\rho'= \sigma\_M(\rho)$ - that is, the image of $\rho$ under the skinning map.


I believe that $\sigma\_M$ is called the "skinning map" because $\rho \sqcup \rho'$ gives the "skin at infinity" of the lifted hyperbolic structure on $S \times \mathbb{R}$.


