Thurston's celebrated compactification of Teichmuller space was first described in his famous Bulletin paper. Teichmuller space is famously homeomorphic to an open disc of some dimension (this can be seen using Fenchel-Nielsen coordinates), which is $6g-6$ for closed surfaces of genus $g\geq 2$.
Thurston embeds Teichmuller space in some infinite-dimensional natural space (the projective space of all real functions on isotopy classes of simple closed curves) and studies its closure there. The closure is realised by adding some points that correspond geometrically to some particular objects (called projective measured foliations). The added points are homeomorphic to a sphere of dimension $6g-7$ and the resulting topological space is just a closed disc, the new points forming its boundary.
The Bulletin paper contains almost no proofs. The only complete proofs I know for this beautiful piece of mathematics is described in the book Travaux de Thurston sur les surfaces of Fathi-Laudenbach-Ponearu (an english translation written by Kim and Margalit is available here). The homeomorphism between the space of projective measured laminations and the sphere $S^{6g-7}$ as explained there is clear and natural, it's obtained by re-adapting the Fenchel-Nielsen coordinates to the context of measured foliations.
The proof that the whole compactified space is homeomorphic to $D^{6g-6}$ is however more involved and less direct. First they study some charts to prove that we get a topological manifold with boundary, and that's ok. The compactification is thus a topological manifold with boundary homeomorphic to $S^{6g-7}$, whose interior is homeomorphic to an open ball of dimension $6g-6$. Are we done to conclude that the compactification is a closed disc? Yes, but only by invoking a couple of deep results: the existence of a collar for topological manifolds, and the topological Schoenflies Theorem in high dimension. That's the argument used in the book.
Is there a more direct description of the homeomorphism between Thurston's compactification and the closed disc $D^{6g-6}$?
Is there in particular a Fenchel-Nielsen-like parametrization of the whole compactification?
Best Answer
One natural attempt to compactify Teichmuller space is by the visual sphere of the Teichuller metric. However, Anna Lenzhen showed that there are Teichmuller geodesics which do not limit to $PMF$ (in fact, I think it was known before by Kerckhoff that the visual compactification is not Thurston's compactification).However, it was shown by Cormac Walsh that if one takes Thurston's Lipschitz (asymmetric) metric on Teichmuller space, and take the horofunction compactification of this metric, one gets Thurston's compactification of Teichmuller space. In fact, he shows in Corollary 1.1 that every geodesic in the Lipschitz metric converges in the forward direction to a point in Thurston's boundary. I think this gives a new proof that Thurston's compactification gives a ball.As Misha points out, it's not clear that the horofunction compactification is a ball.
Another approach was given by Mike Wolf, who gave a compactification in terms of harmonic maps, in The Teichmüller theory of harmonic maps, and showed that this is equivalent to Thurston's compactification (Theorem 4.1 of the paper). Wolf shows that given a Riemann surface $\sigma \in \mathcal{T}_g$, there is a unique harmonic map to any other Riemann surface $\rho \in \mathcal{T}_g$ which has an associated quadratic differential $\Phi(\sigma,\rho) dz^2 \in QD(\sigma)$ ($QD(\sigma)$ is naturally a linear space homeomorphic to $\mathbb{R}^{6g-6}$). Wolf shows that this is a continuous bijection between $\mathcal{T}_g$ and $QD(\sigma)$, and shows that the compactification of $QD(\sigma)$ by rays is homeomorphic to Thurston's compactification $\overline{\mathcal{T}_g}$ in Theorem 4.1. I skimmed through the proof, and as far as I can tell the proof of the homeomorphism does not appeal to the fact that Thurston's compactification is a ball, so I think this might give another proof that it is a ball.