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?