To parametrize Teichmüller space, it suffices to measure the hyperbolic lengths of a finite number of curves. It is well-known that $9g-9$ curves suffice, by a standard pair-of-pants argument given in, for instance Fathi-Laudenbach-Poenaru.


I recall that you need exactly $6g-5$ curves: you cannot achieve it by $6g-6$, because the character variety is not an algebraic subset of $\mathbb{C}^{6g-6}$, but one extra curve suffices. Is this correct, and if so, who proved it?


Likewise for measured foliations, the cone over the boundary at infinity: can you parametrize measured foliations with $6g-5$ curves, and how to see that you cannot do it with $6g-6$?



To find a point in Teichmüller space or measured foliation, how many lengths of curves do you need?
As indicated in my comments, the Teichmüller question is a duplicate of this [question](http://mathoverflow.net/q/243622/1345). 


For the measured lamination case, the fact that $6g-5$ curves suffice was shown by Hamenstädt.


*Hamenstädt, Ursula*, [Parametrizations of Teichmüller space and its Thurston boundary.](https://link.springer.com/chapter/10.1007/978-3-642-55627-2_5), Hildebrandt, Stefan (ed.) et al., Geometric analysis and nonlinear partial differential equations. Berlin: Springer (ISBN 3-540-44051-8/hbk). 81-88 (2003). [ZBL1044.32005](https://zbmath.org/?q=an:1044.32005). [MR2008332](https://mathscinet.ams.org/mathscinet-getitem?mr=2008332) 


To see that $6g-6$ curves do not suffice, suppose we have $6g-6$ curves $(a\_1,\ldots, a\_{6g-6})$ and an embedding $$\mathcal{MF}\_g \hookrightarrow \mathbb{R\_{\geq 0}}^{6g-6}-\{{\bf 0}\}, \lambda \mapsto (i(a\_1,\lambda),\ldots, i(a\_{6g-6},\lambda)).$$ Then we would get an embedding $$\mathcal{PMF}\_g \hookrightarrow \Delta^{6g-7} \subset \mathbb{RP}^{6g-7}.$$ But this is impossible by invariance of domain [since](https://en.wikipedia.org/wiki/Thurston_boundary) $\mathcal{PMF}\_g \cong S^{6g-7}$. 


