# To find a point in Teichmüller space or measured foliation, how many lengths of curves do you need?

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 $$6g-6$$?

For the measured lamination case, the fact that $$6g-5$$ curves suffice was shown by Hamenstädt.
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 $$\mathcal{PMF}_g \cong S^{6g-7}$$.