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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 $6g-6$?

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As indicated in my comments, the Teichmüller question is a duplicate of this question.

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., 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. MR2008332

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 $\mathcal{PMF}_g \cong S^{6g-7}$.

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  • $\begingroup$ Thanks, Ian. I don't know why I missed that earlier question. $\endgroup$ – Dylan Thurston Jul 11 at 12:50

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