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 $6g-6$?


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

| cite | improve this answer | |
  • $\begingroup$ Thanks, Ian. I don't know why I missed that earlier question. $\endgroup$ – Dylan Thurston Jul 11 '19 at 12:50

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.