I asked this question almost a month ago on Math SE. After waiting three weeks for an answer or a comment, I opened a bounty on the question in hope that it might get an answer this way. The bounty expires in a day, and sadly I didn't get any answer or hint on it until now. However, someone said in a comment that this question might be better suited for MathOverflow, and this is why I decided to post it now on MO. I apologize if this question isn't appropriate for MathOverflow.

For better readability, I decided not to fully copy-paste my question and only write the important parts. For the research and attempt I've done on the question and for further references, I kindly refer to the original question found here:

The original question on Math SE

**Setting:** It is well known that the Teichmüller space $T_{g,b}$ of an oriented Riemann surface $S_{g,b}$ of genus $g \geq 2$ with $b \geq 1$ geodesic boundary components (satisfying $2g + b \geq 3$) can be globally parametrized by $6g - 6 + 3b$ geodesic length functions, a number which equals its dimension (see Schmutz [1]).

However, in the case of a closed surface $S_{g,0}$ (i.e. without boundary components), the minimal number of geodesic length functions needed for a parametrization of the corresponding Teichmüller space $T_{g,0}$ is $6g-5$ (first proven by Schmutz [1]), which is greater than its dimension (which is $6g-6$). The fact that $6g-5$ is minimal is not proven in Schmutz' paper. Note that Schmutz' given parametrization is to be understood as embedding of $T_{g,0}$ into $\mathbb{R}^{6g-5}$ via the length functions of $6g-5$ geodesics, and not as homeomorphism between $T_{g,0}$ and $\mathbb{R}^{6g-5}$.

**Question:** The question is why a global parametrization of the Teichmüller space $T_{g,0}$ (of a closed surface $S_{g,0}$) by $6g-6$ geodesic length functions is not possible, i.e. why the number $6g-5$ is minimal.

**What I am looking for:** A proof to the question or a reference where I can find a proof. Also hints that may lead to a proof would be kindly appreciated. I know from several references (see original question on Math SE) that it has something to do with Scott Wolpert's studies on the convexity of the geodesic length functions and that the result might be stated in terms of trace functions for marked Fuchsian groups (these references also state that Wolpert first found the result). Overall, I'd also be interested in the reason why the parametrization can be done by a number of geodesic length functions that equals the dimension of the Teichmüller space in the case of surfaces with boundary components, but not in the case of closed surfaces.

**Edit:** Thanks to a comment of Mr. Agol I noticed a mistake in my initial statement of Hamenstädt's result. What is shown in [2] is that for surfaces with $n \geq 1$ punctures, $6g - 5 + 2n$ geodesic length functions provide coordinates on $T_{g,n}$, and **not** $6g - 6 + 2n$ as I stated. So in case of surfaces with punctures, we have the same situation as for closed surfaces.

**References:**

[1] P. Schmutz, *Die Parametrisierung des Teichmüllerraumes durch geodätische Längenfunktionen*, Comment. Math. Helv. 68, 1993, no. 2, 278-288 (found here in german or here in french, sadly not available in english)

[2] U. Hamenstädt, *Length functions and parametrizations of Teichmüller space for surfaces with cusps*, Ann. Acad. Sci. Fenn. Math. Vol. 28, 2003, 75 - 88 (found here)