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 $b \geq 1$ 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, if the proof is easy, a hint on where to start so I can try to come up with the rest of it by myself. 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.

Attempt: In [3] and [4] it is stated on the first page that this is a consequence of Wolpert's studies on (the convexity of) geodesic length functions, found in [5] and [6]. I tried to show with the results in [5] and [6] that $6g-6$ geodesic length functions can't determine the Fenchel-Nielsen coordinates on $T_{g,0}$, but sadly I couldn't conclude anything.

Edit: I did some more research in the past few days that I haven't mentioned in the attempt, but I decided to add them now. I found the following:

  • In [7], it is stated that Scott Wolpert proved a result which is equivalent to the following: "any $6g - 6$ absolute values of traces of hyperbolic elements in a marked Fuchsian group can not give global (even locally) real analytic coordinates for $T_{g,0}$". Understanding this statement is equivalent to my question since these traces correspond to the lengths of geodesics in the Riemann surface the Fuchsian group corresponds to. However, it is not stated in [3] which result of Wolpert is equivalent to the above statement. I thought that I may find the result in [5] and [6], but I couldn't conclude anything.
  • On the last page in [8], I found the same statement as in [7], but the given reference is a "personal conversation" with Scott Wolpert. It is also stated in Irwin Kra's chapter in the book "Holomorphic Functions and Moduli II" that the result was obtained from Scott Wolpert by "oral communication", so maybe it was never written down explicitly (the statement of Kra is found on the first page in the preview here)
  • Last but not least, I followed the reference Kra gave in his chapter and landed here [9]. I tried to study this paper too, but I have to admit that it was far to advanced for me in order to conclude something.

Edit 2: Thanks to a comment of Mr. Agol in the question on MO, 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.


[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)

[3] Y. Okumura, Global real analytic length parameters for Teichmüller spaces, Hiroshima Math. J. 26, 1996, no. 1, 165–179 (found here)

[4] G. Nakamura and T. Nakanishi, Parametrizations of some Teichmüller spaces by trace functions, Conform. Geom. Dyn. 17, 2013, (found here)

[5] S. A. Wolpert, On the symplectic geometry of deformations of a hyperbolic surface, Ann. of Math. 117, 1983, 207-234 (found here)

[6] S. A. Wolpert, Geodesic length functions and the Nielsen problem, J. Differential Geom. 25, 1987, 275-296 (found here)

[7] Y. Okumura, On the global real analytic coordinates for Teichmüller spaces, J. Math. Soc. Japan 42, 1990, no. 1, 91–101 (found here)

[8] L. Keen, Trace moduli for quasifuchsian groups, J. Math. Kyoto Univ. (JMKYAZ) 26-1, 1986, 81-94 (found here)

[9] I. Kra, Uniformization, automorphic forms and accessory parameters, RIMS (Kyoto Univ.), Kokyuroku 571, 1985, 54–84 (found here)

  • $\begingroup$ It has been almost three weeks now since I asked this question and sadly I didn't get any further. I opened a bounty now and hope that someone sees this question who can help. In case it doesn't get any answers after the bounty, I'd be very happy about a comment if this might be an appropriate question (after editing) for MathOverflow or not (I have difficulties in telling this by myself since I'm an undergraduate student and doesn't know if this question or result might be of interest to other mathematicians as well). $\endgroup$ – user331406 Jun 28 '16 at 9:58
  • $\begingroup$ I think it is because it's unclear what do you really want. Do you have doubts about the proof? Do you want somebody to come up with an alternative argument? $\endgroup$ – Moishe Kohan Jul 1 '16 at 0:07
  • $\begingroup$ Thank you for your comment. I am looking for a proof to my question, or, if it's an easy proof, a hint to it so I can try to come up with the rest of it. I haven't found one in the literature, and the only thing I know is that it has something to do with the convexity of the length functions and that it might be statet in terms of marked Fuchsian groups (as I said under "attempt" and "edit"). I will add a small section to my question in which I say what I am looking for. $\endgroup$ – user331406 Jul 1 '16 at 6:46
  • $\begingroup$ Maybe I should delete the sections "attempt" or "edit" to keep the overall post more readable? $\endgroup$ – user331406 Jul 1 '16 at 7:33
  • $\begingroup$ I think this question would be more suited for MathOverflow. Have you tried posting there as well? $\endgroup$ – rubik Jul 1 '16 at 9:12

As proposed in the comments, I posted this question on MathOverflow and it got an answer there. Here is the link:

The question on MathOverflow (answered)

Thanks to everyone who read the question and thought about it.


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