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I was reading the part on Fenchel-Nielsen coordinates, the proof on page 64, I don't understand when they say:

Since $\theta_j(t_1)=\theta_j(t_2)$, $j=1,\ldots,3g-3$, all $g_k$ can be glued together into marking-preserving homeomorphism, say $h$ of $R_{t_1}$ onto $R_{t_2}$. Since $h$ is holomorphic on $R_{t_1}$ except for a finite number of analytic curves, so is $h$ on the entire $R_{t_1}$ by Painleve's theorem...

My questions are:

  • how do different twisting parameters determine different points in the Teichmüller space?

  • I would like to know some reference for this Painleve's theorem.

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Here is the reference for Painleve's theorem:

http://eom.springer.de/p/p071100.htm (the second paragraph).

I don't understand your first question.

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  • $\begingroup$ My first question is related to this part: $\theta_j(t_1)=\theta_j(t_2),j=1,\ldots,3g-3$ all $g_k$ can be glued together into marking-preserving homeomorphism. Why? $\endgroup$
    – daniel
    Commented Aug 10, 2011 at 13:13
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    $\begingroup$ For those of us who don't have the book in front of us, why don't you explain (preferably in the original question) what the notation means? $\endgroup$
    – Igor Rivin
    Commented Aug 10, 2011 at 15:20
  • $\begingroup$ The link to eom.springer.de is broken, but the article can now be found at encyclopediaofmath.org/wiki/Painlev%C3%A9_theorem. $\endgroup$
    – The Amplitwist
    Commented Jul 20, 2022 at 15:18

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