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.