The Teichmüller space $T\_g$ of genus $g$ Riemann surfaces can be parameterized in terms of Fenchel-Nielsen coordinates, taking values in $\mathbb{R}^{3g-3}\times \mathbb{R}\_+^{3g-3}$. 


The Teichmüller space $T\_g$ also has a natural complex structure. 


Could someone suggest a reference where the complex structure is given in terms of Fenchel-Nielsen coordinates?


A more rough question is: very naively I would combine $\mathbb{R}^{3g-3}\times \mathbb{R}\_+^{3g-3}$ into $\mathbb{H}^{3g-3}$ where $\mathbb{H}$ is the upper half plane. In the natural complex structure, is $T\_g$ the same as $\mathbb{H}^{3g-3}$ as complex manifolds? I guess it's not, but is it proved somewhere?



Complex structure of the Teichmüller space in terms of Fenchel-Nielsen coordinates
Fenchel-Nielsen coordinates are real-analytic but not holomorphic. One indirect way to see this is as follows. Royden proved that besides a few exceptions the group of biholomorphisms of Teichmueller space is equal to the mapping class group. In particular it is discrete and countable. This shows that the Fenchel-Nielsen coordinates are not holomorphic, since the group of biholomorphisms of the polydisc $\mathbb{H}^{3g−3}$ is an uncountable Lie group.


