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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?

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Even though FN coordinates do not give you the standard complex structure, something similar does work, see 'Horocyclic Coordinates for Riemann Surfaces and Moduli Spaces. I: Teichmuller and Riemann Spaces of Kleinian Groups', by Irwin Kra, Journal of the American Mathematical Society Vol. 3, No. 3 (Jul., 1990), pp. 499-578. –  Misha Oct 4 '13 at 3:46
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3 Answers

up vote 9 down vote accepted

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.

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Although Fenchel-Nielsen coordinates do not give a complex-analytic parameterization of Teichmuller space, they do extend to give a complex-analytic parameterization of $T_g\times \overline{T}_g$ on a subset. This follows from Bers' simultaneous uniformization theorem, which parameterizes quasi-fuchsian groups by the Teichmuller spaces of the two components of the domain of discontinuity. In turn, (marked) quasi-fuchsian groups are parameterized by the character variety, which in turn is determined by a subset of complex Fenchel-Nielsen coordinates $\subset \mathbb{C}^{3g-3}\times \mathbb{C}^{* 3g-3}$. Bers then obtained a complex structure on Teichmuller space by fixing a conformal structure $x$ on one domain of discontinuity, and letting the other one vary, to get $T_g\times \{x\}\subset \mathbb{C}^{6g-6}$. He shows that as one varies $x$, the complex structure on $T_g$ does not change, and thus one obtains a well-defined complex structure on Teichmuller space, which is equivalent to the complex structure originally defined by Ahlfors.

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Probably you want to check chapter 6 of this book. An introduction to Teichmüller spaces 1992 Yoichi Imayoshi, Masahiko Taniguchi.

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