# Complex structure of the Teichmüller space in terms of Fenchel-Nielsen coordinates

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?

• 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

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.

• According to Wolpert's magic formula, the Weil-Petersson Kahler form $\omega_{\bf{WP}}$ is known in terms of Fenchel-Nielsen coordinates. This means that the formula $\omega_{\bf{WP}}(X,Y)\equiv g_{\bf{WP}}(I⋅X,Y)$ in which $g_{\bf{WP}}$ is the Weil-Petersson metric in terms of Fenchel-Nielsen coordinates and $X$ and $Y$ are two vector fields (which I think could be taken to be Beltrami differentials), can be used to get the complex structure $I$. Isn't it correct? – QGravity Sep 13 '16 at 5:51

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.

• Could you please suggest a good reference for this? – QGravity Sep 8 '16 at 16:44
• I think this is discussed in Chapter 6 of Hubbard's book: matrixeditions.com/TeichmullerVol1.html – Ian Agol Sep 8 '16 at 16:47
• Thank you! I just have a question: The determinant of the operators on the Riemann surface is defined by the determinant of the corresponding Laplacian which in turn can be written in terms of Selberg zeta function. Is there a notion of holomorphic factorization of Selberg zeta function with respect to the complex structure that you described? In the Selberg zeta function, everything is real, so I was wondering how one can define determinant of Dirac operator directly in terms of Selberg zeta function not through its corresponding Laplacian using this holomorphic factorization? – QGravity Sep 8 '16 at 21:45
• I googled this topic, and found this paper: arxiv.org/abs/math/0505530 – Ian Agol Sep 9 '16 at 3:05
• @QGravity: I suppose that's true, but those statements require proof, which I believe depends on already knowing the complex structure on Teichmüller space, so you have to be sure that your reasoning is not circular. Just knowing the formula for the Weil-Petersson metric and Wolpert's form, you can probably get an almost complex structure. But one must prove it is integrable. – Ian Agol Sep 13 '16 at 5:51

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