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A Fuchsian model for a Riemann surface $X$ is a discrete subgroup $G$ of $PSL_2(\mathbb{R})$ such that there is a biholomorphic map from $U/G$ to $X$.

For a fixed genus $g \geq 2$ one knows from Bers embedding theorem that there is a holomorphic family of all Fuchsian models parametrized by the Teichmueller space $T_g$. More precisely, the mapping $t \mapsto G_t$ depends holomorphically on $t$ where $G_t$ is the Fuchsian model for the Riemann surface represented by $t$.

My question is: does there exist another parametrization of Fuchsian models which is not the one I described above?

Or another question, that is very much connected to my question: is there another method than Bers' embedding to induce a complex structure on the Teichmueller space $T_g$?

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$U=PSL_2(\mathbb R)$ –  Kimball Mar 4 '11 at 4:41
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I think $U$ is the upper half plane. –  S. Carnahan Mar 4 '11 at 9:07
    
Right. I wrote that a little too fast. –  Kimball Mar 4 '11 at 15:44
    
I'll delete my comment if you delete yours. –  S. Carnahan Mar 4 '11 at 17:30
    
The original construction of a complex structure is due to Ahlfors: ams.org/mathscinet-getitem?mr=124486 –  Ian Agol Mar 4 '11 at 22:28
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2 Answers 2

I think the first question is not well-formed. The Bers construction gives a holomorphically varying family of quasi-Fuchsian groups $G(t)$ (in $PSL_2(\mathbb{C})$) each of which has a simply-connected invariant domain $U(t)$ whose quotient is biholomorphic to the Riemann surface represented by $t$. Both the groups and the domains are varying, as is necessary to obtain a holomorphic family.

For the second question, do you care about the actual complex structure (i.e. charts mapping open subsets of $\mathcal{T}_g$ into $\mathbb{C}^n$) or just the almost complex structure (multiplication by $i$ on a tangent space)?

The almost complex structure is easy to see in most models for Teichmuller space, because the tangent space at a point is typically represented by a complex vector space. For example, $T_X \mathcal{T}_g$ can be seen as the linear dual of $Q(X)$, the vector space of holomorphic quadratic differentials on the Riemann surface $X$. Multiplication of differentials by the constant function $i$ induces the almost complex structure.

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Since you seem to be working with curves that don't have punctures, your questions lie in the purview of Grothendieck's 1960-61 Seminaire Cartan paper: Techniques de construction en géométrie analytique. I. Description axiomatique de l'espace de Teichmüller et de ses variantes (I hope you like large pdfs in French). This paper answers your second question in the positive, and even better, describes a moduli functor on complex analytic spaces that is represented by $T_g$.

I don't know enough about Fuchsian models to say anything about your first question.

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