Confusing definitions of Modular Group and Teichmüller space

Question

Notations

1.$\Sigma_g$ is the Reimann surface with genus $g$

2.$M_g$ is the space of all metrics

3.Diff($\Sigma_g$) is the diffeomorphism on $\Sigma_g$

4.$\text{Diff}_0(\Sigma)$ is the connected component of the identity of Diff($\Sigma_g$)

5.Weyl($\Sigma_g$) is the Weyl transfromation on $\Sigma_g$

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Contexts

On the page 470 in Nakahara, the author defines the Moduli space as : $$M_g / \text{Diff}(\Sigma_g)\times \text{Weyl}(\Sigma_g)$$

Teichmüller space as : $$\text{Teich}(\Sigma_g) = M_g / \text{Diff}_0(\Sigma_g)\times \text{Weyl}(\Sigma_g) $$

and Modular Group(MG) or Mapping Class Group (MCG) as : $$\text{Diff}(\Sigma_g)/ \text{Diff}_0(\Sigma_g) $$

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However, as one discusses the torus, there seems to be different definitions of these terminologies. (If I understand correctly )One can define the atlas of the torus by the equivalence relation on $\mathbb{C}$ :

$$z \sim z + n \omega_1 + m \omega_2 \quad ,\quad m,n \in \mathbb{Z}$$

Define $\tau \equiv \frac{\omega_2}{\omega_1}$. In Nakahara p.270, the modular transformation is generated by $\tau \rightarrow \tau +1$ and $\tau \rightarrow -\frac{1}{\tau}$. The torus with the same complex structure can be related by modular transformation. Besides, in Gleb Arutyunov - String lecture note p.90, $\tau$ is the Teichmüller parameter and the region $\text{Im} \tau > 0$ is the Teichmüller space.

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Questions

Are these two kinds of definitions (MG,Teich space and modular transformation)equivalent? Why? It seems to me that they are totally different since the former involves the metric structure; nevertheless, the latter doesn't. However, they seems to be exactly the same in the case of torus.

Answer 1

I'm not sure I've ever seen the Teichmuller space and the modular space defined directly using the full space of metrics modulo Weyl transformations; perhaps that's a physics thing.

Instead, the usual practice in geometric topology and geometric analysis is to apply the Uniformization Theorem, which says that every conformal class of metrics has a unique representative which is normalized in an appropriate manner: when $g \ge 2$ this means that the sectional curvature is $-1$; whereas when $g=1$ this means that the sectional curvature is $0$, and one special geodesic loop on the torus has length equal to $1$. One side effect is that taking the quotient by Weyl transformations is entirely unnecessary.

It just so happens that in the case that $g=1$ a special thing happens: the space of normalized metrics may be defined as you say using lattices in $\mathbb C$. For $g \ge 2$, there's no such special thing happening.