There is a theorem saying that the moduli space of conformal structure on a torus is parametrized by the set $$\mathcal M = \{a+bi \in \mathbb C: -1/2 < a\leq 1/2,\ a^2+ b^2\geq 1, a\geq 0 \text{ when }a^2+b^2=1\}$$
and on each point $c = a+bi\in \mathcal M$, we have the torus $\Sigma_c = \mathbb C/\text{span}_{\mathbb Z} \{1, c\}$. I also heard that there is a compactification of $\mathcal M$ by adding nodal curve which are formed by shrinking some geodesics to a point.
My question is: In the case of $\mathcal M$, if we take $b\to +\infty$, then the "limit" should be some element in the compactification $\overline{\mathcal M}$, but it seems to me that the limit has to be an $\mathbb S^1$ instead of some nodal curves. Is there any misunderstanding?
I have to apologize that the question is somewhat unclear, as I don't have a good understanding of $\overline{\mathcal M}$ in general.
Also, can anyone suggest some readable literatures concerning the compactification of the moduli space of Riemann surfaces of genus $g$? Thanks in advance.
The torus $\mathbb C/\langle 1, c\rangle$ is conformally equivalent to the torus $\mathbb C/\langle 1/b, c/b \rangle,$ and as $b \to \infty$, the latter tends (in some naive sense) to $\mathbb C/\mathbb Z$, which is a cylinder (which is what you get if you delete the node from a nodal cubic).
There is an enormous amount of literature on $\overline{\mathcal M}_g$; one think you can do is just google "Deligne--Mumford compactification" and see what turns up. Precise suggestions are hard to give without knowing more about your background.