In the complex analytic setting, it is easy to see that the moduli space of a sphere with four punctures is $\mathcal{M}=\mathbb{CP}^1 / { 0,1,\infty }$, since I can use a Moebius transformation to send three of the punctures at those location whilst the fourth puncture is free to move.


On the other hand, the moduli space can be obtained as the quotient of the Teichmüller space by the mapping class group. Teichmüller space can be described using e.g. Fenchel-Nielsen coordinates, for each simple closed curve on the sphere we have two coordinates (a length and an angle). The mapping class group can be described using Dehn twists.


I always assumed that the three boundary points ${0,1,\infty}$ could be interpreted as sending to zero the length of three distinct curves separating the punctures into one of the three sets $(12)(34)$, $(23)(41)$ or $(13)(24)$, but I have recently realized that these curves are related to each other by suitable elements of the mapping class group. So I would naively say that the corresponding limits should be identified as well.


How do I reconcile the above picture of the moduli space with the one coming from Teichmüller modulo mapping class group? How do I see three boundary points in this picture?



Moduli, Teichmüller spaces and mapping class group of a sphere with four punctures
In your description of moduli space you say: 



> 
> I can use a Moebius transformation to send three of the punctures at
>  those location whilst the fourth puncture is free to move.
> 
> 
> 


That assumes that the punctures have names. Let's call them $a, b, c, d$ and we will agree to send them to $0, 1, \infty, z$. (I prefer to use letters for the names, as then we can't confuse the punctures with numbers.)


In your discussion of Teichmuller space, you identify three curves. The first curve, call it $\beta$, separates $a$ and $b$ from $c$ and $d$, while the second curve, call it $\gamma$, separates $a$ and $c$ from $b$ and $d$. Now, since these punctures have names, there is no mapping class that sends $b$ to $c$. Thus there is no mapping class that sends $\beta$ to $\gamma$. 


Except there obviously is... 


The solution lies in understanding which mapping class group you are using. In your definition of moduli space you are using the *pure* mapping class group (fixing all punctures). In your definition of Teichmuller space you are using the full mapping class group (of orientation preserving homeomorphisms, up to isotopy). 


