I'm seeking for a discrete representation of $2$-manifolds with a conformal structure (i.e. metric modulo scalar prefactor).
The topology of a $2$-manifold is determined by the combinatorics of a triangulation, so it seems a natural question to ask what kind of extra data one needs to add to the triangulation to describe a conformal structure. As the moduli space of conformal manifolds (for fixed topology) is finite, I'd assume that a finite set of real parameters (such as one number at every edge of the triangulation) would suffice.
There is a unique conformal structure on the disk, plus a $3$-dimensional conformal automorphism group. So by parameter count I'd guess that if we fix $3$ points at the boundary of the disk there is still a unique conformal structure (though only a $0$-dimensional automorphism group). So also for a triangle with fixed angles at the corners (which don't have to be those of a flat triangle) one would expect a unique conformal structure.
Now imagine gluing two triangles at one of their edges yielding a $4$-gon. According to parameter count, there is a one-parameter family of conformal structure on a $4$-gon, so there should be also a one-parameter family of inequivalent ways to glue the two conformal triangles. So if we add these gluing parameters as additional data to the triangulation one could think this is a way to represent conformal manifolds.
Question: 1) Is any of what I've written above wrong or doesn't make sense? 2) Can anyone tell me whether and how the above construction sketch can be made precise? E.g. what extra structure one needs to add to the boundary edges of the triangle such that identifying ("gluing") the edges yields a sensible conformal structure on the glued $n$-gon. 3) Does anyone know other (similar) ways to represent conformal structures combinatorially?
