I "finished" part II, the one where I show the duality between my two graphs. Unfortunately there is little to be gained from that fact, as the structure of these graphs is dictated more by algebra than by graph theory. Still, it might be interesting to see that it's possible.
Fearing that I will have to omit part II in the final thesis, I have combined the theory on $\Gamma^n$ and the commutator subgroup into a more general theory which tries to emulate Kulkarni's results. I gave an exposé of this part III to my prof along with part II and will work out the details far into the beginning of next year.
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