I don't have a full proof yet, but I'm almost sure the following holds:
Let . On any simply-connected open subset of , it is possible to define an inverse of .
In terms of "slit plane", the inverse may thus be defined on the whole plane except for a simple path joining all the points of and going to infinity. The slit could more generally be any "planar tree" containing the elements of and .
I have a few ideas for a proof but there are still parts missing. I'll post it if I can complete it.